id int32 | title string | problem string | question_latex string | question_html string | numerical_answer string | pub_date string | solved_by string | diff_rate string | difficulty string |
|---|---|---|---|---|---|---|---|---|---|
879 | Touch-screen Password | A touch-screen device can be unlocked with a "password" consisting of a sequence of two or more distinct spots that the user selects from a rectangular grid of spots on the screen. The user enters their sequence by touching the first spot, then tracing a straight line segment to the next spot, and so on until the end o... | A touch-screen device can be unlocked with a "password" consisting of a sequence of two or more distinct spots that the user selects from a rectangular grid of spots on the screen. The user enters their sequence by touching the first spot, then tracing a straight line segment to the next spot, and so on until the end o... | <p>A touch-screen device can be unlocked with a "password" consisting of a sequence of two or more distinct spots that the user selects from a rectangular grid of spots on the screen. The user enters their sequence by touching the first spot, then tracing a straight line segment to the next spot, and so on until the en... | 4350069824940 | Saturday, 24th February 2024, 04:00 pm | 439 | 25% | easy |
547 | Distance of Random Points Within Hollow Square Laminae | Assuming that two points are chosen randomly (with uniform distribution) within a rectangle, it is possible to determine the expected value of the distance between these two points.
For example, the expected distance between two random points in a unit square is about $0.521405$, while the expected distance between two... | Assuming that two points are chosen randomly (with uniform distribution) within a rectangle, it is possible to determine the expected value of the distance between these two points.
For example, the expected distance between two random points in a unit square is about $0.521405$, while the expected distance between two... | <p>Assuming that two points are chosen randomly (with <strong>uniform distribution</strong>) within a rectangle, it is possible to determine the <strong>expected value</strong> of the distance between these two points.</p>
<p>For example, the expected distance between two random points in a unit square is about $0.5214... | 11730879.0023 | Sunday, 14th February 2016, 04:00 am | 250 | 70% | hard |
352 | Blood Tests | Each one of the $25$ sheep in a flock must be tested for a rare virus, known to affect $2\%$ of the sheep population.
An accurate and extremely sensitive PCR test exists for blood samples, producing a clear positive / negative result, but it is very time-consuming and expensive.
Because of the high cost, the vet-in-c... | Each one of the $25$ sheep in a flock must be tested for a rare virus, known to affect $2\%$ of the sheep population.
An accurate and extremely sensitive PCR test exists for blood samples, producing a clear positive / negative result, but it is very time-consuming and expensive.
Because of the high cost, the vet-in-c... | <p>
Each one of the $25$ sheep in a flock must be tested for a rare virus, known to affect $2\%$ of the sheep population.
An accurate and extremely sensitive PCR test exists for blood samples, producing a clear positive / negative result, but it is very time-consuming and expensive.
</p>
<p>
Because of the high cost, t... | 378563.260589 | Sunday, 2nd October 2011, 01:00 am | 679 | 55% | medium |
175 | Fractions and Sum of Powers of Two | Define $f(0)=1$ and $f(n)$ to be the number of ways to write $n$ as a sum of powers of $2$ where no power occurs more than twice.
For example, $f(10)=5$ since there are five different ways to express $10$:$10 = 8+2 = 8+1+1 = 4+4+2 = 4+2+2+1+1 = 4+4+1+1.$
It can be shown that for every fraction $p / q$ ($p \gt 0$, $q ... | Define $f(0)=1$ and $f(n)$ to be the number of ways to write $n$ as a sum of powers of $2$ where no power occurs more than twice.
For example, $f(10)=5$ since there are five different ways to express $10$:$10 = 8+2 = 8+1+1 = 4+4+2 = 4+2+2+1+1 = 4+4+1+1.$
It can be shown that for every fraction $p / q$ ($p \gt 0$, $q ... | <p>Define $f(0)=1$ and $f(n)$ to be the number of ways to write $n$ as a sum of powers of $2$ where no power occurs more than twice.</p>
<p>
For example, $f(10)=5$ since there are five different ways to express $10$:<br/>$10 = 8+2 = 8+1+1 = 4+4+2 = 4+2+2+1+1 = 4+4+1+1.$</p>
<p>
It can be shown that for every fraction $... | 1,13717420,8 | Friday, 28th December 2007, 01:00 pm | 2033 | 70% | hard |
715 | Sextuplet Norms | Let $f(n)$ be the number of $6$-tuples $(x_1,x_2,x_3,x_4,x_5,x_6)$ such that:
All $x_i$ are integers with $0 \leq x_i < n$
$\gcd(x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2,\ n^2)=1$
Let $\displaystyle G(n)=\displaystyle\sum_{k=1}^n \frac{f(k)}{k^2\varphi(k)}$
where $\varphi(n)$ is Euler's totient function.
For example, $G(10)... | Let $f(n)$ be the number of $6$-tuples $(x_1,x_2,x_3,x_4,x_5,x_6)$ such that:
All $x_i$ are integers with $0 \leq x_i < n$
$\gcd(x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2,\ n^2)=1$
Let $\displaystyle G(n)=\displaystyle\sum_{k=1}^n \frac{f(k)}{k^2\varphi(k)}$
where $\varphi(n)$ is Euler's totient function.
For example, $G(10)... | <p>Let $f(n)$ be the number of $6$-tuples $(x_1,x_2,x_3,x_4,x_5,x_6)$ such that:</p>
<ul><li>All $x_i$ are integers with $0 \leq x_i < n$</li>
<li>$\gcd(x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2,\ n^2)=1$</li>
</ul><p>Let $\displaystyle G(n)=\displaystyle\sum_{k=1}^n \frac{f(k)}{k^2\varphi(k)}$<br>
where $\varphi(n)$ is E... | 883188017 | Sunday, 10th May 2020, 08:00 am | 215 | 65% | hard |
745 | Sum of Squares II | For a positive integer, $n$, define $g(n)$ to be the maximum perfect square that divides $n$.
For example, $g(18) = 9$, $g(19) = 1$.
Also define
$$\displaystyle S(N) = \sum_{n=1}^N g(n)$$
For example, $S(10) = 24$ and $S(100) = 767$.
Find $S(10^{14})$. Give your answer modulo $1\,000\,000\,007$. | For a positive integer, $n$, define $g(n)$ to be the maximum perfect square that divides $n$.
For example, $g(18) = 9$, $g(19) = 1$.
Also define
$$\displaystyle S(N) = \sum_{n=1}^N g(n)$$
For example, $S(10) = 24$ and $S(100) = 767$.
Find $S(10^{14})$. Give your answer modulo $1\,000\,000\,007$. | <p>
For a positive integer, $n$, define $g(n)$ to be the maximum perfect square that divides $n$.<br>
For example, $g(18) = 9$, $g(19) = 1$.
</br></p>
<p>
Also define
$$\displaystyle S(N) = \sum_{n=1}^N g(n)$$
</p>
<p>
For example, $S(10) = 24$ and $S(100) = 767$.
</p>
<p>
Find $S(10^{14})$. Give your answer modulo $1\... | 94586478 | Sunday, 31st January 2021, 04:00 am | 1361 | 10% | easy |
669 | The King's Banquet | The Knights of the Order of Fibonacci are preparing a grand feast for their king. There are $n$ knights, and each knight is assigned a distinct number from $1$ to $n$.
When the knights sit down at the roundtable for their feast, they follow a peculiar seating rule: two knights can only sit next to each other if their r... | The Knights of the Order of Fibonacci are preparing a grand feast for their king. There are $n$ knights, and each knight is assigned a distinct number from $1$ to $n$.
When the knights sit down at the roundtable for their feast, they follow a peculiar seating rule: two knights can only sit next to each other if their r... | <p>The Knights of the Order of Fibonacci are preparing a grand feast for their king. There are $n$ knights, and each knight is assigned a distinct number from $1$ to $n$.</p>
<p>When the knights sit down at the roundtable for their feast, they follow a peculiar seating rule: two knights can only sit next to each other ... | 56342087360542122 | Saturday, 11th May 2019, 10:00 pm | 338 | 45% | medium |
767 | Window into a Matrix II | A window into a matrix is a contiguous sub matrix.
Consider a $16\times n$ matrix where every entry is either 0 or 1.
Let $B(k,n)$ be the total number of these matrices such that the sum of the entries in every $2\times k$ window is $k$.
You are given that $B(2,4) = 65550$ and $B(3,9) \equiv 87273560 \pmod{1\,000\,000\... | A window into a matrix is a contiguous sub matrix.
Consider a $16\times n$ matrix where every entry is either 0 or 1.
Let $B(k,n)$ be the total number of these matrices such that the sum of the entries in every $2\times k$ window is $k$.
You are given that $B(2,4) = 65550$ and $B(3,9) \equiv 87273560 \pmod{1\,000\,000\... | <p>A window into a matrix is a contiguous sub matrix.</p>
<p>Consider a $16\times n$ matrix where every entry is either 0 or 1.
Let $B(k,n)$ be the total number of these matrices such that the sum of the entries in every $2\times k$ window is $k$.</p>
<p>You are given that $B(2,4) = 65550$ and $B(3,9) \equiv 87273560 \... | 783976175 | Sunday, 10th October 2021, 02:00 am | 188 | 60% | hard |
563 | Robot Welders | A company specialises in producing large rectangular metal sheets, starting from unit square metal plates. The welding is performed by a range of robots of increasing size. Unfortunately, the programming options of these robots are rather limited. Each one can only process up to $25$ identical rectangles of metal, w... | A company specialises in producing large rectangular metal sheets, starting from unit square metal plates. The welding is performed by a range of robots of increasing size. Unfortunately, the programming options of these robots are rather limited. Each one can only process up to $25$ identical rectangles of metal, w... | <p>A company specialises in producing large rectangular metal sheets, starting from unit square metal plates. The welding is performed by a range of robots of increasing size. Unfortunately, the programming options of these robots are rather limited. Each one can only process up to $25$ identical rectangles of metal... | 27186308211734760 | Sunday, 5th June 2016, 04:00 am | 378 | 45% | medium |
882 | Removing Bits | Dr. One and Dr. Zero are playing the following partisan game.
The game begins with one $1$, two $2$'s, three $3$'s, ..., $n$ $n$'s. Starting with Dr. One, they make moves in turn.
Dr. One chooses a number and changes it by removing a $1$ from its binary expansion.
Dr. Zero chooses a number and changes it by removing a ... | Dr. One and Dr. Zero are playing the following partisan game.
The game begins with one $1$, two $2$'s, three $3$'s, ..., $n$ $n$'s. Starting with Dr. One, they make moves in turn.
Dr. One chooses a number and changes it by removing a $1$ from its binary expansion.
Dr. Zero chooses a number and changes it by removing a ... | <p>Dr. One and Dr. Zero are playing the following partisan game.<br/>
The game begins with one $1$, two $2$'s, three $3$'s, ..., $n$ $n$'s. Starting with Dr. One, they make moves in turn.<br/>
Dr. One chooses a number and changes it by removing a $1$ from its binary expansion.<br/>
Dr. Zero chooses a number and changes... | 15800662276 | Sunday, 17th March 2024, 01:00 am | 144 | 75% | hard |
606 | Gozinta Chains II | A gozinta chain for $n$ is a sequence $\{1,a,b,\dots,n\}$ where each element properly divides the next.
For example, there are eight distinct gozinta chains for $12$:
$\{1,12\}$, $\{1,2,12\}$, $\{1,2,4,12\}$, $\{1,2,6,12\}$, $\{1,3,12\}$, $\{1,3,6,12\}$, $\{1,4,12\}$ and $\{1,6,12\}$.
Let $S(n)$ be the sum of all nu... | A gozinta chain for $n$ is a sequence $\{1,a,b,\dots,n\}$ where each element properly divides the next.
For example, there are eight distinct gozinta chains for $12$:
$\{1,12\}$, $\{1,2,12\}$, $\{1,2,4,12\}$, $\{1,2,6,12\}$, $\{1,3,12\}$, $\{1,3,6,12\}$, $\{1,4,12\}$ and $\{1,6,12\}$.
Let $S(n)$ be the sum of all nu... | <p>
A <strong>gozinta chain</strong> for $n$ is a sequence $\{1,a,b,\dots,n\}$ where each element properly divides the next. <br/>
For example, there are eight distinct gozinta chains for $12$:<br/>
$\{1,12\}$, $\{1,2,12\}$, $\{1,2,4,12\}$, $\{1,2,6,12\}$, $\{1,3,12\}$, $\{1,3,6,12\}$, $\{1,4,12\}$ and $\{1,6,12\}$.
</... | 158452775 | Sunday, 4th June 2017, 10:00 am | 417 | 50% | medium |
98 | Anagramic Squares | By replacing each of the letters in the word CARE with $1$, $2$, $9$, and $6$ respectively, we form a square number: $1296 = 36^2$. What is remarkable is that, by using the same digital substitutions, the anagram, RACE, also forms a square number: $9216 = 96^2$. We shall call CARE (and RACE) a square anagram word pair ... | By replacing each of the letters in the word CARE with $1$, $2$, $9$, and $6$ respectively, we form a square number: $1296 = 36^2$. What is remarkable is that, by using the same digital substitutions, the anagram, RACE, also forms a square number: $9216 = 96^2$. We shall call CARE (and RACE) a square anagram word pair ... | <p>By replacing each of the letters in the word CARE with $1$, $2$, $9$, and $6$ respectively, we form a square number: $1296 = 36^2$. What is remarkable is that, by using the same digital substitutions, the anagram, RACE, also forms a square number: $9216 = 96^2$. We shall call CARE (and RACE) a square anagram word pa... | 18769 | Friday, 17th June 2005, 06:00 pm | 12939 | 35% | medium |
434 | Rigid Graphs | Recall that a graph is a collection of vertices and edges connecting the vertices, and that two vertices connected by an edge are called adjacent.
Graphs can be embedded in Euclidean space by associating each vertex with a point in the Euclidean space.
A flexible graph is an embedding of a graph where it is possible to... | Recall that a graph is a collection of vertices and edges connecting the vertices, and that two vertices connected by an edge are called adjacent.
Graphs can be embedded in Euclidean space by associating each vertex with a point in the Euclidean space.
A flexible graph is an embedding of a graph where it is possible to... | <p>Recall that a graph is a collection of vertices and edges connecting the vertices, and that two vertices connected by an edge are called adjacent.<br/>
Graphs can be embedded in Euclidean space by associating each vertex with a point in the Euclidean space.<br/>
A <strong>flexible</strong> graph is an embedding of a... | 863253606 | Saturday, 29th June 2013, 07:00 pm | 365 | 75% | hard |
709 | Even Stevens | Every day for the past $n$ days Even Stevens brings home his groceries in a plastic bag. He stores these plastic bags in a cupboard. He either puts the plastic bag into the cupboard with the rest, or else he takes an even number of the existing bags (which may either be empty or previously filled with other bags themse... | Every day for the past $n$ days Even Stevens brings home his groceries in a plastic bag. He stores these plastic bags in a cupboard. He either puts the plastic bag into the cupboard with the rest, or else he takes an even number of the existing bags (which may either be empty or previously filled with other bags themse... | <p>Every day for the past $n$ days Even Stevens brings home his groceries in a plastic bag. He stores these plastic bags in a cupboard. He either puts the plastic bag into the cupboard with the rest, or else he takes an <b>even</b> number of the existing bags (which may either be empty or previously filled with other b... | 773479144 | Saturday, 4th April 2020, 05:00 pm | 989 | 15% | easy |
212 | Combined Volume of Cuboids | An axis-aligned cuboid, specified by parameters $\{(x_0, y_0, z_0), (dx, dy, dz)\}$, consists of all points $(X,Y,Z)$ such that $x_0 \le X \le x_0 + dx$, $y_0 \le Y \le y_0 + dy$ and $z_0 \le Z \le z_0 + dz$. The volume of the cuboid is the product, $dx \times dy \times dz$. The combined volume of a collection of cub... | An axis-aligned cuboid, specified by parameters $\{(x_0, y_0, z_0), (dx, dy, dz)\}$, consists of all points $(X,Y,Z)$ such that $x_0 \le X \le x_0 + dx$, $y_0 \le Y \le y_0 + dy$ and $z_0 \le Z \le z_0 + dz$. The volume of the cuboid is the product, $dx \times dy \times dz$. The combined volume of a collection of cub... | <p>An <dfn>axis-aligned cuboid</dfn>, specified by parameters $\{(x_0, y_0, z_0), (dx, dy, dz)\}$, consists of all points $(X,Y,Z)$ such that $x_0 \le X \le x_0 + dx$, $y_0 \le Y \le y_0 + dy$ and $z_0 \le Z \le z_0 + dz$. The volume of the cuboid is the product, $dx \times dy \times dz$. The <dfn>combined volume</df... | 328968937309 | Saturday, 11th October 2008, 06:00 am | 1561 | 70% | hard |
310 | Nim Square | Alice and Bob play the game Nim Square.
Nim Square is just like ordinary three-heap normal play Nim, but the players may only remove a square number of stones from a heap.
The number of stones in the three heaps is represented by the ordered triple $(a,b,c)$.
If $0 \le a \le b \le c \le 29$ then the number of losing po... | Alice and Bob play the game Nim Square.
Nim Square is just like ordinary three-heap normal play Nim, but the players may only remove a square number of stones from a heap.
The number of stones in the three heaps is represented by the ordered triple $(a,b,c)$.
If $0 \le a \le b \le c \le 29$ then the number of losing po... | <p>
Alice and Bob play the game Nim Square.<br/>
Nim Square is just like ordinary three-heap normal play Nim, but the players may only remove a square number of stones from a heap.<br/>
The number of stones in the three heaps is represented by the ordered triple $(a,b,c)$.<br/>
If $0 \le a \le b \le c \le 29$ then the ... | 2586528661783 | Saturday, 13th November 2010, 07:00 pm | 1308 | 40% | medium |
458 | Permutations of Project | Consider the alphabet $A$ made out of the letters of the word "$\text{project}$": $A=\{\text c,\text e,\text j,\text o,\text p,\text r,\text t\}$.
Let $T(n)$ be the number of strings of length $n$ consisting of letters from $A$ that do not have a substring that is one of the $5040$ permutations of "$\text{project}$".
... | Consider the alphabet $A$ made out of the letters of the word "$\text{project}$": $A=\{\text c,\text e,\text j,\text o,\text p,\text r,\text t\}$.
Let $T(n)$ be the number of strings of length $n$ consisting of letters from $A$ that do not have a substring that is one of the $5040$ permutations of "$\text{project}$".
... | <p>
Consider the alphabet $A$ made out of the letters of the word "$\text{project}$": $A=\{\text c,\text e,\text j,\text o,\text p,\text r,\text t\}$.<br/>
Let $T(n)$ be the number of strings of length $n$ consisting of letters from $A$ that do not have a substring that is one of the $5040$ permutations of "$\text{proj... | 423341841 | Sunday, 9th February 2014, 07:00 am | 1049 | 30% | easy |
4 | Largest Palindrome Product | A palindromic number reads the same both ways. The largest palindrome made from the product of two $2$-digit numbers is $9009 = 91 \times 99$.
Find the largest palindrome made from the product of two $3$-digit numbers. | A palindromic number reads the same both ways. The largest palindrome made from the product of two $2$-digit numbers is $9009 = 91 \times 99$.
Find the largest palindrome made from the product of two $3$-digit numbers. | <p>A palindromic number reads the same both ways. The largest palindrome made from the product of two $2$-digit numbers is $9009 = 91 \times 99$.</p>
<p>Find the largest palindrome made from the product of two $3$-digit numbers.</p> | 906609 | Friday, 16th November 2001, 06:00 pm | 519815 | 5% | easy |
475 | Music Festival | $12n$ musicians participate at a music festival. On the first day, they form $3n$ quartets and practice all day.
It is a disaster. At the end of the day, all musicians decide they will never again agree to play with any member of their quartet.
On the second day, they form $4n$ trios, with every musician avoiding any p... | $12n$ musicians participate at a music festival. On the first day, they form $3n$ quartets and practice all day.
It is a disaster. At the end of the day, all musicians decide they will never again agree to play with any member of their quartet.
On the second day, they form $4n$ trios, with every musician avoiding any p... | <p>$12n$ musicians participate at a music festival. On the first day, they form $3n$ quartets and practice all day.</p>
<p>It is a disaster. At the end of the day, all musicians decide they will never again agree to play with any member of their quartet.</p>
<p>On the second day, they form $4n$ trios, with every musici... | 75780067 | Sunday, 8th June 2014, 10:00 am | 495 | 50% | medium |
621 | Expressing an Integer as the Sum of Triangular Numbers | Gauss famously proved that every positive integer can be expressed as the sum of three triangular numbers (including $0$ as the lowest triangular number). In fact most numbers can be expressed as a sum of three triangular numbers in several ways.
Let $G(n)$ be the number of ways of expressing $n$ as the sum of three t... | Gauss famously proved that every positive integer can be expressed as the sum of three triangular numbers (including $0$ as the lowest triangular number). In fact most numbers can be expressed as a sum of three triangular numbers in several ways.
Let $G(n)$ be the number of ways of expressing $n$ as the sum of three t... | <p>Gauss famously proved that every positive integer can be expressed as the sum of three <strong>triangular numbers</strong> (including $0$ as the lowest triangular number). In fact most numbers can be expressed as a sum of three triangular numbers in several ways.</p>
<p>
Let $G(n)$ be the number of ways of expressin... | 11429712 | Sunday, 25th February 2018, 04:00 am | 613 | 35% | medium |
193 | Squarefree Numbers | A positive integer $n$ is called squarefree, if no square of a prime divides $n$, thus $1, 2, 3, 5, 6, 7, 10, 11$ are squarefree, but not $4, 8, 9, 12$.
How many squarefree numbers are there below $2^{50}$? | A positive integer $n$ is called squarefree, if no square of a prime divides $n$, thus $1, 2, 3, 5, 6, 7, 10, 11$ are squarefree, but not $4, 8, 9, 12$.
How many squarefree numbers are there below $2^{50}$? | <p>A positive integer $n$ is called squarefree, if no square of a prime divides $n$, thus $1, 2, 3, 5, 6, 7, 10, 11$ are squarefree, but not $4, 8, 9, 12$.</p>
<p>How many squarefree numbers are there below $2^{50}$?</p> | 684465067343069 | Saturday, 10th May 2008, 01:00 pm | 3801 | 55% | medium |
671 | Colouring a Loop | A certain type of flexible tile comes in three different sizes - $1 \times 1$, $1 \times 2$, and $1 \times 3$ - and in $k$ different colours. There is an unlimited number of tiles available in each combination of size and colour.
These are used to tile a closed loop of width $2$ and length (circumference) $n$, where $n... | A certain type of flexible tile comes in three different sizes - $1 \times 1$, $1 \times 2$, and $1 \times 3$ - and in $k$ different colours. There is an unlimited number of tiles available in each combination of size and colour.
These are used to tile a closed loop of width $2$ and length (circumference) $n$, where $n... | <p>A certain type of flexible tile comes in three different sizes - $1 \times 1$, $1 \times 2$, and $1 \times 3$ - and in $k$ different colours. There is an unlimited number of tiles available in each combination of size and colour.</p>
<p>These are used to tile a closed loop of width $2$ and length (circumference) $n$... | 946106780 | Sunday, 19th May 2019, 01:00 am | 195 | 80% | hard |
81 | Path Sum: Two Ways | In the $5$ by $5$ matrix below, the minimal path sum from the top left to the bottom right, by only moving to the right and down, is indicated in bold red and is equal to $2427$.
$$
\begin{pmatrix}
\color{red}{131} & 673 & 234 & 103 & 18\\
\color{red}{201} & \color{red}{96} & \color{red}{342} & 965 & 150\\
630 & 803 &... | In the $5$ by $5$ matrix below, the minimal path sum from the top left to the bottom right, by only moving to the right and down, is indicated in bold red and is equal to $2427$.
$$
\begin{pmatrix}
\color{red}{131} & 673 & 234 & 103 & 18\\
\color{red}{201} & \color{red}{96} & \color{red}{342} & 965 & 150\\
630 & 803 &... | <p>In the $5$ by $5$ matrix below, the minimal path sum from the top left to the bottom right, by <b>only moving to the right and down</b>, is indicated in bold red and is equal to $2427$.</p>
<div class="center">
$$
\begin{pmatrix}
\color{red}{131} & 673 & 234 & 103 & 18\\
\color{red}{201} & \color... | 427337 | Friday, 22nd October 2004, 06:00 pm | 37695 | 10% | easy |
914 | Triangles inside Circles | For a given integer $R$ consider all primitive Pythagorean triangles that can fit inside, without touching, a circle with radius $R$. Define $F(R)$ to be the largest inradius of those triangles. You are given $F(100) = 36$.
Find $F(10^{18})$. | For a given integer $R$ consider all primitive Pythagorean triangles that can fit inside, without touching, a circle with radius $R$. Define $F(R)$ to be the largest inradius of those triangles. You are given $F(100) = 36$.
Find $F(10^{18})$. | <p>
For a given integer $R$ consider all primitive Pythagorean triangles that can fit inside, without touching, a circle with radius $R$. Define $F(R)$ to be the largest inradius of those triangles. You are given $F(100) = 36$.</p>
<p>
Find $F(10^{18})$.</p> | 414213562371805310 | Saturday, 26th October 2024, 05:00 pm | 474 | 20% | easy |
483 | Repeated Permutation | We define a permutation as an operation that rearranges the order of the elements $\{1, 2, 3, ..., n\}$.
There are $n!$ such permutations, one of which leaves the elements in their initial order.
For $n = 3$ we have $3! = 6$ permutations:
$P_1 =$ keep the initial order
$P_2 =$ exchange the 1st and 2nd elements
$P_3 =$... | We define a permutation as an operation that rearranges the order of the elements $\{1, 2, 3, ..., n\}$.
There are $n!$ such permutations, one of which leaves the elements in their initial order.
For $n = 3$ we have $3! = 6$ permutations:
$P_1 =$ keep the initial order
$P_2 =$ exchange the 1st and 2nd elements
$P_3 =$... | <p>
We define a <dfn>permutation</dfn> as an operation that rearranges the order of the elements $\{1, 2, 3, ..., n\}$.
There are $n!$ such permutations, one of which leaves the elements in their initial order.
For $n = 3$ we have $3! = 6$ permutations:
</p><ul>
<li>$P_1 =$ keep the initial order</li>
<li>$P_2 =$ excha... | 4.993401567e22 | Sunday, 5th October 2014, 10:00 am | 291 | 100% | hard |
604 | Convex Path in Square | Let $F(N)$ be the maximum number of lattice points in an axis-aligned $N\times N$ square that the graph of a single strictly convex increasing function can pass through.
You are given that $F(1) = 2$, $F(3) = 3$, $F(9) = 6$, $F(11) = 7$, $F(100) = 30$ and $F(50000) = 1898$.
Below is the graph of a function reaching... | Let $F(N)$ be the maximum number of lattice points in an axis-aligned $N\times N$ square that the graph of a single strictly convex increasing function can pass through.
You are given that $F(1) = 2$, $F(3) = 3$, $F(9) = 6$, $F(11) = 7$, $F(100) = 30$ and $F(50000) = 1898$.
Below is the graph of a function reaching... | <p>
Let $F(N)$ be the maximum number of lattice points in an axis-aligned $N\times N$ square that the graph of a single <strong>strictly convex</strong> increasing function can pass through.
</p>
<p>
You are given that $F(1) = 2$, $F(3) = 3$, $F(9) = 6$, $F(11) = 7$, $F(100) = 30$ and $F(50000) = 1898$.<br/>
Below is... | 1398582231101 | Sunday, 21st May 2017, 04:00 am | 556 | 40% | medium |
856 | Waiting for a Pair | A standard 52-card deck comprises 13 ranks in four suits. A pair is a set of two cards of the same rank.
Cards are drawn, without replacement, from a well shuffled 52-card deck waiting for consecutive cards that form a pair. For example, the probability of finding a pair in the first two draws is $\frac{1}{17}$.
Cards ... | A standard 52-card deck comprises 13 ranks in four suits. A pair is a set of two cards of the same rank.
Cards are drawn, without replacement, from a well shuffled 52-card deck waiting for consecutive cards that form a pair. For example, the probability of finding a pair in the first two draws is $\frac{1}{17}$.
Cards ... | <p>A standard 52-card deck comprises 13 ranks in four suits. A <i>pair</i> is a set of two cards of the same rank.</p>
<p>Cards are drawn, without replacement, from a well shuffled 52-card deck waiting for consecutive cards that form a pair. For example, the probability of finding a pair in the first two draws is $\fra... | 17.09661501 | Saturday, 23rd September 2023, 11:00 pm | 630 | 20% | easy |
546 | The Floor's Revenge | Define $f_k(n) = \sum_{i=0}^n f_k(\lfloor\frac i k \rfloor)$ where $f_k(0) = 1$ and $\lfloor x \rfloor$ denotes the floor function.
For example, $f_5(10) = 18$, $f_7(100) = 1003$, and $f_2(10^3) = 264830889564$.
Find $(\sum_{k=2}^{10} f_k(10^{14})) \bmod (10^9+7)$. | Define $f_k(n) = \sum_{i=0}^n f_k(\lfloor\frac i k \rfloor)$ where $f_k(0) = 1$ and $\lfloor x \rfloor$ denotes the floor function.
For example, $f_5(10) = 18$, $f_7(100) = 1003$, and $f_2(10^3) = 264830889564$.
Find $(\sum_{k=2}^{10} f_k(10^{14})) \bmod (10^9+7)$. | <p>Define $f_k(n) = \sum_{i=0}^n f_k(\lfloor\frac i k \rfloor)$ where $f_k(0) = 1$ and $\lfloor x \rfloor$ denotes the floor function.</p>
<p>For example, $f_5(10) = 18$, $f_7(100) = 1003$, and $f_2(10^3) = 264830889564$.</p>
<p>Find $(\sum_{k=2}^{10} f_k(10^{14})) \bmod (10^9+7)$.</p> | 215656873 | Sunday, 7th February 2016, 01:00 am | 269 | 85% | hard |
209 | Circular Logic | A $k$-input binary truth table is a map from $k$ input bits (binary digits, $0$ [false] or $1$ [true]) to $1$ output bit. For example, the $2$-input binary truth tables for the logical $\mathbin{\text{AND}}$ and $\mathbin{\text{XOR}}$ functions are:
$x$
$y$
$x \mathbin{\text{AND}} y$
$0$$0$$0$$0$$1$$0$$1$$0$$0$$1$$1$$... | A $k$-input binary truth table is a map from $k$ input bits (binary digits, $0$ [false] or $1$ [true]) to $1$ output bit. For example, the $2$-input binary truth tables for the logical $\mathbin{\text{AND}}$ and $\mathbin{\text{XOR}}$ functions are:
$x$
$y$
$x \mathbin{\text{AND}} y$
$0$$0$$0$$0$$1$$0$$1$$0$$0$$1$$1$$... | <p>A $k$-input <strong>binary truth table</strong> is a map from $k$ input bits (binary digits, $0$ [false] or $1$ [true]) to $1$ output bit. For example, the $2$-input binary truth tables for the logical $\mathbin{\text{AND}}$ and $\mathbin{\text{XOR}}$ functions are:</p>
<div style="float:left;margin:10px 50px;text-a... | 15964587728784 | Friday, 19th September 2008, 06:00 pm | 2772 | 60% | hard |
346 | Strong Repunits | The number $7$ is special, because $7$ is $111$ written in base $2$, and $11$ written in base $6$ (i.e. $7_{10} = 11_6 = 111_2$). In other words, $7$ is a repunit in at least two bases $b \gt 1$.
We shall call a positive integer with this property a strong repunit. It can be verified that there are $8$ strong repuni... | The number $7$ is special, because $7$ is $111$ written in base $2$, and $11$ written in base $6$ (i.e. $7_{10} = 11_6 = 111_2$). In other words, $7$ is a repunit in at least two bases $b \gt 1$.
We shall call a positive integer with this property a strong repunit. It can be verified that there are $8$ strong repuni... | <p>
The number $7$ is special, because $7$ is $111$ written in base $2$, and $11$ written in base $6$ (i.e. $7_{10} = 11_6 = 111_2$). In other words, $7$ is a repunit in at least two bases $b \gt 1$.
</p>
<p>
We shall call a positive integer with this property a strong repunit. It can be verified that there are $8$ st... | 336108797689259276 | Saturday, 3rd September 2011, 04:00 pm | 4816 | 15% | easy |
257 | Angular Bisectors | Given is an integer sided triangle $ABC$ with sides $a \le b \le c$.
($AB = c$, $BC = a$ and $AC = b$.)
The angular bisectors of the triangle intersect the sides at points $E$, $F$ and $G$ (see picture below).
The segments $EF$, $EG$ and $FG$ partition the triangle $ABC$ into four smaller triangles: $AEG$, $BFE$, ... | Given is an integer sided triangle $ABC$ with sides $a \le b \le c$.
($AB = c$, $BC = a$ and $AC = b$.)
The angular bisectors of the triangle intersect the sides at points $E$, $F$ and $G$ (see picture below).
The segments $EF$, $EG$ and $FG$ partition the triangle $ABC$ into four smaller triangles: $AEG$, $BFE$, ... | <p>Given is an integer sided triangle $ABC$ with sides $a \le b \le c$.
($AB = c$, $BC = a$ and $AC = b$.)<br/>
The angular bisectors of the triangle intersect the sides at points $E$, $F$ and $G$ (see picture below).
</p>
<div align="center">
<img alt="0257_bisector.gif" class="dark_img" src="resources/images/0257_bi... | 139012411 | Saturday, 26th September 2009, 05:00 am | 787 | 85% | hard |
566 | Cake Icing Puzzle | Adam plays the following game with his birthday cake.
He cuts a piece forming a circular sector of $60$ degrees and flips the piece upside down, with the icing on the bottom.
He then rotates the cake by $60$ degrees counterclockwise, cuts an adjacent $60$ degree piece and flips it upside down.
He keeps repeating this, ... | Adam plays the following game with his birthday cake.
He cuts a piece forming a circular sector of $60$ degrees and flips the piece upside down, with the icing on the bottom.
He then rotates the cake by $60$ degrees counterclockwise, cuts an adjacent $60$ degree piece and flips it upside down.
He keeps repeating this, ... | <p>Adam plays the following game with his birthday cake.</p>
<p>He cuts a piece forming a circular sector of $60$ degrees and flips the piece upside down, with the icing on the bottom.<br/>
He then rotates the cake by $60$ degrees counterclockwise, cuts an adjacent $60$ degree piece and flips it upside down.<br/>
He ke... | 329569369413585 | Saturday, 25th June 2016, 01:00 pm | 213 | 100% | hard |
134 | Prime Pair Connection | Consider the consecutive primes $p_1 = 19$ and $p_2 = 23$. It can be verified that $1219$ is the smallest number such that the last digits are formed by $p_1$ whilst also being divisible by $p_2$.
In fact, with the exception of $p_1 = 3$ and $p_2 = 5$, for every pair of consecutive primes, $p_2 \gt p_1$, there exist va... | Consider the consecutive primes $p_1 = 19$ and $p_2 = 23$. It can be verified that $1219$ is the smallest number such that the last digits are formed by $p_1$ whilst also being divisible by $p_2$.
In fact, with the exception of $p_1 = 3$ and $p_2 = 5$, for every pair of consecutive primes, $p_2 \gt p_1$, there exist va... | <p>Consider the consecutive primes $p_1 = 19$ and $p_2 = 23$. It can be verified that $1219$ is the smallest number such that the last digits are formed by $p_1$ whilst also being divisible by $p_2$.</p>
<p>In fact, with the exception of $p_1 = 3$ and $p_2 = 5$, for every pair of consecutive primes, $p_2 \gt p_1$, ther... | 18613426663617118 | Friday, 15th December 2006, 06:00 pm | 7860 | 45% | medium |
913 | Row-major vs Column-major | The numbers from $1$ to $12$ can be arranged into a $3 \times 4$ matrix in either row-major or column-major order:
$$R=\begin{pmatrix}
1 & 2 & 3 & 4\\
5 & 6 & 7 & 8\\
9 & 10 & 11 & 12\end{pmatrix}, C=\begin{pmatrix}
1 & 4 & 7 & 10\\
2 & 5 & 8 & 11\\
3 & 6 & 9 & 12\end{pmatrix}$$
By swapping two entries at a time, at le... | The numbers from $1$ to $12$ can be arranged into a $3 \times 4$ matrix in either row-major or column-major order:
$$R=\begin{pmatrix}
1 & 2 & 3 & 4\\
5 & 6 & 7 & 8\\
9 & 10 & 11 & 12\end{pmatrix}, C=\begin{pmatrix}
1 & 4 & 7 & 10\\
2 & 5 & 8 & 11\\
3 & 6 & 9 & 12\end{pmatrix}$$
By swapping two entries at a time, at le... | <p>
The numbers from $1$ to $12$ can be arranged into a $3 \times 4$ matrix in either <strong>row-major</strong> or <strong>column-major</strong> order:
$$R=\begin{pmatrix}
1 & 2 & 3 & 4\\
5 & 6 & 7 & 8\\
9 & 10 & 11 & 12\end{pmatrix}, C=\begin{pmatrix}
1 & 4 & 7 & 10\\
2... | 2101925115560555020 | Saturday, 19th October 2024, 02:00 pm | 140 | 60% | hard |
146 | Investigating a Prime Pattern | The smallest positive integer $n$ for which the numbers $n^2 + 1$, $n^2 + 3$, $n^2 + 7$, $n^2 + 9$, $n^2 + 13$, and $n^2 + 27$ are consecutive primes is $10$. The sum of all such integers $n$ below one-million is $1242490$.
What is the sum of all such integers $n$ below $150$ million? | The smallest positive integer $n$ for which the numbers $n^2 + 1$, $n^2 + 3$, $n^2 + 7$, $n^2 + 9$, $n^2 + 13$, and $n^2 + 27$ are consecutive primes is $10$. The sum of all such integers $n$ below one-million is $1242490$.
What is the sum of all such integers $n$ below $150$ million? | <p>The smallest positive integer $n$ for which the numbers $n^2 + 1$, $n^2 + 3$, $n^2 + 7$, $n^2 + 9$, $n^2 + 13$, and $n^2 + 27$ are consecutive primes is $10$. The sum of all such integers $n$ below one-million is $1242490$.</p>
<p>What is the sum of all such integers $n$ below $150$ million?</p> | 676333270 | Saturday, 24th March 2007, 09:00 am | 5740 | 50% | medium |
754 | Product of Gauss Factorials | The Gauss Factorial of a number $n$ is defined as the product of all positive numbers $\leq n$ that are relatively prime to $n$. For example $g(10)=1\times 3\times 7\times 9 = 189$.
Also we define
$$\displaystyle G(n) = \prod_{i=1}^{n}g(i)$$
You are given $G(10) = 23044331520000$.
Find $G(10^8)$. Give your answer modu... | The Gauss Factorial of a number $n$ is defined as the product of all positive numbers $\leq n$ that are relatively prime to $n$. For example $g(10)=1\times 3\times 7\times 9 = 189$.
Also we define
$$\displaystyle G(n) = \prod_{i=1}^{n}g(i)$$
You are given $G(10) = 23044331520000$.
Find $G(10^8)$. Give your answer modu... | <p>The <strong>Gauss Factorial</strong> of a number $n$ is defined as the product of all positive numbers $\leq n$ that are relatively prime to $n$. For example $g(10)=1\times 3\times 7\times 9 = 189$. </p>
<p>Also we define
$$\displaystyle G(n) = \prod_{i=1}^{n}g(i)$$</p>
<p>You are given $G(10) = 23044331520000$.</p>... | 785845900 | Sunday, 4th April 2021, 08:00 am | 784 | 20% | easy |
63 | Powerful Digit Counts | The $5$-digit number, $16807=7^5$, is also a fifth power. Similarly, the $9$-digit number, $134217728=8^9$, is a ninth power.
How many $n$-digit positive integers exist which are also an $n$th power? | The $5$-digit number, $16807=7^5$, is also a fifth power. Similarly, the $9$-digit number, $134217728=8^9$, is a ninth power.
How many $n$-digit positive integers exist which are also an $n$th power? | <p>The $5$-digit number, $16807=7^5$, is also a fifth power. Similarly, the $9$-digit number, $134217728=8^9$, is a ninth power.</p>
<p>How many $n$-digit positive integers exist which are also an $n$th power?</p> | 49 | Friday, 13th February 2004, 06:00 pm | 47358 | 5% | easy |
872 | Recursive Tree | A sequence of rooted trees $T_n$ is constructed such that $T_n$ has $n$ nodes numbered $1$ to $n$.
The sequence starts at $T_1$, a tree with a single node as a root with the number $1$.
For $n > 1$, $T_n$ is constructed from $T_{n-1}$ using the following procedure:
Trace a path from the root of $T_{n-1}$ to a leaf by ... | A sequence of rooted trees $T_n$ is constructed such that $T_n$ has $n$ nodes numbered $1$ to $n$.
The sequence starts at $T_1$, a tree with a single node as a root with the number $1$.
For $n > 1$, $T_n$ is constructed from $T_{n-1}$ using the following procedure:
Trace a path from the root of $T_{n-1}$ to a leaf by ... | <p>A sequence of rooted trees $T_n$ is constructed such that $T_n$ has $n$ nodes numbered $1$ to $n$.</p>
<p>The sequence starts at $T_1$, a tree with a single node as a root with the number $1$.</p>
<p>For $n > 1$, $T_n$ is constructed from $T_{n-1}$ using the following procedure:
</p><ol>
<li>Trace a path from the... | 2903144925319290239 | Saturday, 13th January 2024, 10:00 pm | 1028 | 5% | easy |
51 | Prime Digit Replacements | By replacing the 1st digit of the 2-digit number *3, it turns out that six of the nine possible values: 13, 23, 43, 53, 73, and 83, are all prime.
By replacing the 3rd and 4th digits of 56**3 with the same digit, this 5-digit number is the first example having seven primes among the ten generated numbers, yielding the ... | By replacing the 1st digit of the 2-digit number *3, it turns out that six of the nine possible values: 13, 23, 43, 53, 73, and 83, are all prime.
By replacing the 3rd and 4th digits of 56**3 with the same digit, this 5-digit number is the first example having seven primes among the ten generated numbers, yielding the ... | <p>By replacing the 1<sup>st</sup> digit of the 2-digit number *3, it turns out that six of the nine possible values: 13, 23, 43, 53, 73, and 83, are all prime.</p>
<p>By replacing the 3<sup>rd</sup> and 4<sup>th</sup> digits of 56**3 with the same digit, this 5-digit number is the first example having seven primes amo... | 121313 | Friday, 29th August 2003, 06:00 pm | 38149 | 15% | easy |
648 | Skipping Squares | For some fixed $\rho \in [0, 1]$, we begin a sum $s$ at $0$ and repeatedly apply a process: With probability $\rho$, we add $1$ to $s$, otherwise we add $2$ to $s$.
The process ends when either $s$ is a perfect square or $s$ exceeds $10^{18}$, whichever occurs first. For example, if $s$ goes through $0, 2, 3, 5, 7, 9$,... | For some fixed $\rho \in [0, 1]$, we begin a sum $s$ at $0$ and repeatedly apply a process: With probability $\rho$, we add $1$ to $s$, otherwise we add $2$ to $s$.
The process ends when either $s$ is a perfect square or $s$ exceeds $10^{18}$, whichever occurs first. For example, if $s$ goes through $0, 2, 3, 5, 7, 9$,... | <p>For some fixed $\rho \in [0, 1]$, we begin a sum $s$ at $0$ and repeatedly apply a process: With probability $\rho$, we add $1$ to $s$, otherwise we add $2$ to $s$.</p>
<p>The process ends when either $s$ is a perfect square or $s$ exceeds $10^{18}$, whichever occurs first. For example, if $s$ goes through $0, 2, 3,... | 301483197 | Sunday, 23rd December 2018, 10:00 am | 319 | 45% | medium |
91 | Right Triangles with Integer Coordinates | The points $P(x_1, y_1)$ and $Q(x_2, y_2)$ are plotted at integer co-ordinates and are joined to the origin, $O(0,0)$, to form $\triangle OPQ$.
There are exactly fourteen triangles containing a right angle that can be formed when each co-ordinate lies between $0$ and $2$ inclusive; that is, $0 \le x_1, y_1, x_2, y_2 ... | The points $P(x_1, y_1)$ and $Q(x_2, y_2)$ are plotted at integer co-ordinates and are joined to the origin, $O(0,0)$, to form $\triangle OPQ$.
There are exactly fourteen triangles containing a right angle that can be formed when each co-ordinate lies between $0$ and $2$ inclusive; that is, $0 \le x_1, y_1, x_2, y_2 ... | <p>The points $P(x_1, y_1)$ and $Q(x_2, y_2)$ are plotted at integer co-ordinates and are joined to the origin, $O(0,0)$, to form $\triangle OPQ$.</p>
<div class="center">
<img alt="" class="dark_img" src="resources/images/0091_1.png?1678992052"/><br/></div>
<p>There are exactly fourteen triangles containing a right an... | 14234 | Friday, 18th March 2005, 06:00 pm | 17353 | 25% | easy |
497 | Drunken Tower of Hanoi | Bob is very familiar with the famous mathematical puzzle/game, "Tower of Hanoi," which consists of three upright rods and disks of different sizes that can slide onto any of the rods. The game begins with a stack of $n$ disks placed on the leftmost rod in descending order by size. The objective of the game is to move a... | Bob is very familiar with the famous mathematical puzzle/game, "Tower of Hanoi," which consists of three upright rods and disks of different sizes that can slide onto any of the rods. The game begins with a stack of $n$ disks placed on the leftmost rod in descending order by size. The objective of the game is to move a... | <p>Bob is very familiar with the famous mathematical puzzle/game, "Tower of Hanoi," which consists of three upright rods and disks of different sizes that can slide onto any of the rods. The game begins with a stack of $n$ disks placed on the leftmost rod in descending order by size. The objective of the game is to mov... | 684901360 | Sunday, 11th January 2015, 04:00 am | 609 | 40% | medium |
860 | Gold and Silver Coin Game | Gary and Sally play a game using gold and silver coins arranged into a number of vertical stacks, alternating turns. On Gary's turn he chooses a gold coin and removes it from the game along with any other coins sitting on top. Sally does the same on her turn by removing a silver coin. The first player unable to make a ... | Gary and Sally play a game using gold and silver coins arranged into a number of vertical stacks, alternating turns. On Gary's turn he chooses a gold coin and removes it from the game along with any other coins sitting on top. Sally does the same on her turn by removing a silver coin. The first player unable to make a ... | <p>
Gary and Sally play a game using gold and silver coins arranged into a number of vertical stacks, alternating turns. On Gary's turn he chooses a gold coin and removes it from the game along with any other coins sitting on top. Sally does the same on her turn by removing a silver coin. The first player unable to mak... | 958666903 | Sunday, 22nd October 2023, 11:00 am | 491 | 20% | easy |
73 | Counting Fractions in a Range | Consider the fraction, $\dfrac n d$, where $n$ and $d$ are positive integers. If $n \lt d$ and $\operatorname{HCF}(n, d)=1$, it is called a reduced proper fraction.
If we list the set of reduced proper fractions for $d \le 8$ in ascending order of size, we get:
$$\frac 1 8, \frac 1 7, \frac 1 6, \frac 1 5, \frac 1 4, \... | Consider the fraction, $\dfrac n d$, where $n$ and $d$ are positive integers. If $n \lt d$ and $\operatorname{HCF}(n, d)=1$, it is called a reduced proper fraction.
If we list the set of reduced proper fractions for $d \le 8$ in ascending order of size, we get:
$$\frac 1 8, \frac 1 7, \frac 1 6, \frac 1 5, \frac 1 4, \... | <p>Consider the fraction, $\dfrac n d$, where $n$ and $d$ are positive integers. If $n \lt d$ and $\operatorname{HCF}(n, d)=1$, it is called a reduced proper fraction.</p>
<p>If we list the set of reduced proper fractions for $d \le 8$ in ascending order of size, we get:
$$\frac 1 8, \frac 1 7, \frac 1 6, \frac 1 5, \f... | 7295372 | Friday, 2nd July 2004, 06:00 pm | 27754 | 15% | easy |
109 | Darts | In the game of darts a player throws three darts at a target board which is split into twenty equal sized sections numbered one to twenty.
The score of a dart is determined by the number of the region that the dart lands in. A dart landing outside the red/green outer ring scores zero. The black and cream regions insi... | In the game of darts a player throws three darts at a target board which is split into twenty equal sized sections numbered one to twenty.
The score of a dart is determined by the number of the region that the dart lands in. A dart landing outside the red/green outer ring scores zero. The black and cream regions insi... | <p>In the game of darts a player throws three darts at a target board which is split into twenty equal sized sections numbered one to twenty.</p>
<div class="center">
<img alt="" class="dark_img" src="project/images/p109.png"><br/></img></div>
<p>The score of a dart is determined by the number of the region that the da... | 38182 | Friday, 18th November 2005, 06:00 pm | 9162 | 45% | medium |
405 | A Rectangular Tiling | We wish to tile a rectangle whose length is twice its width.
Let $T(0)$ be the tiling consisting of a single rectangle.
For $n \gt 0$, let $T(n)$ be obtained from $T(n-1)$ by replacing all tiles in the following manner:
The following animation demonstrates the tilings $T(n)$ for $n$ from $0$ to $5$:
Let $f(n)$ ... | We wish to tile a rectangle whose length is twice its width.
Let $T(0)$ be the tiling consisting of a single rectangle.
For $n \gt 0$, let $T(n)$ be obtained from $T(n-1)$ by replacing all tiles in the following manner:
The following animation demonstrates the tilings $T(n)$ for $n$ from $0$ to $5$:
Let $f(n)$ ... | <p>
We wish to tile a rectangle whose length is twice its width.<br/>
Let $T(0)$ be the tiling consisting of a single rectangle.<br/>
For $n \gt 0$, let $T(n)$ be obtained from $T(n-1)$ by replacing all tiles in the following manner:
</p>
<div align="center">
<img alt="0405_tile1.png" src="resources/images/0405_tile1.p... | 237696125 | Sunday, 9th December 2012, 04:00 am | 694 | 40% | medium |
339 | Peredur Fab Efrawg | "And he came towards a valley, through which ran a river; and the borders of the valley were wooded, and on each side of the river were level meadows. And on one side of the river he saw a flock of white sheep, and on the other a flock of black sheep. And whenever one of the white sheep bleated, one of the black sheep ... | "And he came towards a valley, through which ran a river; and the borders of the valley were wooded, and on each side of the river were level meadows. And on one side of the river he saw a flock of white sheep, and on the other a flock of black sheep. And whenever one of the white sheep bleated, one of the black sheep ... | <p>
<i>"And he came towards a valley, through which ran a river; and the borders of the valley were wooded, and on each side of the river were level meadows. And on one side of the river he saw a flock of white sheep, and on the other a flock of black sheep. And whenever one of the white sheep bleated, one of the black... | 19823.542204 | Sunday, 22nd May 2011, 04:00 am | 622 | 70% | hard |
69 | Totient Maximum | Euler's totient function, $\phi(n)$ [sometimes called the phi function], is defined as the number of positive integers not exceeding $n$ which are relatively prime to $n$. For example, as $1$, $2$, $4$, $5$, $7$, and $8$, are all less than or equal to nine and relatively prime to nine, $\phi(9)=6$.
$n$
Relatively Prim... | Euler's totient function, $\phi(n)$ [sometimes called the phi function], is defined as the number of positive integers not exceeding $n$ which are relatively prime to $n$. For example, as $1$, $2$, $4$, $5$, $7$, and $8$, are all less than or equal to nine and relatively prime to nine, $\phi(9)=6$.
$n$
Relatively Prim... | <p>Euler's totient function, $\phi(n)$ [sometimes called the phi function], is defined as the number of positive integers not exceeding $n$ which are relatively prime to $n$. For example, as $1$, $2$, $4$, $5$, $7$, and $8$, are all less than or equal to nine and relatively prime to nine, $\phi(9)=6$.</p>
<div class="c... | 510510 | Friday, 7th May 2004, 06:00 pm | 38706 | 10% | easy |
636 | Restricted Factorisations | Consider writing a natural number as product of powers of natural numbers with given exponents, additionally requiring different base numbers for each power.
For example, $256$ can be written as a product of a square and a fourth power in three ways such that the base numbers are different.
That is, $256=1^2\times 4^4=... | Consider writing a natural number as product of powers of natural numbers with given exponents, additionally requiring different base numbers for each power.
For example, $256$ can be written as a product of a square and a fourth power in three ways such that the base numbers are different.
That is, $256=1^2\times 4^4=... | <p>Consider writing a natural number as product of powers of natural numbers with given exponents, additionally requiring different base numbers for each power.</p>
<p>For example, $256$ can be written as a product of a square and a fourth power in three ways such that the base numbers are different.<br>
That is, $256=... | 888316 | Saturday, 8th September 2018, 10:00 pm | 217 | 90% | hard |
455 | Powers with Trailing Digits | Let $f(n)$ be the largest positive integer $x$ less than $10^9$ such that the last $9$ digits of $n^x$ form the number $x$ (including leading zeros), or zero if no such integer exists.
For example:
$f(4) = 411728896$ ($4^{411728896} = \cdots 490\underline{411728896}$)
$f(10) = 0$
$f(157) = 743757$ ($157^{743757} = \cd... | Let $f(n)$ be the largest positive integer $x$ less than $10^9$ such that the last $9$ digits of $n^x$ form the number $x$ (including leading zeros), or zero if no such integer exists.
For example:
$f(4) = 411728896$ ($4^{411728896} = \cdots 490\underline{411728896}$)
$f(10) = 0$
$f(157) = 743757$ ($157^{743757} = \cd... | <p>Let $f(n)$ be the largest positive integer $x$ less than $10^9$ such that the last $9$ digits of $n^x$ form the number $x$ (including leading zeros), or zero if no such integer exists.</p>
<p>For example:</p>
<ul><li>$f(4) = 411728896$ ($4^{411728896} = \cdots 490\underline{411728896}$) </li>
<li>$f(10) = 0$</li>
<l... | 450186511399999 | Saturday, 18th January 2014, 10:00 pm | 802 | 40% | medium |
849 | The Tournament | In a tournament there are $n$ teams and each team plays each other team twice. A team gets two points for a win, one point for a draw and no points for a loss.
With two teams there are three possible outcomes for the total points. $(4,0)$ where a team wins twice, $(3,1)$ where a team wins and draws, and $(2,2)$ where ... | In a tournament there are $n$ teams and each team plays each other team twice. A team gets two points for a win, one point for a draw and no points for a loss.
With two teams there are three possible outcomes for the total points. $(4,0)$ where a team wins twice, $(3,1)$ where a team wins and draws, and $(2,2)$ where ... | <p>
In a tournament there are $n$ teams and each team plays each other team twice. A team gets two points for a win, one point for a draw and no points for a loss.</p>
<p>
With two teams there are three possible outcomes for the total points. $(4,0)$ where a team wins twice, $(3,1)$ where a team wins and draws, and $(2... | 936203459 | Sunday, 25th June 2023, 05:00 am | 270 | 45% | medium |
717 | Summation of a Modular Formula | For an odd prime $p$, define $f(p) = \left\lfloor\frac{2^{(2^p)}}{p}\right\rfloor\bmod{2^p}$
For example, when $p=3$, $\lfloor 2^8/3\rfloor = 85 \equiv 5 \pmod 8$ and so $f(3) = 5$.
Further define $g(p) = f(p)\bmod p$. You are given $g(31) = 17$.
Now define $G(N)$ to be the summation of $g(p)$ for all odd primes less t... | For an odd prime $p$, define $f(p) = \left\lfloor\frac{2^{(2^p)}}{p}\right\rfloor\bmod{2^p}$
For example, when $p=3$, $\lfloor 2^8/3\rfloor = 85 \equiv 5 \pmod 8$ and so $f(3) = 5$.
Further define $g(p) = f(p)\bmod p$. You are given $g(31) = 17$.
Now define $G(N)$ to be the summation of $g(p)$ for all odd primes less t... | <p>For an odd prime $p$, define $f(p) = \left\lfloor\frac{2^{(2^p)}}{p}\right\rfloor\bmod{2^p}$<br/>
For example, when $p=3$, $\lfloor 2^8/3\rfloor = 85 \equiv 5 \pmod 8$ and so $f(3) = 5$.</p>
<p>Further define $g(p) = f(p)\bmod p$. You are given $g(31) = 17$.</p>
<p>Now define $G(N)$ to be the summation of $g(p)$ for... | 1603036763131 | Saturday, 23rd May 2020, 02:00 pm | 551 | 25% | easy |
756 | Approximating a Sum | Consider a function $f(k)$ defined for all positive integers $k>0$. Let $S$ be the sum of the first $n$ values of $f$. That is,
$$S=f(1)+f(2)+f(3)+\cdots+f(n)=\sum_{k=1}^n f(k).$$
In this problem, we employ randomness to approximate this sum. That is, we choose a random, uniformly distributed, $m$-tuple of positive int... | Consider a function $f(k)$ defined for all positive integers $k>0$. Let $S$ be the sum of the first $n$ values of $f$. That is,
$$S=f(1)+f(2)+f(3)+\cdots+f(n)=\sum_{k=1}^n f(k).$$
In this problem, we employ randomness to approximate this sum. That is, we choose a random, uniformly distributed, $m$-tuple of positive int... | <p>Consider a function $f(k)$ defined for all positive integers $k>0$. Let $S$ be the sum of the first $n$ values of $f$. That is,
$$S=f(1)+f(2)+f(3)+\cdots+f(n)=\sum_{k=1}^n f(k).$$</p>
<p>In this problem, we employ randomness to approximate this sum. That is, we choose a random, uniformly distributed, $m$-tuple of... | 607238.610661 | Saturday, 1st May 2021, 02:00 pm | 363 | 30% | easy |
570 | Snowflakes | A snowflake of order $n$ is formed by overlaying an equilateral triangle (rotated by $180$ degrees) onto each equilateral triangle of the same size in a snowflake of order $n-1$. A snowflake of order $1$ is a single equilateral triangle.
Some areas of the snowflake are overlaid repeatedly. In the above picture, blue... | A snowflake of order $n$ is formed by overlaying an equilateral triangle (rotated by $180$ degrees) onto each equilateral triangle of the same size in a snowflake of order $n-1$. A snowflake of order $1$ is a single equilateral triangle.
Some areas of the snowflake are overlaid repeatedly. In the above picture, blue... | <p>A snowflake of order $n$ is formed by overlaying an equilateral triangle (rotated by $180$ degrees) onto each equilateral triangle of the same size in a snowflake of order $n-1$. A snowflake of order $1$ is a single equilateral triangle.</p>
<div> <img alt="0570-snowflakes.png" src="resources/images/0570-snowflakes.... | 271197444 | Saturday, 17th September 2016, 10:00 pm | 296 | 55% | medium |
198 | Ambiguous Numbers | A best approximation to a real number $x$ for the denominator bound $d$ is a rational number $\frac r s$ (in reduced form) with $s \le d$, so that any rational number $\frac p q$ which is closer to $x$ than $\frac r s$ has $q \gt d$.
Usually the best approximation to a real number is uniquely determined for all denomin... | A best approximation to a real number $x$ for the denominator bound $d$ is a rational number $\frac r s$ (in reduced form) with $s \le d$, so that any rational number $\frac p q$ which is closer to $x$ than $\frac r s$ has $q \gt d$.
Usually the best approximation to a real number is uniquely determined for all denomin... | <p>A best approximation to a real number $x$ for the denominator bound $d$ is a rational number $\frac r s$ (in reduced form) with $s \le d$, so that any rational number $\frac p q$ which is closer to $x$ than $\frac r s$ has $q \gt d$.</p>
<p>Usually the best approximation to a real number is uniquely determined for a... | 52374425 | Saturday, 14th June 2008, 02:00 am | 1295 | 80% | hard |
867 | Tiling Dodecagon | There are 5 ways to tile a regular dodecagon of side 1 with regular polygons of side 1.
Let $T(n)$ be the number of ways to tile a regular dodecagon of side $n$ with regular polygons of side 1. Then $T(1) = 5$. You are also given $T(2) = 48$.
Find $T(10)$. Give your answer modulo $10^9+7$. | There are 5 ways to tile a regular dodecagon of side 1 with regular polygons of side 1.
Let $T(n)$ be the number of ways to tile a regular dodecagon of side $n$ with regular polygons of side 1. Then $T(1) = 5$. You are also given $T(2) = 48$.
Find $T(10)$. Give your answer modulo $10^9+7$. | <p>
There are 5 ways to tile a regular dodecagon of side 1 with regular polygons of side 1.</p>
<img alt="0867_DodecaDiagram.jpg" src="resources/images/0867_DodecaDiagram.jpg?1700512497"/>
<p>
Let $T(n)$ be the number of ways to tile a regular dodecagon of side $n$ with regular polygons of side 1. Then $T(1) = 5$. You ... | 870557257 | Sunday, 10th December 2023, 07:00 am | 177 | 55% | medium |
156 | Counting Digits | Starting from zero the natural numbers are written down in base $10$ like this:
$0\,1\,2\,3\,4\,5\,6\,7\,8\,9\,10\,11\,12\cdots$
Consider the digit $d=1$. After we write down each number $n$, we will update the number of ones that have occurred and call this number $f(n,1)$. The first values for $f(n,1)$, then, are a... | Starting from zero the natural numbers are written down in base $10$ like this:
$0\,1\,2\,3\,4\,5\,6\,7\,8\,9\,10\,11\,12\cdots$
Consider the digit $d=1$. After we write down each number $n$, we will update the number of ones that have occurred and call this number $f(n,1)$. The first values for $f(n,1)$, then, are a... | <p>Starting from zero the natural numbers are written down in base $10$ like this:
<br/>
$0\,1\,2\,3\,4\,5\,6\,7\,8\,9\,10\,11\,12\cdots$
</p>
<p>Consider the digit $d=1$. After we write down each number $n$, we will update the number of ones that have occurred and call this number $f(n,1)$. The first values for $f(n,1... | 21295121502550 | Friday, 25th May 2007, 10:00 pm | 2771 | 70% | hard |
469 | Empty Chairs | In a room $N$ chairs are placed around a round table.
Knights enter the room one by one and choose at random an available empty chair.
To have enough elbow room the knights always leave at least one empty chair between each other.
When there aren't any suitable chairs left, the fraction $C$ of empty chairs is determi... | In a room $N$ chairs are placed around a round table.
Knights enter the room one by one and choose at random an available empty chair.
To have enough elbow room the knights always leave at least one empty chair between each other.
When there aren't any suitable chairs left, the fraction $C$ of empty chairs is determi... | <p>
In a room $N$ chairs are placed around a round table.<br/>
Knights enter the room one by one and choose at random an available empty chair.<br/>
To have enough elbow room the knights always leave at least one empty chair between each other.
</p>
<p>
When there aren't any suitable chairs left, the fraction $C$ of em... | 0.56766764161831 | Saturday, 26th April 2014, 04:00 pm | 828 | 40% | medium |
663 | Sums of Subarrays | Let $t_k$ be the tribonacci numbers defined as:
$\quad t_0 = t_1 = 0$;
$\quad t_2 = 1$;
$\quad t_k = t_{k-1} + t_{k-2} + t_{k-3} \quad \text{ for } k \ge 3$.
For a given integer $n$, let $A_n$ be an array of length $n$ (indexed from $0$ to $n-1$), that is initially filled with zeros.
The array is changed iterati... | Let $t_k$ be the tribonacci numbers defined as:
$\quad t_0 = t_1 = 0$;
$\quad t_2 = 1$;
$\quad t_k = t_{k-1} + t_{k-2} + t_{k-3} \quad \text{ for } k \ge 3$.
For a given integer $n$, let $A_n$ be an array of length $n$ (indexed from $0$ to $n-1$), that is initially filled with zeros.
The array is changed iterati... | <p>Let $t_k$ be the <b>tribonacci numbers</b> defined as: <br/>
$\quad t_0 = t_1 = 0$;<br/>
$\quad t_2 = 1$; <br/>
$\quad t_k = t_{k-1} + t_{k-2} + t_{k-3} \quad \text{ for } k \ge 3$.</p>
<p>For a given integer $n$, let $A_n$ be an array of length $n$ (indexed from $0$ to $n-1$), that is initially filled with zer... | 1884138010064752 | Sunday, 31st March 2019, 04:00 am | 404 | 35% | medium |
131 | Prime Cube Partnership | There are some prime values, $p$, for which there exists a positive integer, $n$, such that the expression $n^3 + n^2p$ is a perfect cube.
For example, when $p = 19$, $8^3 + 8^2 \times 19 = 12^3$.
What is perhaps most surprising is that for each prime with this property the value of $n$ is unique, and there are only fo... | There are some prime values, $p$, for which there exists a positive integer, $n$, such that the expression $n^3 + n^2p$ is a perfect cube.
For example, when $p = 19$, $8^3 + 8^2 \times 19 = 12^3$.
What is perhaps most surprising is that for each prime with this property the value of $n$ is unique, and there are only fo... | <p>There are some prime values, $p$, for which there exists a positive integer, $n$, such that the expression $n^3 + n^2p$ is a perfect cube.</p>
<p>For example, when $p = 19$, $8^3 + 8^2 \times 19 = 12^3$.</p>
<p>What is perhaps most surprising is that for each prime with this property the value of $n$ is unique, and ... | 173 | Friday, 10th November 2006, 06:00 pm | 8246 | 40% | medium |
221 | Alexandrian Integers | We shall call a positive integer $A$ an "Alexandrian integer", if there exist integers $p, q, r$ such that:
$$A = p \cdot q \cdot r$$
and
$$\dfrac{1}{A} = \dfrac{1}{p} + \dfrac{1}{q} + \dfrac{1}{r}.$$
For example, $630$ is an Alexandrian integer ($p = 5, q = -7, r = -18$).
In fact, $630$ is the $6$th Alexandrian intege... | We shall call a positive integer $A$ an "Alexandrian integer", if there exist integers $p, q, r$ such that:
$$A = p \cdot q \cdot r$$
and
$$\dfrac{1}{A} = \dfrac{1}{p} + \dfrac{1}{q} + \dfrac{1}{r}.$$
For example, $630$ is an Alexandrian integer ($p = 5, q = -7, r = -18$).
In fact, $630$ is the $6$th Alexandrian intege... | <p>We shall call a positive integer $A$ an "Alexandrian integer", if there exist integers $p, q, r$ such that:</p>
<p class="center">$$A = p \cdot q \cdot r$$
and
$$\dfrac{1}{A} = \dfrac{1}{p} + \dfrac{1}{q} + \dfrac{1}{r}.$$</p>
<p>For example, $630$ is an Alexandrian integer ($p = 5, q = -7, r = -18$).
In fact, $630$... | 1884161251122450 | Saturday, 13th December 2008, 01:00 pm | 2340 | 65% | hard |
679 | Freefarea | Let $S$ be the set consisting of the four letters $\{\texttt{`A'},\texttt{`E'},\texttt{`F'},\texttt{`R'}\}$.
For $n\ge 0$, let $S^*(n)$ denote the set of words of length $n$ consisting of letters belonging to $S$.
We designate the words $\texttt{FREE}, \texttt{FARE}, \texttt{AREA}, \texttt{REEF}$ as keywords.
Let $f(n)... | Let $S$ be the set consisting of the four letters $\{\texttt{`A'},\texttt{`E'},\texttt{`F'},\texttt{`R'}\}$.
For $n\ge 0$, let $S^*(n)$ denote the set of words of length $n$ consisting of letters belonging to $S$.
We designate the words $\texttt{FREE}, \texttt{FARE}, \texttt{AREA}, \texttt{REEF}$ as keywords.
Let $f(n)... | <p>Let $S$ be the set consisting of the four letters $\{\texttt{`A'},\texttt{`E'},\texttt{`F'},\texttt{`R'}\}$.<br>
For $n\ge 0$, let $S^*(n)$ denote the set of words of length $n$ consisting of letters belonging to $S$.<br/>
We designate the words $\texttt{FREE}, \texttt{FARE}, \texttt{AREA}, \texttt{REEF}$ as <i>keyw... | 644997092988678 | Sunday, 15th September 2019, 01:00 am | 993 | 20% | easy |
34 | Digit Factorials | $145$ is a curious number, as $1! + 4! + 5! = 1 + 24 + 120 = 145$.
Find the sum of all numbers which are equal to the sum of the factorial of their digits.
Note: As $1! = 1$ and $2! = 2$ are not sums they are not included. | $145$ is a curious number, as $1! + 4! + 5! = 1 + 24 + 120 = 145$.
Find the sum of all numbers which are equal to the sum of the factorial of their digits.
Note: As $1! = 1$ and $2! = 2$ are not sums they are not included. | <p>$145$ is a curious number, as $1! + 4! + 5! = 1 + 24 + 120 = 145$.</p>
<p>Find the sum of all numbers which are equal to the sum of the factorial of their digits.</p>
<p class="smaller">Note: As $1! = 1$ and $2! = 2$ are not sums they are not included.</p> | 40730 | Friday, 3rd January 2003, 06:00 pm | 102509 | 5% | easy |
195 | $60$-degree Triangle Inscribed Circles | Let's call an integer sided triangle with exactly one angle of $60$ degrees a $60$-degree triangle.
Let $r$ be the radius of the inscribed circle of such a $60$-degree triangle.
There are $1234$ $60$-degree triangles for which $r \le 100$.
Let $T(n)$ be the number of $60$-degree triangles for which $r \le n$, so
$T(100... | Let's call an integer sided triangle with exactly one angle of $60$ degrees a $60$-degree triangle.
Let $r$ be the radius of the inscribed circle of such a $60$-degree triangle.
There are $1234$ $60$-degree triangles for which $r \le 100$.
Let $T(n)$ be the number of $60$-degree triangles for which $r \le n$, so
$T(100... | <p>Let's call an integer sided triangle with exactly one angle of $60$ degrees a $60$-degree triangle.<br/>
Let $r$ be the radius of the inscribed circle of such a $60$-degree triangle.</p>
<p>There are $1234$ $60$-degree triangles for which $r \le 100$.
<br/>Let $T(n)$ be the number of $60$-degree triangles for which ... | 75085391 | Friday, 23rd May 2008, 02:00 pm | 1616 | 75% | hard |
851 | SOP and POS | Let $n$ be a positive integer and let $E_n$ be the set of $n$-tuples of strictly positive integers.
For $u = (u_1, \cdots, u_n)$ and $v = (v_1, \cdots, v_n)$ two elements of $E_n$, we define:
the Sum Of Products of $u$ and $v$, denoted by $\langle u, v\rangle$, as the sum $\displaystyle\sum_{i = 1}^n u_i v_i$;
the Pr... | Let $n$ be a positive integer and let $E_n$ be the set of $n$-tuples of strictly positive integers.
For $u = (u_1, \cdots, u_n)$ and $v = (v_1, \cdots, v_n)$ two elements of $E_n$, we define:
the Sum Of Products of $u$ and $v$, denoted by $\langle u, v\rangle$, as the sum $\displaystyle\sum_{i = 1}^n u_i v_i$;
the Pr... | <p>
Let $n$ be a positive integer and let $E_n$ be the set of $n$-tuples of strictly positive integers.</p>
<p>
For $u = (u_1, \cdots, u_n)$ and $v = (v_1, \cdots, v_n)$ two elements of $E_n$, we define:</p>
<ul>
<li>the <dfn>Sum Of Products</dfn> of $u$ and $v$, denoted by $\langle u, v\rangle$, as the sum $\displayst... | 726358482 | Sunday, 9th July 2023, 11:00 am | 151 | 85% | hard |
514 | Geoboard Shapes | A geoboard (of order $N$) is a square board with equally-spaced pins protruding from the surface, representing an integer point lattice for coordinates $0 \le x, y \le N$.
John begins with a pinless geoboard. Each position on the board is a hole that can be filled with a pin. John decides to generate a random integer b... | A geoboard (of order $N$) is a square board with equally-spaced pins protruding from the surface, representing an integer point lattice for coordinates $0 \le x, y \le N$.
John begins with a pinless geoboard. Each position on the board is a hole that can be filled with a pin. John decides to generate a random integer b... | <p>A <strong>geoboard</strong> (of order $N$) is a square board with equally-spaced pins protruding from the surface, representing an integer point lattice for coordinates $0 \le x, y \le N$.</p>
<p>John begins with a pinless geoboard. Each position on the board is a hole that can be filled with a pin. John decides to ... | 8986.86698 | Sunday, 3rd May 2015, 04:00 am | 248 | 90% | hard |
479 | Roots on the Rise | Let $a_k$, $b_k$, and $c_k$ represent the three solutions (real or complex numbers) to the equation
$\frac 1 x = (\frac k x)^2(k+x^2)-k x$.
For instance, for $k=5$, we see that $\{a_5, b_5, c_5 \}$ is approximately $\{5.727244, -0.363622+2.057397i, -0.363622-2.057397i\}$.
Let $\displaystyle S(n) = \sum_{p=1}^n\sum_{k=1... | Let $a_k$, $b_k$, and $c_k$ represent the three solutions (real or complex numbers) to the equation
$\frac 1 x = (\frac k x)^2(k+x^2)-k x$.
For instance, for $k=5$, we see that $\{a_5, b_5, c_5 \}$ is approximately $\{5.727244, -0.363622+2.057397i, -0.363622-2.057397i\}$.
Let $\displaystyle S(n) = \sum_{p=1}^n\sum_{k=1... | <p>Let $a_k$, $b_k$, and $c_k$ represent the three solutions (real or complex numbers) to the equation
$\frac 1 x = (\frac k x)^2(k+x^2)-k x$.</p>
<p>For instance, for $k=5$, we see that $\{a_5, b_5, c_5 \}$ is approximately $\{5.727244, -0.363622+2.057397i, -0.363622-2.057397i\}$.</p>
<p>Let $\displaystyle S(n) = \sum... | 191541795 | Saturday, 6th September 2014, 10:00 pm | 1444 | 25% | easy |
658 | Incomplete Words II | In the context of formal languages, any finite sequence of letters of a given alphabet $\Sigma$ is called a word over $\Sigma$. We call a word incomplete if it does not contain every letter of $\Sigma$.
For example, using the alphabet $\Sigma=\{ a, b, c\}$, '$ab$', '$abab$' and '$\,$' (the empty word) are incomplete w... | In the context of formal languages, any finite sequence of letters of a given alphabet $\Sigma$ is called a word over $\Sigma$. We call a word incomplete if it does not contain every letter of $\Sigma$.
For example, using the alphabet $\Sigma=\{ a, b, c\}$, '$ab$', '$abab$' and '$\,$' (the empty word) are incomplete w... | <p>In the context of <strong>formal languages</strong>, any finite sequence of letters of a given <strong>alphabet</strong> $\Sigma$ is called a <strong>word</strong> over $\Sigma$. We call a word <dfn>incomplete</dfn> if it does not contain every letter of $\Sigma$.</p>
<p>
For example, using the alphabet $\Sigma=\{ a... | 958280177 | Saturday, 23rd February 2019, 01:00 pm | 265 | 55% | medium |
783 | Urns | Given $n$ and $k$ two positive integers we begin with an urn that contains $kn$ white balls. We then proceed through $n$ turns where on each turn $k$ black balls are added to the urn and then $2k$ random balls are removed from the urn.
We let $B_t(n,k)$ be the number of black balls that are removed on turn $t$.
Furth... | Given $n$ and $k$ two positive integers we begin with an urn that contains $kn$ white balls. We then proceed through $n$ turns where on each turn $k$ black balls are added to the urn and then $2k$ random balls are removed from the urn.
We let $B_t(n,k)$ be the number of black balls that are removed on turn $t$.
Furth... | <p>
Given $n$ and $k$ two positive integers we begin with an urn that contains $kn$ white balls. We then proceed through $n$ turns where on each turn $k$ black balls are added to the urn and then $2k$ random balls are removed from the urn.</p>
<p>
We let $B_t(n,k)$ be the number of black balls that are removed on turn ... | 136666597 | Sunday, 30th January 2022, 01:00 am | 212 | 55% | medium |
583 | Heron Envelopes | A standard envelope shape is a convex figure consisting of an isosceles triangle (the flap) placed on top of a rectangle. An example of an envelope with integral sides is shown below. Note that to form a sensible envelope, the perpendicular height of the flap ($BCD$) must be smaller than the height of the rectangle (... | A standard envelope shape is a convex figure consisting of an isosceles triangle (the flap) placed on top of a rectangle. An example of an envelope with integral sides is shown below. Note that to form a sensible envelope, the perpendicular height of the flap ($BCD$) must be smaller than the height of the rectangle (... | <p>
A standard envelope shape is a convex figure consisting of an isosceles triangle (the flap) placed on top of a rectangle. An example of an envelope with integral sides is shown below. Note that to form a sensible envelope, the perpendicular height of the flap ($BCD$) must be smaller than the height of the rectang... | 1174137929000 | Saturday, 24th December 2016, 01:00 pm | 410 | 50% | medium |
59 | XOR Decryption | Each character on a computer is assigned a unique code and the preferred standard is ASCII (American Standard Code for Information Interchange). For example, uppercase A = 65, asterisk (*) = 42, and lowercase k = 107.
A modern encryption method is to take a text file, convert the bytes to ASCII, then XOR each byte with... | Each character on a computer is assigned a unique code and the preferred standard is ASCII (American Standard Code for Information Interchange). For example, uppercase A = 65, asterisk (*) = 42, and lowercase k = 107.
A modern encryption method is to take a text file, convert the bytes to ASCII, then XOR each byte with... | <p>Each character on a computer is assigned a unique code and the preferred standard is ASCII (American Standard Code for Information Interchange). For example, uppercase A = 65, asterisk (*) = 42, and lowercase k = 107.</p>
<p>A modern encryption method is to take a text file, convert the bytes to ASCII, then XOR each... | 129448 | Friday, 19th December 2003, 06:00 pm | 45196 | 5% | easy |
831 | Triple Product | Let $g(m)$ be the integer defined by the following double sum of products of binomial coefficients:
$$\sum_{j=0}^m\sum_{i = 0}^j (-1)^{j-i}\binom mj \binom ji \binom{j+5+6i}{j+5}.$$
You are given that $g(10) = 127278262644918$. Its first (most significant) five digits are $12727$.
Find the first ten digits of $g(14... | Let $g(m)$ be the integer defined by the following double sum of products of binomial coefficients:
$$\sum_{j=0}^m\sum_{i = 0}^j (-1)^{j-i}\binom mj \binom ji \binom{j+5+6i}{j+5}.$$
You are given that $g(10) = 127278262644918$. Its first (most significant) five digits are $12727$.
Find the first ten digits of $g(14... | <p>Let $g(m)$ be the integer defined by the following double sum of products of binomial coefficients:</p>
<p>
$$\sum_{j=0}^m\sum_{i = 0}^j (-1)^{j-i}\binom mj \binom ji \binom{j+5+6i}{j+5}.$$
</p>
<p>
You are given that $g(10) = 127278262644918$.<br/> Its first (most significant) five digits are $12727$.<br/>
Find th... | 5226432553 | Sunday, 26th February 2023, 01:00 am | 197 | 60% | hard |
830 | Binomials and Powers | Let $\displaystyle S(n)=\sum\limits_{k=0}^{n}\binom{n}{k}k^n$.
You are given, $S(10)=142469423360$.
Find $S(10^{18})$. Submit your answer modulo $83^3 89^3 97^3$. | Let $\displaystyle S(n)=\sum\limits_{k=0}^{n}\binom{n}{k}k^n$.
You are given, $S(10)=142469423360$.
Find $S(10^{18})$. Submit your answer modulo $83^3 89^3 97^3$. | <p>
Let $\displaystyle S(n)=\sum\limits_{k=0}^{n}\binom{n}{k}k^n$.</p>
<p>
You are given, $S(10)=142469423360$.</p>
<p>
Find $S(10^{18})$. Submit your answer modulo $83^3 89^3 97^3$.</p> | 254179446930484376 | Saturday, 18th February 2023, 10:00 pm | 161 | 75% | hard |
773 | Ruff Numbers | Let $S_k$ be the set containing $2$ and $5$ and the first $k$ primes that end in $7$. For example, $S_3 = \{2,5,7,17,37\}$.
Define a $k$-Ruff number to be one that is not divisible by any element in $S_k$.
If $N_k$ is the product of the numbers in $S_k$ then define $F(k)$ to be the sum of all $k$-Ruff numbers less th... | Let $S_k$ be the set containing $2$ and $5$ and the first $k$ primes that end in $7$. For example, $S_3 = \{2,5,7,17,37\}$.
Define a $k$-Ruff number to be one that is not divisible by any element in $S_k$.
If $N_k$ is the product of the numbers in $S_k$ then define $F(k)$ to be the sum of all $k$-Ruff numbers less th... | <p>
Let $S_k$ be the set containing $2$ and $5$ and the first $k$ primes that end in $7$. For example, $S_3 = \{2,5,7,17,37\}$.</p>
<p>
Define a <dfn>$k$-Ruff</dfn> number to be one that is not divisible by any element in $S_k$.</p>
<p>
If $N_k$ is the product of the numbers in $S_k$ then define $F(k)$ to be the sum of... | 556206950 | Saturday, 20th November 2021, 07:00 pm | 213 | 50% | medium |
859 | Cookie Game | Odd and Even are playing a game with $N$ cookies.
The game begins with the $N$ cookies divided into one or more piles, not necessarily of the same size. They then make moves in turn, starting with Odd.
Odd's turn: Odd may choose any pile with an odd number of cookies, eat one and divide the remaining (if any) into two... | Odd and Even are playing a game with $N$ cookies.
The game begins with the $N$ cookies divided into one or more piles, not necessarily of the same size. They then make moves in turn, starting with Odd.
Odd's turn: Odd may choose any pile with an odd number of cookies, eat one and divide the remaining (if any) into two... | <p>
Odd and Even are playing a game with $N$ cookies.</p>
<p>
The game begins with the $N$ cookies divided into one or more piles, not necessarily of the same size. They then make moves in turn, starting with Odd.<br/>
Odd's turn: Odd may choose any pile with an <b>odd</b> number of cookies, eat one and divide the rema... | 1527162658488196 | Sunday, 15th October 2023, 08:00 am | 241 | 55% | medium |
44 | Pentagon Numbers | Pentagonal numbers are generated by the formula, $P_n=n(3n-1)/2$. The first ten pentagonal numbers are:
$$1, 5, 12, 22, 35, 51, 70, 92, 117, 145, \dots$$
It can be seen that $P_4 + P_7 = 22 + 70 = 92 = P_8$. However, their difference, $70 - 22 = 48$, is not pentagonal.
Find the pair of pentagonal numbers, $P_j$ and $P_... | Pentagonal numbers are generated by the formula, $P_n=n(3n-1)/2$. The first ten pentagonal numbers are:
$$1, 5, 12, 22, 35, 51, 70, 92, 117, 145, \dots$$
It can be seen that $P_4 + P_7 = 22 + 70 = 92 = P_8$. However, their difference, $70 - 22 = 48$, is not pentagonal.
Find the pair of pentagonal numbers, $P_j$ and $P_... | <p>Pentagonal numbers are generated by the formula, $P_n=n(3n-1)/2$. The first ten pentagonal numbers are:
$$1, 5, 12, 22, 35, 51, 70, 92, 117, 145, \dots$$</p>
<p>It can be seen that $P_4 + P_7 = 22 + 70 = 92 = P_8$. However, their difference, $70 - 22 = 48$, is not pentagonal.</p>
<p>Find the pair of pentagonal numbe... | 5482660 | Friday, 23rd May 2003, 06:00 pm | 64423 | 5% | easy |
262 | Mountain Range | The following equation represents the continuous topography of a mountainous region, giving the elevationheight above sea level $h$ at any point $(x, y)$:
$$h = \left(5000 - \frac{x^2 + y^2 + xy}{200} + \frac{25(x + y)}2\right) \cdot e^{-\left|\frac{x^2 + y^2}{1000000} - \frac{3(x + y)}{2000} + \frac 7 {10}\right|}.$$
... | The following equation represents the continuous topography of a mountainous region, giving the elevationheight above sea level $h$ at any point $(x, y)$:
$$h = \left(5000 - \frac{x^2 + y^2 + xy}{200} + \frac{25(x + y)}2\right) \cdot e^{-\left|\frac{x^2 + y^2}{1000000} - \frac{3(x + y)}{2000} + \frac 7 {10}\right|}.$$
... | <p>The following equation represents the <i>continuous</i> topography of a mountainous region, giving the <strong class="tooltip">elevation<span class="tooltiptext">height above sea level</span></strong> $h$ at any point $(x, y)$:
$$h = \left(5000 - \frac{x^2 + y^2 + xy}{200} + \frac{25(x + y)}2\right) \cdot e^{-\left|... | 2531.205 | Friday, 30th October 2009, 09:00 pm | 815 | 80% | hard |
399 | Squarefree Fibonacci Numbers | The first $15$ Fibonacci numbers are:
$1,1,2,3,5,8,13,21,34,55,89,144,233,377,610$.
It can be seen that $8$ and $144$ are not squarefree: $8$ is divisible by $4$ and $144$ is divisible by $4$ and by $9$.
So the first $13$ squarefree Fibonacci numbers are:
$1,1,2,3,5,13,21,34,55,89,233,377$ and $610$.
The $200$th squ... | The first $15$ Fibonacci numbers are:
$1,1,2,3,5,8,13,21,34,55,89,144,233,377,610$.
It can be seen that $8$ and $144$ are not squarefree: $8$ is divisible by $4$ and $144$ is divisible by $4$ and by $9$.
So the first $13$ squarefree Fibonacci numbers are:
$1,1,2,3,5,13,21,34,55,89,233,377$ and $610$.
The $200$th squ... | <p>
The first $15$ Fibonacci numbers are:<br/>
$1,1,2,3,5,8,13,21,34,55,89,144,233,377,610$.<br/>
It can be seen that $8$ and $144$ are not squarefree: $8$ is divisible by $4$ and $144$ is divisible by $4$ and by $9$.<br/>
So the first $13$ squarefree Fibonacci numbers are:<br/>
$1,1,2,3,5,13,21,34,55,89,233,377$ and ... | 1508395636674243,6.5e27330467 | Sunday, 21st October 2012, 11:00 am | 638 | 45% | medium |
687 | Shuffling Cards | A standard deck of $52$ playing cards, which consists of thirteen ranks (Ace, Two, ..., Ten, King, Queen and Jack) each in four suits (Clubs, Diamonds, Hearts and Spades), is randomly shuffled. Let us call a rank perfect if no two cards of that same rank appear next to each other after the shuffle.
It can be seen tha... | A standard deck of $52$ playing cards, which consists of thirteen ranks (Ace, Two, ..., Ten, King, Queen and Jack) each in four suits (Clubs, Diamonds, Hearts and Spades), is randomly shuffled. Let us call a rank perfect if no two cards of that same rank appear next to each other after the shuffle.
It can be seen tha... | <p>A standard deck of $52$ playing cards, which consists of thirteen ranks (Ace, Two, ..., Ten, King, Queen and Jack) each in four suits (Clubs, Diamonds, Hearts and Spades), is randomly shuffled. Let us call a rank <dfn>perfect</dfn> if no two cards of that same rank appear next to each other after the shuffle.</p>
<... | 0.3285320869 | Saturday, 2nd November 2019, 10:00 pm | 345 | 45% | medium |
571 | Super Pandigital Numbers | A positive number is pandigital in base $b$ if it contains all digits from $0$ to $b - 1$ at least once when written in base $b$.
An $n$-super-pandigital number is a number that is simultaneously pandigital in all bases from $2$ to $n$ inclusively.
For example $978 = 1111010010_2 = 1100020_3 = 33102_4 = 12403_5$ is the... | A positive number is pandigital in base $b$ if it contains all digits from $0$ to $b - 1$ at least once when written in base $b$.
An $n$-super-pandigital number is a number that is simultaneously pandigital in all bases from $2$ to $n$ inclusively.
For example $978 = 1111010010_2 = 1100020_3 = 33102_4 = 12403_5$ is the... | <p>A positive number is <strong>pandigital</strong> in base $b$ if it contains all digits from $0$ to $b - 1$ at least once when written in base $b$.</p>
<p>An <dfn>$n$-super-pandigital</dfn> number is a number that is simultaneously pandigital in all bases from $2$ to $n$ inclusively.<br/>
For example $978 = 111101001... | 30510390701978 | Sunday, 25th September 2016, 01:00 am | 1224 | 25% | easy |
253 | Tidying Up A | A small child has a “number caterpillar” consisting of forty jigsaw pieces, each with one number on it, which, when connected together in a line, reveal the numbers $1$ to $40$ in order.
Every night, the child's father has to pick up the pieces of the caterpillar that have been scattered across the play room. He picks ... | A small child has a “number caterpillar” consisting of forty jigsaw pieces, each with one number on it, which, when connected together in a line, reveal the numbers $1$ to $40$ in order.
Every night, the child's father has to pick up the pieces of the caterpillar that have been scattered across the play room. He picks ... | <p>A small child has a “number caterpillar” consisting of forty jigsaw pieces, each with one number on it, which, when connected together in a line, reveal the numbers $1$ to $40$ in order.</p>
<p>Every night, the child's father has to pick up the pieces of the caterpillar that have been scattered across the play room.... | 11.492847 | Friday, 28th August 2009, 01:00 pm | 1173 | 75% | hard |
825 | Chasing Game | Two cars are on a circular track of total length $2n$, facing the same direction, initially distance $n$ apart.
They move in turn. At each turn, the moving car will advance a distance of $1$, $2$ or $3$, with equal probabilities.
The chase ends when the moving car reaches or goes beyond the position of the other car. T... | Two cars are on a circular track of total length $2n$, facing the same direction, initially distance $n$ apart.
They move in turn. At each turn, the moving car will advance a distance of $1$, $2$ or $3$, with equal probabilities.
The chase ends when the moving car reaches or goes beyond the position of the other car. T... | <p>Two cars are on a circular track of total length $2n$, facing the same direction, initially distance $n$ apart.<br>
They move in turn. At each turn, the moving car will advance a distance of $1$, $2$ or $3$, with equal probabilities.<br/>
The chase ends when the moving car reaches or goes beyond the position of the ... | 32.34481054 | Sunday, 15th January 2023, 07:00 am | 167 | 60% | hard |
699 | Triffle Numbers | Let $\sigma(n)$ be the sum of all the divisors of the positive integer $n$, for example:
$\sigma(10) = 1+2+5+10 = 18$.
Define $T(N)$ to be the sum of all numbers $n \le N$ such that when the fraction $\frac{\sigma(n)}{n}$ is written in its lowest form $\frac ab$, the denominator is a power of 3 i.e. $b = 3^k, k > 0$.... | Let $\sigma(n)$ be the sum of all the divisors of the positive integer $n$, for example:
$\sigma(10) = 1+2+5+10 = 18$.
Define $T(N)$ to be the sum of all numbers $n \le N$ such that when the fraction $\frac{\sigma(n)}{n}$ is written in its lowest form $\frac ab$, the denominator is a power of 3 i.e. $b = 3^k, k > 0$.... | <p>
Let $\sigma(n)$ be the sum of all the divisors of the positive integer $n$, for example:<br>
$\sigma(10) = 1+2+5+10 = 18$.
</br></p>
<p>
Define $T(N)$ to be the sum of all numbers $n \le N$ such that when the fraction $\frac{\sigma(n)}{n}$ is written in its lowest form $\frac ab$, the denominator is a power of 3 i.... | 37010438774467572 | Sunday, 26th January 2020, 10:00 am | 209 | 80% | hard |
460 | An Ant on the Move | On the Euclidean plane, an ant travels from point $A(0, 1)$ to point $B(d, 1)$ for an integer $d$.
In each step, the ant at point $(x_0, y_0)$ chooses one of the lattice points $(x_1, y_1)$ which satisfy $x_1 \ge 0$ and $y_1 \ge 1$ and goes straight to $(x_1, y_1)$ at a constant velocity $v$. The value of $v$ depends... | On the Euclidean plane, an ant travels from point $A(0, 1)$ to point $B(d, 1)$ for an integer $d$.
In each step, the ant at point $(x_0, y_0)$ chooses one of the lattice points $(x_1, y_1)$ which satisfy $x_1 \ge 0$ and $y_1 \ge 1$ and goes straight to $(x_1, y_1)$ at a constant velocity $v$. The value of $v$ depends... | <p>
On the Euclidean plane, an ant travels from point $A(0, 1)$ to point $B(d, 1)$ for an integer $d$.
</p>
<p>
In each step, the ant at point $(x_0, y_0)$ chooses one of the lattice points $(x_1, y_1)$ which satisfy $x_1 \ge 0$ and $y_1 \ge 1$ and goes straight to $(x_1, y_1)$ at a constant velocity $v$. The value of ... | 18.420738199 | Saturday, 22nd February 2014, 01:00 pm | 358 | 60% | hard |
573 | Unfair Race | $n$ runners in very different training states want to compete in a race. Each one of them is given a different starting number $k$ $(1\leq k \leq n)$ according to the runner's (constant) individual racing speed being $v_k=\frac{k}{n}$.
In order to give the slower runners a chance to win the race, $n$ different starting... | $n$ runners in very different training states want to compete in a race. Each one of them is given a different starting number $k$ $(1\leq k \leq n)$ according to the runner's (constant) individual racing speed being $v_k=\frac{k}{n}$.
In order to give the slower runners a chance to win the race, $n$ different starting... | <p>$n$ runners in very different training states want to compete in a race. Each one of them is given a different starting number $k$ $(1\leq k \leq n)$ according to the runner's (constant) individual racing speed being $v_k=\frac{k}{n}$.<br/>
In order to give the slower runners a chance to win the race, $n$ different ... | 1252.9809 | Sunday, 9th October 2016, 07:00 am | 232 | 80% | hard |
651 | Patterned Cylinders | An infinitely long cylinder has its curved surface fully covered with different coloured but otherwise identical rectangular stickers, without overlapping. The stickers are aligned with the cylinder, so two of their edges are parallel with the cylinder's axis, with four stickers meeting at each corner.
Let $a>0$ and su... | An infinitely long cylinder has its curved surface fully covered with different coloured but otherwise identical rectangular stickers, without overlapping. The stickers are aligned with the cylinder, so two of their edges are parallel with the cylinder's axis, with four stickers meeting at each corner.
Let $a>0$ and su... | <p>An infinitely long cylinder has its curved surface fully covered with different coloured but otherwise identical rectangular stickers, without overlapping. The stickers are aligned with the cylinder, so two of their edges are parallel with the cylinder's axis, with four stickers meeting at each corner.</p>
<p>Let $a... | 448233151 | Saturday, 12th January 2019, 07:00 pm | 188 | 70% | hard |
214 | Totient Chains | Let $\phi$ be Euler's totient function, i.e. for a natural number $n$,
$\phi(n)$ is the number of $k$, $1 \le k \le n$, for which $\gcd(k, n) = 1$.
By iterating $\phi$, each positive integer generates a decreasing chain of numbers ending in $1$.
E.g. if we start with $5$ the sequence $5,4,2,1$ is generated.
Here is a l... | Let $\phi$ be Euler's totient function, i.e. for a natural number $n$,
$\phi(n)$ is the number of $k$, $1 \le k \le n$, for which $\gcd(k, n) = 1$.
By iterating $\phi$, each positive integer generates a decreasing chain of numbers ending in $1$.
E.g. if we start with $5$ the sequence $5,4,2,1$ is generated.
Here is a l... | <p>Let $\phi$ be Euler's totient function, i.e. for a natural number $n$,
$\phi(n)$ is the number of $k$, $1 \le k \le n$, for which $\gcd(k, n) = 1$.</p>
<p>By iterating $\phi$, each positive integer generates a decreasing chain of numbers ending in $1$.<br/>
E.g. if we start with $5$ the sequence $5,4,2,1$ is generat... | 1677366278943 | Saturday, 25th October 2008, 02:00 pm | 5717 | 40% | medium |
481 | Chef Showdown | A group of chefs (numbered #$1$, #$2$, etc) participate in a turn-based strategic cooking competition. On each chef's turn, he/she cooks up a dish to the best of his/her ability and gives it to a separate panel of judges for taste-testing. Let $S(k)$ represent chef #$k$'s skill level (which is publicly known). More spe... | A group of chefs (numbered #$1$, #$2$, etc) participate in a turn-based strategic cooking competition. On each chef's turn, he/she cooks up a dish to the best of his/her ability and gives it to a separate panel of judges for taste-testing. Let $S(k)$ represent chef #$k$'s skill level (which is publicly known). More spe... | <p>A group of chefs (numbered #$1$, #$2$, etc) participate in a turn-based strategic cooking competition. On each chef's turn, he/she cooks up a dish to the best of his/her ability and gives it to a separate panel of judges for taste-testing. Let $S(k)$ represent chef #$k$'s skill level (which is publicly known). More ... | 729.12106947 | Sunday, 21st September 2014, 04:00 am | 280 | 70% | hard |
126 | Cuboid Layers | The minimum number of cubes to cover every visible face on a cuboid measuring $3 \times 2 \times 1$ is twenty-two.
If we then add a second layer to this solid it would require forty-six cubes to cover every visible face, the third layer would require seventy-eight cubes, and the fourth layer would require one-hundred... | The minimum number of cubes to cover every visible face on a cuboid measuring $3 \times 2 \times 1$ is twenty-two.
If we then add a second layer to this solid it would require forty-six cubes to cover every visible face, the third layer would require seventy-eight cubes, and the fourth layer would require one-hundred... | <p>The minimum number of cubes to cover every visible face on a cuboid measuring $3 \times 2 \times 1$ is twenty-two.</p>
<div class="center">
<img alt="" class="dark_img" src="resources/images/0126.png?1678992052"/><br/></div>
<p>If we then add a second layer to this solid it would require forty-six cubes to cover eve... | 18522 | Friday, 18th August 2006, 06:00 pm | 5374 | 55% | medium |
840 | Sum of Products | A partition of $n$ is a set of positive integers for which the sum equals $n$.
The partitions of 5 are:
$\{5\},\{1,4\},\{2,3\},\{1,1,3\},\{1,2,2\},\{1,1,1,2\}$ and $\{1,1,1,1,1\}$.
Further we define the function $D(p)$ as:
$$
\begin{align}
\begin{split}
D(1) &= 1 \\
D(p) &= 1, \text{ for any prime } p \\
D(pq) &= D(p... | A partition of $n$ is a set of positive integers for which the sum equals $n$.
The partitions of 5 are:
$\{5\},\{1,4\},\{2,3\},\{1,1,3\},\{1,2,2\},\{1,1,1,2\}$ and $\{1,1,1,1,1\}$.
Further we define the function $D(p)$ as:
$$
\begin{align}
\begin{split}
D(1) &= 1 \\
D(p) &= 1, \text{ for any prime } p \\
D(pq) &= D(p... | <p>A <strong>partition</strong> of $n$ is a set of positive integers for which the sum equals $n$.<br/>
The partitions of 5 are:<br/>
$\{5\},\{1,4\},\{2,3\},\{1,1,3\},\{1,2,2\},\{1,1,1,2\}$ and $\{1,1,1,1,1\}$.
</p>
<p>
Further we define the function $D(p)$ as:<br/>
$$
\begin{align}
\begin{split}
D(1) &= 1 \\
D(p) ... | 194396971 | Sunday, 23rd April 2023, 02:00 am | 400 | 25% | easy |
236 | Luxury Hampers | Suppliers 'A' and 'B' provided the following numbers of products for the luxury hamper market:
Product'A''B'Beluga Caviar5248640Christmas Cake13121888Gammon Joint26243776Vintage Port57603776Champagne Truffles39365664
Although the suppliers try very hard to ship their goods in perfect condition, there is inevitably some... | Suppliers 'A' and 'B' provided the following numbers of products for the luxury hamper market:
Product'A''B'Beluga Caviar5248640Christmas Cake13121888Gammon Joint26243776Vintage Port57603776Champagne Truffles39365664
Although the suppliers try very hard to ship their goods in perfect condition, there is inevitably some... | <p>Suppliers 'A' and 'B' provided the following numbers of products for the luxury hamper market:</p>
<p></p><center><table class="p236"><tr><th>Product</th><th class="center">'A'</th><th class="center">'B'</th></tr><tr><td>Beluga Caviar</td><td>5248</td><td>640</td></tr><tr><td>Christmas Cake</td><td>1312</td><td>1888... | 123/59 | Saturday, 14th March 2009, 09:00 am | 1054 | 80% | hard |
281 | Pizza Toppings | You are given a pizza (perfect circle) that has been cut into $m \cdot n$ equal pieces and you want to have exactly one topping on each slice.
Let $f(m, n)$ denote the number of ways you can have toppings on the pizza with $m$ different toppings ($m \ge 2$), using each topping on exactly $n$ slices ($n \ge 1$).Reflecti... | You are given a pizza (perfect circle) that has been cut into $m \cdot n$ equal pieces and you want to have exactly one topping on each slice.
Let $f(m, n)$ denote the number of ways you can have toppings on the pizza with $m$ different toppings ($m \ge 2$), using each topping on exactly $n$ slices ($n \ge 1$).Reflecti... | <p>You are given a pizza (perfect circle) that has been cut into $m \cdot n$ equal pieces and you want to have exactly one topping on each slice.</p>
<p>Let $f(m, n)$ denote the number of ways you can have toppings on the pizza with $m$ different toppings ($m \ge 2$), using each topping on exactly $n$ slices ($n \ge 1$... | 1485776387445623 | Friday, 5th March 2010, 01:00 pm | 1109 | 55% | medium |
154 | Exploring Pascal's Pyramid | A triangular pyramid is constructed using spherical balls so that each ball rests on exactly three balls of the next lower level.
Then, we calculate the number of paths leading from the apex to each position:
A path starts at the apex and progresses downwards to any of the three spheres directly below the current posi... | A triangular pyramid is constructed using spherical balls so that each ball rests on exactly three balls of the next lower level.
Then, we calculate the number of paths leading from the apex to each position:
A path starts at the apex and progresses downwards to any of the three spheres directly below the current posi... | <p>A triangular pyramid is constructed using spherical balls so that each ball rests on exactly three balls of the next lower level.</p>
<div class="center"><img alt="" class="dark_img" src="resources/images/0154_pyramid.png?1678992052"/></div>
<p>Then, we calculate the number of paths leading from the apex to each pos... | 479742450 | Saturday, 12th May 2007, 06:00 am | 2975 | 65% | hard |
848 | Guessing with Sets | Two players play a game. At the start of the game each player secretly chooses an integer; the first player from $1,...,n$ and the second player from $1,...,m$. Then they take alternate turns, starting with the first player. The player, whose turn it is, displays a set of numbers and the other player tells whether thei... | Two players play a game. At the start of the game each player secretly chooses an integer; the first player from $1,...,n$ and the second player from $1,...,m$. Then they take alternate turns, starting with the first player. The player, whose turn it is, displays a set of numbers and the other player tells whether thei... | <p>Two players play a game. At the start of the game each player secretly chooses an integer; the first player from $1,...,n$ and the second player from $1,...,m$. Then they take alternate turns, starting with the first player. The player, whose turn it is, displays a set of numbers and the other player tells whether t... | 188.45503259 | Sunday, 18th June 2023, 02:00 am | 213 | 45% | medium |
540 | Counting Primitive Pythagorean Triples | A Pythagorean triple consists of three positive integers $a, b$ and $c$ satisfying $a^2+b^2=c^2$.
The triple is called primitive if $a, b$ and $c$ are relatively prime.
Let $P(n)$ be the number of primitive Pythagorean triples with $a \lt b \lt c \le n$.
For example $P(20) = 3$, since there are three triples: $(3,4,5)$... | A Pythagorean triple consists of three positive integers $a, b$ and $c$ satisfying $a^2+b^2=c^2$.
The triple is called primitive if $a, b$ and $c$ are relatively prime.
Let $P(n)$ be the number of primitive Pythagorean triples with $a \lt b \lt c \le n$.
For example $P(20) = 3$, since there are three triples: $(3,4,5)$... | <p>
A <strong>Pythagorean triple</strong> consists of three positive integers $a, b$ and $c$ satisfying $a^2+b^2=c^2$.<br/>
The triple is called primitive if $a, b$ and $c$ are relatively prime.<br/>
Let $P(n)$ be the number of <strong>primitive Pythagorean triples</strong> with $a \lt b \lt c \le n$.<br/>
For example ... | 500000000002845 | Sunday, 27th December 2015, 07:00 am | 736 | 30% | easy |
672 | One More One | Consider the following process that can be applied recursively to any positive integer $n$:
if $n = 1$ do nothing and the process stops,
if $n$ is divisible by $7$ divide it by $7$,
otherwise add $1$.
Define $g(n)$ to be the number of $1$'s that must be added before the process ends. For example:
$125\xrightarrow{\sc... | Consider the following process that can be applied recursively to any positive integer $n$:
if $n = 1$ do nothing and the process stops,
if $n$ is divisible by $7$ divide it by $7$,
otherwise add $1$.
Define $g(n)$ to be the number of $1$'s that must be added before the process ends. For example:
$125\xrightarrow{\sc... | <p>Consider the following process that can be applied recursively to any positive integer $n$:</p>
<ul>
<li>if $n = 1$ do nothing and the process stops,</li>
<li>if $n$ is divisible by $7$ divide it by $7$,</li>
<li>otherwise add $1$.</li>
</ul>
<p>Define $g(n)$ to be the number of $1$'s that must be added before the p... | 91627537 | Sunday, 26th May 2019, 04:00 am | 292 | 50% | medium |
294 | Sum of Digits - Experience #23 | For a positive integer $k$, define $d(k)$ as the sum of the digits of $k$ in its usual decimal representation.
Thus $d(42) = 4+2 = 6$.
For a positive integer $n$, define $S(n)$ as the number of positive integers $k \lt 10^n$ with the following properties :
$k$ is divisible by $23$ and
$d(k) = 23$.
You are given that... | For a positive integer $k$, define $d(k)$ as the sum of the digits of $k$ in its usual decimal representation.
Thus $d(42) = 4+2 = 6$.
For a positive integer $n$, define $S(n)$ as the number of positive integers $k \lt 10^n$ with the following properties :
$k$ is divisible by $23$ and
$d(k) = 23$.
You are given that... | <p>
For a positive integer $k$, define $d(k)$ as the sum of the digits of $k$ in its usual decimal representation.
Thus $d(42) = 4+2 = 6$.
</p>
<p>
For a positive integer $n$, define $S(n)$ as the number of positive integers $k \lt 10^n$ with the following properties :
</p><ul><li>$k$ is divisible by $23$ and
</li><li>... | 789184709 | Saturday, 29th May 2010, 09:00 am | 1044 | 45% | medium |
375 | Minimum of Subsequences | Let $S_n$ be an integer sequence produced with the following pseudo-random number generator:
\begin{align}
S_0 & = 290797 \\
S_{n+1} & = S_n^2 \bmod 50515093
\end{align}
Let $A(i, j)$ be the minimum of the numbers $S_i, S_{i+1}, \dots, S_j$ for $i\le j$.
Let $M(N) = \sum A(i, j)$ for $1 \le i \le j \le N$.
We can ver... | Let $S_n$ be an integer sequence produced with the following pseudo-random number generator:
\begin{align}
S_0 & = 290797 \\
S_{n+1} & = S_n^2 \bmod 50515093
\end{align}
Let $A(i, j)$ be the minimum of the numbers $S_i, S_{i+1}, \dots, S_j$ for $i\le j$.
Let $M(N) = \sum A(i, j)$ for $1 \le i \le j \le N$.
We can ver... | <p>Let $S_n$ be an integer sequence produced with the following pseudo-random number generator:</p>
\begin{align}
S_0 & = 290797 \\
S_{n+1} & = S_n^2 \bmod 50515093
\end{align}
<p>
Let $A(i, j)$ be the minimum of the numbers $S_i, S_{i+1}, \dots, S_j$ for $i\le j$.<br/>
Let $M(N) = \sum A(i, j)$ for $1 \le i \le j \le... | 7435327983715286168 | Saturday, 10th March 2012, 10:00 pm | 979 | 40% | medium |
358 | Cyclic Numbers | A cyclic number with $n$ digits has a very interesting property:
When it is multiplied by $1, 2, 3, 4, \dots, n$, all the products have exactly the same digits, in the same order, but rotated in a circular fashion!
The smallest cyclic number is the $6$-digit number $142857$:
$142857 \times 1 = 142857$
$142857 \times ... | A cyclic number with $n$ digits has a very interesting property:
When it is multiplied by $1, 2, 3, 4, \dots, n$, all the products have exactly the same digits, in the same order, but rotated in a circular fashion!
The smallest cyclic number is the $6$-digit number $142857$:
$142857 \times 1 = 142857$
$142857 \times ... | <p>A <strong>cyclic number</strong> with $n$ digits has a very interesting property:<br/>
When it is multiplied by $1, 2, 3, 4, \dots, n$, all the products have exactly the same digits, in the same order, but rotated in a circular fashion!
</p>
<p>
The smallest cyclic number is the $6$-digit number $142857$:<br/>
$1428... | 3284144505 | Saturday, 12th November 2011, 07:00 pm | 1785 | 25% | easy |
27 | Quadratic Primes | Euler discovered the remarkable quadratic formula:
$n^2 + n + 41$
It turns out that the formula will produce $40$ primes for the consecutive integer values $0 \le n \le 39$. However, when $n = 40, 40^2 + 40 + 41 = 40(40 + 1) + 41$ is divisible by $41$, and certainly when $n = 41, 41^2 + 41 + 41$ is clearly divisible by... | Euler discovered the remarkable quadratic formula:
$n^2 + n + 41$
It turns out that the formula will produce $40$ primes for the consecutive integer values $0 \le n \le 39$. However, when $n = 40, 40^2 + 40 + 41 = 40(40 + 1) + 41$ is divisible by $41$, and certainly when $n = 41, 41^2 + 41 + 41$ is clearly divisible by... | <p>Euler discovered the remarkable quadratic formula:</p>
<p class="center">$n^2 + n + 41$</p>
<p>It turns out that the formula will produce $40$ primes for the consecutive integer values $0 \le n \le 39$. However, when $n = 40, 40^2 + 40 + 41 = 40(40 + 1) + 41$ is divisible by $41$, and certainly when $n = 41, 41^2 + ... | -59231 | Friday, 27th September 2002, 06:00 pm | 95475 | 5% | easy |
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