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 |
|---|---|---|---|---|---|---|---|---|---|
630 | Crossed Lines | Given a set, $L$, of unique lines, let $M(L)$ be the number of lines in the set and let $S(L)$ be the sum over every line of the number of times that line is crossed by another line in the set. For example, two sets of three lines are shown below:
In both cases $M(L)$ is $3$ and $S(L)$ is $6$: each of the three lin... | Given a set, $L$, of unique lines, let $M(L)$ be the number of lines in the set and let $S(L)$ be the sum over every line of the number of times that line is crossed by another line in the set. For example, two sets of three lines are shown below:
In both cases $M(L)$ is $3$ and $S(L)$ is $6$: each of the three lin... | <p>
Given a set, $L$, of unique lines, let $M(L)$ be the number of lines in the set and let $S(L)$ be the sum over every line of the number of times that line is crossed by another line in the set. For example, two sets of three lines are shown below:
</p>
<div align="center"><img alt="crossed lines" src="resources/im... | 9669182880384 | Sunday, 1st July 2018, 07:00 am | 1128 | 20% | easy |
403 | Lattice Points Enclosed by Parabola and Line | For integers $a$ and $b$, we define $D(a, b)$ as the domain enclosed by the parabola $y = x^2$ and the line $y = a\cdot x + b$:$D(a, b) = \{(x, y) \mid x^2 \leq y \leq a\cdot x + b \}$.
$L(a, b)$ is defined as the number of lattice points contained in $D(a, b)$.
For example, $L(1, 2) = 8$ and $L(2, -1) = 1$.
We als... | For integers $a$ and $b$, we define $D(a, b)$ as the domain enclosed by the parabola $y = x^2$ and the line $y = a\cdot x + b$:$D(a, b) = \{(x, y) \mid x^2 \leq y \leq a\cdot x + b \}$.
$L(a, b)$ is defined as the number of lattice points contained in $D(a, b)$.
For example, $L(1, 2) = 8$ and $L(2, -1) = 1$.
We als... | <p>
For integers $a$ and $b$, we define $D(a, b)$ as the domain enclosed by the parabola $y = x^2$ and the line $y = a\cdot x + b$:<br/>$D(a, b) = \{(x, y) \mid x^2 \leq y \leq a\cdot x + b \}$.
</p>
<p>
$L(a, b)$ is defined as the number of lattice points contained in $D(a, b)$.<br/>
For example, $L(1, 2) = 8$ and $L(... | 18224771 | Saturday, 24th November 2012, 10:00 pm | 398 | 55% | medium |
15 | Lattice Paths | Starting in the top left corner of a $2 \times 2$ grid, and only being able to move to the right and down, there are exactly $6$ routes to the bottom right corner.
How many such routes are there through a $20 \times 20$ grid? | Starting in the top left corner of a $2 \times 2$ grid, and only being able to move to the right and down, there are exactly $6$ routes to the bottom right corner.
How many such routes are there through a $20 \times 20$ grid? | <p>Starting in the top left corner of a $2 \times 2$ grid, and only being able to move to the right and down, there are exactly $6$ routes to the bottom right corner.</p>
<div class="center">
<img alt="" class="dark_img" src="resources/images/0015.png?1678992052"/></div>
<p>How many such routes are there through a $20 ... | 137846528820 | Friday, 19th April 2002, 06:00 pm | 202032 | 5% | easy |
559 | Permuted Matrices | An ascent of a column $j$ in a matrix occurs if the value of column $j$ is smaller than the value of column $j + 1$ in all rows.
Let $P(k, r, n)$ be the number of $r \times n$ matrices with the following properties:
The rows are permutations of $\{1, 2, 3, \dots, n\}$.
Numbering the first column as $1$, a column asce... | An ascent of a column $j$ in a matrix occurs if the value of column $j$ is smaller than the value of column $j + 1$ in all rows.
Let $P(k, r, n)$ be the number of $r \times n$ matrices with the following properties:
The rows are permutations of $\{1, 2, 3, \dots, n\}$.
Numbering the first column as $1$, a column asce... | <p>An <dfn>ascent</dfn> of a column $j$ in a matrix occurs if the value of column $j$ is smaller than the value of column $j + 1$ in all rows.
</p><p>
Let $P(k, r, n)$ be the number of $r \times n$ matrices with the following properties:</p>
<ul><li>The rows are permutations of $\{1, 2, 3, \dots, n\}$.</li>
<li> Number... | 684724920 | Saturday, 7th May 2016, 04:00 pm | 211 | 100% | hard |
427 | $n$-sequences | A sequence of integers $S = \{s_i\}$ is called an $n$-sequence if it has $n$ elements and each element $s_i$ satisfies $1 \leq s_i \leq n$. Thus there are $n^n$ distinct $n$-sequences in total.
For example, the sequence $S = \{1, 5, 5, 10, 7, 7, 7, 2, 3, 7\}$ is a $10$-sequence.
For any sequence $S$, let $L(S)$ be the ... | A sequence of integers $S = \{s_i\}$ is called an $n$-sequence if it has $n$ elements and each element $s_i$ satisfies $1 \leq s_i \leq n$. Thus there are $n^n$ distinct $n$-sequences in total.
For example, the sequence $S = \{1, 5, 5, 10, 7, 7, 7, 2, 3, 7\}$ is a $10$-sequence.
For any sequence $S$, let $L(S)$ be the ... | <p>A sequence of integers $S = \{s_i\}$ is called an $n$-sequence if it has $n$ elements and each element $s_i$ satisfies $1 \leq s_i \leq n$. Thus there are $n^n$ distinct $n$-sequences in total.
For example, the sequence $S = \{1, 5, 5, 10, 7, 7, 7, 2, 3, 7\}$ is a $10$-sequence.</p>
<p>For any sequence $S$, let $L(S... | 97138867 | Saturday, 11th May 2013, 10:00 pm | 352 | 95% | hard |
341 | Golomb's Self-describing Sequence | The Golomb's self-describing sequence $(G(n))$ is the only nondecreasing sequence of natural numbers such that $n$ appears exactly $G(n)$ times in the sequence. The values of $G(n)$ for the first few $n$ are
\[
\begin{matrix}
n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & \ldots \\
G(n) & 1 & 2 ... | The Golomb's self-describing sequence $(G(n))$ is the only nondecreasing sequence of natural numbers such that $n$ appears exactly $G(n)$ times in the sequence. The values of $G(n)$ for the first few $n$ are
\[
\begin{matrix}
n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & \ldots \\
G(n) & 1 & 2 ... | <p>The <b>Golomb's self-describing sequence</b> $(G(n))$ is the only nondecreasing sequence of natural numbers such that $n$ appears exactly $G(n)$ times in the sequence. The values of $G(n)$ for the first few $n$ are</p>
<p></p><div align="center">
\[
\begin{matrix}
n & 1 & 2 & 3 & 4 & 5 & 6 &a... | 56098610614277014 | Sunday, 5th June 2011, 10:00 am | 1041 | 45% | medium |
177 | Integer Angled Quadrilaterals | Let $ABCD$ be a convex quadrilateral, with diagonals $AC$ and $BD$. At each vertex the diagonal makes an angle with each of the two sides, creating eight corner angles.
For example, at vertex $A$, the two angles are $CAD$, $CAB$.
We call such a quadrilateral for which all eight corner angles have integer values when m... | Let $ABCD$ be a convex quadrilateral, with diagonals $AC$ and $BD$. At each vertex the diagonal makes an angle with each of the two sides, creating eight corner angles.
For example, at vertex $A$, the two angles are $CAD$, $CAB$.
We call such a quadrilateral for which all eight corner angles have integer values when m... | <p>Let $ABCD$ be a convex quadrilateral, with diagonals $AC$ and $BD$. At each vertex the diagonal makes an angle with each of the two sides, creating eight corner angles.</p>
<p style="text-align:center;"><img alt="" class="dark_img" src="resources/images/0177_quad.gif?1678992055"/></p>
<p>For example, at vertex $A$, ... | 129325 | Friday, 11th January 2008, 09:00 pm | 1463 | 80% | hard |
174 | Hollow Square Laminae II | We shall define a square lamina to be a square outline with a square "hole" so that the shape possesses vertical and horizontal symmetry.
Given eight tiles it is possible to form a lamina in only one way: $3 \times 3$ square with a $1 \times 1$ hole in the middle. However, using thirty-two tiles it is possible to form ... | We shall define a square lamina to be a square outline with a square "hole" so that the shape possesses vertical and horizontal symmetry.
Given eight tiles it is possible to form a lamina in only one way: $3 \times 3$ square with a $1 \times 1$ hole in the middle. However, using thirty-two tiles it is possible to form ... | <p>We shall define a square lamina to be a square outline with a square "hole" so that the shape possesses vertical and horizontal symmetry.</p>
<p>Given eight tiles it is possible to form a lamina in only one way: $3 \times 3$ square with a $1 \times 1$ hole in the middle. However, using thirty-two tiles it is possibl... | 209566 | Saturday, 22nd December 2007, 01:00 pm | 6507 | 40% | medium |
489 | Common Factors Between Two Sequences | Let $G(a, b)$ be the smallest non-negative integer $n$ for which $\operatorname{\mathbf{gcd}}$Greatest common divisor$(n^3 + b, (n + a)^3 + b)$ is maximized.
For example, $G(1, 1) = 5$ because $\gcd(n^3 + 1, (n + 1)^3 + 1)$ reaches its maximum value of $7$ for $n = 5$, and is smaller for $0 \le n \lt 5$.
Let $H(m, n) =... | Let $G(a, b)$ be the smallest non-negative integer $n$ for which $\operatorname{\mathbf{gcd}}$Greatest common divisor$(n^3 + b, (n + a)^3 + b)$ is maximized.
For example, $G(1, 1) = 5$ because $\gcd(n^3 + 1, (n + 1)^3 + 1)$ reaches its maximum value of $7$ for $n = 5$, and is smaller for $0 \le n \lt 5$.
Let $H(m, n) =... | <p>Let $G(a, b)$ be the smallest non-negative integer $n$ for which <strong class="tooltip">$\operatorname{\mathbf{gcd}}$<span class="tooltiptext">Greatest common divisor</span></strong>$(n^3 + b, (n + a)^3 + b)$ is maximized.<br/>
For example, $G(1, 1) = 5$ because $\gcd(n^3 + 1, (n + 1)^3 + 1)$ reaches its maximum va... | 1791954757162 | Sunday, 16th November 2014, 04:00 am | 281 | 100% | hard |
532 | Nanobots on Geodesics | Bob is a manufacturer of nanobots and wants to impress his customers by giving them a ball coloured by his new nanobots as a present.
His nanobots can be programmed to select and locate exactly one other bot precisely and, after activation, move towards this bot along the shortest possible path and draw a coloured line... | Bob is a manufacturer of nanobots and wants to impress his customers by giving them a ball coloured by his new nanobots as a present.
His nanobots can be programmed to select and locate exactly one other bot precisely and, after activation, move towards this bot along the shortest possible path and draw a coloured line... | <p>Bob is a manufacturer of nanobots and wants to impress his customers by giving them a ball coloured by his new nanobots as a present.</p>
<p>His nanobots can be programmed to select and locate exactly one other bot precisely and, after activation, move towards this bot along the shortest possible path and draw a col... | 827306.56 | Sunday, 1st November 2015, 07:00 am | 336 | 55% | medium |
418 | Factorisation Triples | Let $n$ be a positive integer. An integer triple $(a, b, c)$ is called a factorisation triple of $n$ if:$1 \leq a \leq b \leq c$$a \cdot b \cdot c = n$.
Define $f(n)$ to be $a + b + c$ for the factorisation triple $(a, b, c)$ of $n$ which minimises $c / a$. One can show that this triple is unique.
For example, $f(16... | Let $n$ be a positive integer. An integer triple $(a, b, c)$ is called a factorisation triple of $n$ if:$1 \leq a \leq b \leq c$$a \cdot b \cdot c = n$.
Define $f(n)$ to be $a + b + c$ for the factorisation triple $(a, b, c)$ of $n$ which minimises $c / a$. One can show that this triple is unique.
For example, $f(16... | <p>
Let $n$ be a positive integer. An integer triple $(a, b, c)$ is called a <dfn>factorisation triple</dfn> of $n$ if:</p><ul><li>$1 \leq a \leq b \leq c$</li><li>$a \cdot b \cdot c = n$.
</li></ul><p>
Define $f(n)$ to be $a + b + c$ for the factorisation triple $(a, b, c)$ of $n$ which minimises $c / a$. One can show... | 1177163565297340320 | Saturday, 9th March 2013, 07:00 pm | 758 | 40% | medium |
620 | Planetary Gears | A circle $C$ of circumference $c$ centimetres has a smaller circle $S$ of circumference $s$ centimetres lying off-centre within it. Four other distinct circles, which we call "planets", with circumferences $p$, $p$, $q$, $q$ centimetres respectively ($p<q$), are inscribed within $C$ but outside $S$, with each planet to... | A circle $C$ of circumference $c$ centimetres has a smaller circle $S$ of circumference $s$ centimetres lying off-centre within it. Four other distinct circles, which we call "planets", with circumferences $p$, $p$, $q$, $q$ centimetres respectively ($p<q$), are inscribed within $C$ but outside $S$, with each planet to... | <p>A circle $C$ of circumference $c$ centimetres has a smaller circle $S$ of circumference $s$ centimetres lying off-centre within it. Four other distinct circles, which we call "planets", with circumferences $p$, $p$, $q$, $q$ centimetres respectively ($p<q$), are inscribed within $C$ but outside $S$, with each pla... | 1470337306 | Sunday, 11th February 2018, 01:00 am | 185 | 80% | hard |
160 | Factorial Trailing Digits | For any $N$, let $f(N)$ be the last five digits before the trailing zeroes in $N!$.
For example,
$9! = 362880$ so $f(9)=36288$
$10! = 3628800$ so $f(10)=36288$
$20! = 2432902008176640000$ so $f(20)=17664$
Find $f(1\,000\,000\,000\,000)$. | For any $N$, let $f(N)$ be the last five digits before the trailing zeroes in $N!$.
For example,
$9! = 362880$ so $f(9)=36288$
$10! = 3628800$ so $f(10)=36288$
$20! = 2432902008176640000$ so $f(20)=17664$
Find $f(1\,000\,000\,000\,000)$. | <p>For any $N$, let $f(N)$ be the last five digits before the trailing zeroes in $N!$.<br/>
For example,</p>
<ul style="list-style-type:none;">
<li>$9! = 362880$ so $f(9)=36288$</li>
<li>$10! = 3628800$ so $f(10)=36288$</li>
<li>$20! = 2432902008176640000$ so $f(20)=17664$</li></ul>
<p>Find $f(1\,000\,000\,000\,000)$.<... | 16576 | Friday, 7th September 2007, 02:00 pm | 3948 | 60% | hard |
423 | Consecutive Die Throws | Let $n$ be a positive integer.
A 6-sided die is thrown $n$ times. Let $c$ be the number of pairs of consecutive throws that give the same value.
For example, if $n = 7$ and the values of the die throws are (1,1,5,6,6,6,3), then the following pairs of consecutive throws give the same value:
(1,1,5,6,6,6,3)
(1,1,5,6,6,6,... | Let $n$ be a positive integer.
A 6-sided die is thrown $n$ times. Let $c$ be the number of pairs of consecutive throws that give the same value.
For example, if $n = 7$ and the values of the die throws are (1,1,5,6,6,6,3), then the following pairs of consecutive throws give the same value:
(1,1,5,6,6,6,3)
(1,1,5,6,6,6,... | <p>Let $n$ be a positive integer.<br/>
A 6-sided die is thrown $n$ times. Let $c$ be the number of pairs of consecutive throws that give the same value.</p>
<p>For example, if $n = 7$ and the values of the die throws are (1,1,5,6,6,6,3), then the following pairs of consecutive throws give the same value:<br/>
(<u>1,1</... | 653972374 | Sunday, 14th April 2013, 10:00 am | 540 | 45% | medium |
9 | Special Pythagorean Triplet | A Pythagorean triplet is a set of three natural numbers, $a \lt b \lt c$, for which,
$$a^2 + b^2 = c^2.$$
For example, $3^2 + 4^2 = 9 + 16 = 25 = 5^2$.
There exists exactly one Pythagorean triplet for which $a + b + c = 1000$.Find the product $abc$. | A Pythagorean triplet is a set of three natural numbers, $a \lt b \lt c$, for which,
$$a^2 + b^2 = c^2.$$
For example, $3^2 + 4^2 = 9 + 16 = 25 = 5^2$.
There exists exactly one Pythagorean triplet for which $a + b + c = 1000$.Find the product $abc$. | <p>A Pythagorean triplet is a set of three natural numbers, $a \lt b \lt c$, for which,
$$a^2 + b^2 = c^2.$$</p>
<p>For example, $3^2 + 4^2 = 9 + 16 = 25 = 5^2$.</p>
<p>There exists exactly one Pythagorean triplet for which $a + b + c = 1000$.<br/>Find the product $abc$.</p> | 31875000 | Friday, 25th January 2002, 06:00 pm | 382130 | 5% | easy |
464 | Möbius Function and Intervals | The Möbius function, denoted $\mu(n)$, is defined as:
$\mu(n) = (-1)^{\omega(n)}$ if $n$ is squarefree (where $\omega(n)$ is the number of distinct prime factors of $n$)
$\mu(n) = 0$ if $n$ is not squarefree.
Let $P(a, b)$ be the number of integers $n$ in the interval $[a, b]$ such that $\mu(n) = 1$.
Let $N(a, b)$ be ... | The Möbius function, denoted $\mu(n)$, is defined as:
$\mu(n) = (-1)^{\omega(n)}$ if $n$ is squarefree (where $\omega(n)$ is the number of distinct prime factors of $n$)
$\mu(n) = 0$ if $n$ is not squarefree.
Let $P(a, b)$ be the number of integers $n$ in the interval $[a, b]$ such that $\mu(n) = 1$.
Let $N(a, b)$ be ... | <p>
The <strong>Möbius function</strong>, denoted $\mu(n)$, is defined as:
</p><ul><li>$\mu(n) = (-1)^{\omega(n)}$ if $n$ is squarefree (where $\omega(n)$ is the number of distinct prime factors of $n$)</li>
<li>$\mu(n) = 0$ if $n$ is not squarefree.</li>
</ul><p>
Let $P(a, b)$ be the number of integers $n$ in the inte... | 198775297232878 | Sunday, 23rd March 2014, 01:00 am | 361 | 60% | hard |
299 | Three Similar Triangles | Four points with integer coordinates are selected:$A(a, 0)$, $B(b, 0)$, $C(0, c)$ and $D(0, d)$, with $0 \lt a \lt b$ and $0 \lt c \lt d$.
Point $P$, also with integer coordinates, is chosen on the line $AC$ so that the three triangles $ABP$, $CDP$ and $BDP$ are all similarHave equal angles.
It is easy to prove that t... | Four points with integer coordinates are selected:$A(a, 0)$, $B(b, 0)$, $C(0, c)$ and $D(0, d)$, with $0 \lt a \lt b$ and $0 \lt c \lt d$.
Point $P$, also with integer coordinates, is chosen on the line $AC$ so that the three triangles $ABP$, $CDP$ and $BDP$ are all similarHave equal angles.
It is easy to prove that t... | <p>Four points with integer coordinates are selected:<br/>$A(a, 0)$, $B(b, 0)$, $C(0, c)$ and $D(0, d)$, with $0 \lt a \lt b$ and $0 \lt c \lt d$.<br/>
Point $P$, also with integer coordinates, is chosen on the line $AC$ so that the three triangles $ABP$, $CDP$ and $BDP$ are all <dfn class="tooltip">similar<span class=... | 549936643 | Saturday, 3rd July 2010, 01:00 am | 697 | 60% | hard |
199 | Iterative Circle Packing | Three circles of equal radius are placed inside a larger circle such that each pair of circles is tangent to one another and the inner circles do not overlap. There are four uncovered "gaps" which are to be filled iteratively with more tangent circles.
At each iteration, a maximally sized circle is placed in each ga... | Three circles of equal radius are placed inside a larger circle such that each pair of circles is tangent to one another and the inner circles do not overlap. There are four uncovered "gaps" which are to be filled iteratively with more tangent circles.
At each iteration, a maximally sized circle is placed in each ga... | <p>Three circles of equal radius are placed inside a larger circle such that each pair of circles is tangent to one another and the inner circles do not overlap. There are four uncovered "gaps" which are to be filled iteratively with more tangent circles.</p>
<div class="center">
<img alt="" class="dark_img" src="resou... | 0.00396087 | Saturday, 21st June 2008, 06:00 am | 2198 | 70% | hard |
61 | Cyclical Figurate Numbers | Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are generated by the following formulae:
Triangle
$P_{3,n}=n(n+1)/2$
$1, 3, 6, 10, 15, \dots$
Square
$P_{4,n}=n^2$
$1, 4, 9, 16, 25, \dots$
Pentagonal
$P_{5,n}=n(3n-1)/2$
$1, 5, 12, 22, 35, \d... | Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are generated by the following formulae:
Triangle
$P_{3,n}=n(n+1)/2$
$1, 3, 6, 10, 15, \dots$
Square
$P_{4,n}=n^2$
$1, 4, 9, 16, 25, \dots$
Pentagonal
$P_{5,n}=n(3n-1)/2$
$1, 5, 12, 22, 35, \d... | <p>Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are generated by the following formulae:</p>
<table><tr><td>Triangle</td>
<td> </td>
<td>$P_{3,n}=n(n+1)/2$</td>
<td> </td>
<td>$1, 3, 6, 10, 15, \dots$</td>
</tr><tr><td>Square</td>
<td> </td>
<td>$P_... | 28684 | Friday, 16th January 2004, 06:00 pm | 28541 | 20% | easy |
290 | Digital Signature | How many integers $0 \le n \lt 10^{18}$ have the property that the sum of the digits of $n$ equals the sum of digits of $137n$? | How many integers $0 \le n \lt 10^{18}$ have the property that the sum of the digits of $n$ equals the sum of digits of $137n$? | <p>
How many integers $0 \le n \lt 10^{18}$ have the property that the sum of the digits of $n$ equals the sum of digits of $137n$?
</p> | 20444710234716473 | Friday, 30th April 2010, 05:00 pm | 1098 | 50% | medium |
381 | $(\text{prime}-k)$ Factorial | For a prime $p$ let $S(p) = (\sum (p-k)!) \bmod (p)$ for $1 \le k \le 5$.
For example, if $p=7$,
$(7-1)! + (7-2)! + (7-3)! + (7-4)! + (7-5)! = 6! + 5! + 4! + 3! + 2! = 720+120+24+6+2 = 872$.
As $872 \bmod (7) = 4$, $S(7) = 4$.
It can be verified that $\sum S(p) = 480$ for $5 \le p \lt 100$.
Find $\sum S(p)$ for ... | For a prime $p$ let $S(p) = (\sum (p-k)!) \bmod (p)$ for $1 \le k \le 5$.
For example, if $p=7$,
$(7-1)! + (7-2)! + (7-3)! + (7-4)! + (7-5)! = 6! + 5! + 4! + 3! + 2! = 720+120+24+6+2 = 872$.
As $872 \bmod (7) = 4$, $S(7) = 4$.
It can be verified that $\sum S(p) = 480$ for $5 \le p \lt 100$.
Find $\sum S(p)$ for ... | <p>
For a prime $p$ let $S(p) = (\sum (p-k)!) \bmod (p)$ for $1 \le k \le 5$.
</p>
<p>
For example, if $p=7$,<br/>
$(7-1)! + (7-2)! + (7-3)! + (7-4)! + (7-5)! = 6! + 5! + 4! + 3! + 2! = 720+120+24+6+2 = 872$.<br/>
As $872 \bmod (7) = 4$, $S(7) = 4$.
</p>
<p>
It can be verified that $\sum S(p) = 480$ for $5 \le p \lt 1... | 139602943319822 | Saturday, 21st April 2012, 05:00 pm | 4767 | 15% | easy |
331 | Cross Flips | $N \times N$ disks are placed on a square game board. Each disk has a black side and white side.
At each turn, you may choose a disk and flip all the disks in the same row and the same column as this disk: thus $2 \times N - 1$ disks are flipped. The game ends when all disks show their white side. The following example... | $N \times N$ disks are placed on a square game board. Each disk has a black side and white side.
At each turn, you may choose a disk and flip all the disks in the same row and the same column as this disk: thus $2 \times N - 1$ disks are flipped. The game ends when all disks show their white side. The following example... | <p>$N \times N$ disks are placed on a square game board. Each disk has a black side and white side.</p>
<p>At each turn, you may choose a disk and flip all the disks in the same row and the same column as this disk: thus $2 \times N - 1$ disks are flipped. The game ends when all disks show their white side. The followi... | 467178235146843549 | Sunday, 3rd April 2011, 08:00 am | 471 | 100% | hard |
581 | $47$-smooth Triangular Numbers | A number is $p$-smooth if it has no prime factors larger than $p$.
Let $T$ be the sequence of triangular numbers, i.e. $T(n)=n(n+1)/2$.
Find the sum of all indices $n$ such that $T(n)$ is $47$-smooth. | A number is $p$-smooth if it has no prime factors larger than $p$.
Let $T$ be the sequence of triangular numbers, i.e. $T(n)=n(n+1)/2$.
Find the sum of all indices $n$ such that $T(n)$ is $47$-smooth. | <p>
A number is $p$-smooth if it has no prime factors larger than $p$.<br/>
Let $T$ be the sequence of triangular numbers, i.e. $T(n)=n(n+1)/2$.<br/>
Find the sum of all indices $n$ such that $T(n)$ is $47$-smooth.
</p> | 2227616372734 | Sunday, 11th December 2016, 07:00 am | 974 | 30% | easy |
710 | One Million Members | On Sunday 5 April 2020 the Project Euler membership first exceeded one million members. We would like to present this problem to celebrate that milestone. Thank you to everyone for being a part of Project Euler.
The number 6 can be written as a palindromic sum in exactly eight different ways:
$$(1, 1, 1, 1, 1, 1), (1, ... | On Sunday 5 April 2020 the Project Euler membership first exceeded one million members. We would like to present this problem to celebrate that milestone. Thank you to everyone for being a part of Project Euler.
The number 6 can be written as a palindromic sum in exactly eight different ways:
$$(1, 1, 1, 1, 1, 1), (1, ... | <h4>On Sunday 5 April 2020 the Project Euler membership first exceeded one million members. We would like to present this problem to celebrate that milestone. Thank you to everyone for being a part of Project Euler.</h4>
<p>The number 6 can be written as a palindromic sum in exactly eight different ways:</p>
$$(1, 1, 1... | 1275000 | Saturday, 11th April 2020, 08:00 pm | 1426 | 10% | easy |
510 | Tangent Circles | Circles $A$ and $B$ are tangent to each other and to line $L$ at three distinct points.
Circle $C$ is inside the space between $A$, $B$ and $L$, and tangent to all three.
Let $r_A$, $r_B$ and $r_C$ be the radii of $A$, $B$ and $C$ respectively.
Let $S(n) = \sum r_A + r_B + r_C$, for $0 \lt r_A \le r_B \le n$ where $r_A... | Circles $A$ and $B$ are tangent to each other and to line $L$ at three distinct points.
Circle $C$ is inside the space between $A$, $B$ and $L$, and tangent to all three.
Let $r_A$, $r_B$ and $r_C$ be the radii of $A$, $B$ and $C$ respectively.
Let $S(n) = \sum r_A + r_B + r_C$, for $0 \lt r_A \le r_B \le n$ where $r_A... | <p>Circles $A$ and $B$ are tangent to each other and to line $L$ at three distinct points.<br/>
Circle $C$ is inside the space between $A$, $B$ and $L$, and tangent to all three.<br/>
Let $r_A$, $r_B$ and $r_C$ be the radii of $A$, $B$ and $C$ respectively.<br/></p><div align="center"><img alt="0510_tangent_circles.png... | 315306518862563689 | Saturday, 4th April 2015, 04:00 pm | 1128 | 30% | easy |
238 | Infinite String Tour | Create a sequence of numbers using the "Blum Blum Shub" pseudo-random number generator:
\begin{align}
s_0 &= 14025256\\
s_{n + 1} &= s_n^2 \bmod 20300713
\end{align}
Concatenate these numbers $s_0s_1s_2\cdots$ to create a string $w$ of infinite length.
Then, $w = {\color{blue}14025256741014958470038053646\cdots}$
For... | Create a sequence of numbers using the "Blum Blum Shub" pseudo-random number generator:
\begin{align}
s_0 &= 14025256\\
s_{n + 1} &= s_n^2 \bmod 20300713
\end{align}
Concatenate these numbers $s_0s_1s_2\cdots$ to create a string $w$ of infinite length.
Then, $w = {\color{blue}14025256741014958470038053646\cdots}$
For... | <p>Create a sequence of numbers using the "Blum Blum Shub" pseudo-random number generator:</p>
\begin{align}
s_0 &= 14025256\\
s_{n + 1} &= s_n^2 \bmod 20300713
\end{align}
<p>Concatenate these numbers $s_0s_1s_2\cdots$ to create a string $w$ of infinite length.<br/>
Then, $w = {\color{blue}14025256741014958470038053... | 9922545104535661 | Sunday, 29th March 2009, 03:00 pm | 1085 | 75% | hard |
705 | Total Inversion Count of Divided Sequences | The inversion count of a sequence of digits is the smallest number of adjacent pairs that must be swapped to sort the sequence.
For example, $34214$ has inversion count of $5$:
$34214 \to 32414 \to 23414 \to 23144 \to 21344 \to12344$.
If each digit of a sequence is replaced by one of its divisors a divided sequence i... | The inversion count of a sequence of digits is the smallest number of adjacent pairs that must be swapped to sort the sequence.
For example, $34214$ has inversion count of $5$:
$34214 \to 32414 \to 23414 \to 23144 \to 21344 \to12344$.
If each digit of a sequence is replaced by one of its divisors a divided sequence i... | <p>
The <dfn>inversion count</dfn> of a sequence of digits is the smallest number of adjacent pairs that must be swapped to sort the sequence.<br/>
For example, $34214$ has inversion count of $5$:
$34214 \to 32414 \to 23414 \to 23144 \to 21344 \to12344$.
</p>
<p>
If each digit of a sequence is replaced by one of its di... | 480440153 | Sunday, 8th March 2020, 04:00 am | 490 | 25% | easy |
819 | Iterative Sampling | Given an $n$-tuple of numbers another $n$-tuple is created where each element of the new $n$-tuple is chosen randomly from the numbers in the previous $n$-tuple. For example, given $(2,2,3)$ the probability that $2$ occurs in the first position in the next 3-tuple is $2/3$. The probability of getting all $2$'s would be... | Given an $n$-tuple of numbers another $n$-tuple is created where each element of the new $n$-tuple is chosen randomly from the numbers in the previous $n$-tuple. For example, given $(2,2,3)$ the probability that $2$ occurs in the first position in the next 3-tuple is $2/3$. The probability of getting all $2$'s would be... | <p>Given an $n$-tuple of numbers another $n$-tuple is created where each element of the new $n$-tuple is chosen randomly from the numbers in the previous $n$-tuple. For example, given $(2,2,3)$ the probability that $2$ occurs in the first position in the next 3-tuple is $2/3$. The probability of getting all $2$'s would... | 1995.975556 | Saturday, 3rd December 2022, 01:00 pm | 232 | 45% | medium |
681 | Maximal Area | Given positive integers $a \le b \le c \le d$, it may be possible to form quadrilaterals with edge lengths $a,b,c,d$ (in any order). When this is the case, let $M(a,b,c,d)$ denote the maximal area of such a quadrilateral. For example, $M(2,2,3,3)=6$, attained e.g. by a $2\times 3$ rectangle.
Let $SP(n)$ be the sum of... | Given positive integers $a \le b \le c \le d$, it may be possible to form quadrilaterals with edge lengths $a,b,c,d$ (in any order). When this is the case, let $M(a,b,c,d)$ denote the maximal area of such a quadrilateral. For example, $M(2,2,3,3)=6$, attained e.g. by a $2\times 3$ rectangle.
Let $SP(n)$ be the sum of... | <p>
Given positive integers $a \le b \le c \le d$, it may be possible to form quadrilaterals with edge lengths $a,b,c,d$ (in any order). When this is the case, let $M(a,b,c,d)$ denote the maximal area of such a quadrilateral.<br> For example, $M(2,2,3,3)=6$, attained e.g. by a $2\times 3$ rectangle.
</br></p>
<p>
Let $... | 2611227421428 | Sunday, 29th September 2019, 07:00 am | 250 | 50% | medium |
741 | Binary Grid Colouring | Let $f(n)$ be the number of ways an $n\times n$ square grid can be coloured, each cell either black or white, such that each row and each column contains exactly two black cells.
For example, $f(4)=90$, $f(7) = 3110940$ and $f(8) = 187530840$.
Let $g(n)$ be the number of colourings in $f(n)$ that are unique up to rot... | Let $f(n)$ be the number of ways an $n\times n$ square grid can be coloured, each cell either black or white, such that each row and each column contains exactly two black cells.
For example, $f(4)=90$, $f(7) = 3110940$ and $f(8) = 187530840$.
Let $g(n)$ be the number of colourings in $f(n)$ that are unique up to rot... | <p>
Let $f(n)$ be the number of ways an $n\times n$ square grid can be coloured, each cell either black or white, such that each row and each column contains exactly two black cells.<br/>
For example, $f(4)=90$, $f(7) = 3110940$ and $f(8) = 187530840$.
</p>
<p>
Let $g(n)$ be the number of colourings in $f(n)$ that are ... | 512895223 | Saturday, 2nd January 2021, 04:00 pm | 170 | 80% | hard |
21 | Amicable Numbers | Let $d(n)$ be defined as the sum of proper divisors of $n$ (numbers less than $n$ which divide evenly into $n$).
If $d(a) = b$ and $d(b) = a$, where $a \ne b$, then $a$ and $b$ are an amicable pair and each of $a$ and $b$ are called amicable numbers.
For example, the proper divisors of $220$ are $1, 2, 4, 5, 10, 11, 20... | Let $d(n)$ be defined as the sum of proper divisors of $n$ (numbers less than $n$ which divide evenly into $n$).
If $d(a) = b$ and $d(b) = a$, where $a \ne b$, then $a$ and $b$ are an amicable pair and each of $a$ and $b$ are called amicable numbers.
For example, the proper divisors of $220$ are $1, 2, 4, 5, 10, 11, 20... | <p>Let $d(n)$ be defined as the sum of proper divisors of $n$ (numbers less than $n$ which divide evenly into $n$).<br/>
If $d(a) = b$ and $d(b) = a$, where $a \ne b$, then $a$ and $b$ are an amicable pair and each of $a$ and $b$ are called amicable numbers.</p>
<p>For example, the proper divisors of $220$ are $1, 2, 4... | 31626 | Friday, 5th July 2002, 06:00 pm | 158401 | 5% | easy |
834 | Add and Divide | A sequence is created by starting with a positive integer $n$ and incrementing by $(n+m)$ at the $m^{th}$ step.
If $n=10$, the resulting sequence will be $21,33,46,60,75,91,108,126,\ldots$.
Let $S(n)$ be the set of indices $m$, for which the $m^{th}$ term in the sequence is divisible by $(n+m)$.
For example, $S(10)=... | A sequence is created by starting with a positive integer $n$ and incrementing by $(n+m)$ at the $m^{th}$ step.
If $n=10$, the resulting sequence will be $21,33,46,60,75,91,108,126,\ldots$.
Let $S(n)$ be the set of indices $m$, for which the $m^{th}$ term in the sequence is divisible by $(n+m)$.
For example, $S(10)=... | <p>
A sequence is created by starting with a positive integer $n$ and incrementing by $(n+m)$ at the $m^{th}$ step.
If $n=10$, the resulting sequence will be $21,33,46,60,75,91,108,126,\ldots$.</p>
<p>
Let $S(n)$ be the set of indices $m$, for which the $m^{th}$ term in the sequence is divisible by $(n+m)$.<br/>
For ... | 1254404167198752370 | Sunday, 19th March 2023, 10:00 am | 439 | 25% | easy |
272 | Modular Cubes, Part 2 | For a positive number $n$, define $C(n)$ as the number of the integers $x$, for which $1 \lt x \lt n$ and$x^3 \equiv 1 \bmod n$.
When $n=91$, there are $8$ possible values for $x$, namely: $9, 16, 22, 29, 53, 74, 79, 81$.
Thus, $C(91)=8$.
Find the sum of the positive numbers $n \le 10^{11}$ for which $C(n)=242$. | For a positive number $n$, define $C(n)$ as the number of the integers $x$, for which $1 \lt x \lt n$ and$x^3 \equiv 1 \bmod n$.
When $n=91$, there are $8$ possible values for $x$, namely: $9, 16, 22, 29, 53, 74, 79, 81$.
Thus, $C(91)=8$.
Find the sum of the positive numbers $n \le 10^{11}$ for which $C(n)=242$. | <p>
For a positive number $n$, define $C(n)$ as the number of the integers $x$, for which $1 \lt x \lt n$ and<br/>$x^3 \equiv 1 \bmod n$.
</p>
<p>
When $n=91$, there are $8$ possible values for $x$, namely: $9, 16, 22, 29, 53, 74, 79, 81$.<br/>
Thus, $C(91)=8$.</p>
<p>
Find the sum of the positive numbers $n \le 10^{11... | 8495585919506151122 | Saturday, 2nd January 2010, 05:00 am | 1102 | 80% | hard |
463 | A Weird Recurrence Relation | The function $f$ is defined for all positive integers as follows:
$f(1)=1$
$f(3)=3$
$f(2n)=f(n)$
$f(4n + 1)=2f(2n + 1) - f(n)$
$f(4n + 3)=3f(2n + 1) - 2f(n)$
The function $S(n)$ is defined as $\sum_{i=1}^{n}f(i)$.
$S(8)=22$ and $S(100)=3604$.
Find $S(3^{37})$. Give the last $9$ digits of your answer. | The function $f$ is defined for all positive integers as follows:
$f(1)=1$
$f(3)=3$
$f(2n)=f(n)$
$f(4n + 1)=2f(2n + 1) - f(n)$
$f(4n + 3)=3f(2n + 1) - 2f(n)$
The function $S(n)$ is defined as $\sum_{i=1}^{n}f(i)$.
$S(8)=22$ and $S(100)=3604$.
Find $S(3^{37})$. Give the last $9$ digits of your answer. | <p>
The function $f$ is defined for all positive integers as follows:
</p><ul><li>$f(1)=1$
</li><li>$f(3)=3$
</li><li>$f(2n)=f(n)$
</li><li>$f(4n + 1)=2f(2n + 1) - f(n)$
</li><li>$f(4n + 3)=3f(2n + 1) - 2f(n)$
</li>
</ul><p>The function $S(n)$ is defined as $\sum_{i=1}^{n}f(i)$.</p>
<p>$S(8)=22$ and $S(100)=3604$.</p>
... | 808981553 | Saturday, 15th March 2014, 10:00 pm | 1239 | 35% | medium |
446 | Retractions B | For every integer $n>1$, the family of functions $f_{n,a,b}$ is defined
by
$f_{n,a,b}(x)\equiv a x + b \mod n\,\,\, $ for $a,b,x$ integer and $0< a <n, 0 \le b < n,0 \le x < n$.
We will call $f_{n,a,b}$ a retraction if $\,\,\, f_{n,a,b}(f_{n,a,b}(x)) \equiv f_{n,a,b}(x) \mod n \,\,\,$ for every $0 \le x < n$.
Let... | For every integer $n>1$, the family of functions $f_{n,a,b}$ is defined
by
$f_{n,a,b}(x)\equiv a x + b \mod n\,\,\, $ for $a,b,x$ integer and $0< a <n, 0 \le b < n,0 \le x < n$.
We will call $f_{n,a,b}$ a retraction if $\,\,\, f_{n,a,b}(f_{n,a,b}(x)) \equiv f_{n,a,b}(x) \mod n \,\,\,$ for every $0 \le x < n$.
Let... | <p>
For every integer $n>1$, the family of functions $f_{n,a,b}$ is defined
by <br>
$f_{n,a,b}(x)\equiv a x + b \mod n\,\,\, $ for $a,b,x$ integer and $0< a <n, 0 \le b < n,0 \le x < n$. </br></p>
<p>
We will call $f_{n,a,b}$ a <i>retraction</i> if $\,\,\, f_{n,a,b}(f_{n,a,b}(x)) \equiv f_{n,a,b}(x) \... | 907803852 | Saturday, 16th November 2013, 10:00 pm | 423 | 60% | hard |
711 | Binary Blackboard | Oscar and Eric play the following game. First, they agree on a positive integer $n$, and they begin by writing its binary representation on a blackboard. They then take turns, with Oscar going first, to write a number on the blackboard in binary representation, such that the sum of all written numbers does not exceed $... | Oscar and Eric play the following game. First, they agree on a positive integer $n$, and they begin by writing its binary representation on a blackboard. They then take turns, with Oscar going first, to write a number on the blackboard in binary representation, such that the sum of all written numbers does not exceed $... | <p>Oscar and Eric play the following game. First, they agree on a positive integer $n$, and they begin by writing its binary representation on a blackboard. They then take turns, with Oscar going first, to write a number on the blackboard in binary representation, such that the sum of all written numbers does not excee... | 541510990 | Saturday, 11th April 2020, 08:00 pm | 363 | 35% | medium |
916 | Restricted Permutations | Let $P(n)$ be the number of permutations of $\{1,2,3,\ldots,2n\}$ such that:
1. There is no ascending subsequence with more than $n+1$ elements, and
2. There is no descending subsequence with more than two elements.
Note that subsequences need not be contiguous. For example, the permutation $(4,1,3,2)$ is not counte... | Let $P(n)$ be the number of permutations of $\{1,2,3,\ldots,2n\}$ such that:
1. There is no ascending subsequence with more than $n+1$ elements, and
2. There is no descending subsequence with more than two elements.
Note that subsequences need not be contiguous. For example, the permutation $(4,1,3,2)$ is not counte... | <p>Let $P(n)$ be the number of permutations of $\{1,2,3,\ldots,2n\}$ such that:
<br/>
1. There is no ascending subsequence with more than $n+1$ elements, and
<br/>
2. There is no descending subsequence with more than two elements.
</p>
<p>Note that subsequences need not be contiguous. For example, the permutation $(4,1... | 877789135 | Saturday, 9th November 2024, 10:00 pm | 143 | 55% | medium |
184 | Triangles Containing the Origin | Consider the set $I_r$ of points $(x,y)$ with integer co-ordinates in the interior of the circle with radius $r$, centered at the origin, i.e. $x^2 + y^2 \lt r^2$.
For a radius of $2$, $I_2$ contains the nine points $(0,0)$, $(1,0)$, $(1,1)$, $(0,1)$, $(-1,1)$, $(-1,0)$, $(-1,-1)$, $(0,-1)$ and $(1,-1)$. There are eigh... | Consider the set $I_r$ of points $(x,y)$ with integer co-ordinates in the interior of the circle with radius $r$, centered at the origin, i.e. $x^2 + y^2 \lt r^2$.
For a radius of $2$, $I_2$ contains the nine points $(0,0)$, $(1,0)$, $(1,1)$, $(0,1)$, $(-1,1)$, $(-1,0)$, $(-1,-1)$, $(0,-1)$ and $(1,-1)$. There are eigh... | <p>Consider the set $I_r$ of points $(x,y)$ with integer co-ordinates in the interior of the circle with radius $r$, centered at the origin, i.e. $x^2 + y^2 \lt r^2$.</p>
<p>For a radius of $2$, $I_2$ contains the nine points $(0,0)$, $(1,0)$, $(1,1)$, $(0,1)$, $(-1,1)$, $(-1,0)$, $(-1,-1)$, $(0,-1)$ and $(1,-1)$. Ther... | 1725323624056 | Friday, 29th February 2008, 09:00 pm | 1870 | 75% | hard |
141 | Square Progressive Numbers | A positive integer, $n$, is divided by $d$ and the quotient and remainder are $q$ and $r$ respectively. In addition $d$, $q$, and $r$ are consecutive positive integer terms in a geometric sequence, but not necessarily in that order.
For example, $58$ divided by $6$ has quotient $9$ and remainder $4$. It can also be see... | A positive integer, $n$, is divided by $d$ and the quotient and remainder are $q$ and $r$ respectively. In addition $d$, $q$, and $r$ are consecutive positive integer terms in a geometric sequence, but not necessarily in that order.
For example, $58$ divided by $6$ has quotient $9$ and remainder $4$. It can also be see... | <p>A positive integer, $n$, is divided by $d$ and the quotient and remainder are $q$ and $r$ respectively. In addition $d$, $q$, and $r$ are consecutive positive integer terms in a geometric sequence, but not necessarily in that order.</p>
<p>For example, $58$ divided by $6$ has quotient $9$ and remainder $4$. It can a... | 878454337159 | Saturday, 17th February 2007, 05:00 am | 4486 | 60% | hard |
203 | Squarefree Binomial Coefficients | The binomial coefficients $\displaystyle \binom n k$ can be arranged in triangular form, Pascal's triangle, like this:
111121133114641151010511615201561172135352171
.........
It can be seen that the first eight rows of Pascal's triangle contain twelve distinct numbers: 1, 2, 3, 4, 5, 6, 7, 10, 15, 20, 21 and 35.
A po... | The binomial coefficients $\displaystyle \binom n k$ can be arranged in triangular form, Pascal's triangle, like this:
111121133114641151010511615201561172135352171
.........
It can be seen that the first eight rows of Pascal's triangle contain twelve distinct numbers: 1, 2, 3, 4, 5, 6, 7, 10, 15, 20, 21 and 35.
A po... | <p>The binomial coefficients $\displaystyle \binom n k$ can be arranged in triangular form, Pascal's triangle, like this:</p>
<div class="center">
<table align="center"><tr><td colspan="7"></td><td>1</td><td colspan="7"></td></tr><tr><td colspan="6"></td><td>1</td><td></td><td>1</td><td colspan="6"></td></tr><tr><td co... | 34029210557338 | Saturday, 6th September 2008, 02:00 pm | 9928 | 25% | easy |
590 | Sets with a Given Least Common Multiple | Let $H(n)$ denote the number of sets of positive integers such that the least common multiple of the integers in the set equals $n$.
E.g.:
The integers in the following ten sets all have a least common multiple of $6$:
$\{2,3\}$, $\{1,2,3\}$, $\{6\}$, $\{1,6\}$, $\{2,6\}$, $\{1,2,6\}$, $\{3,6\}$, $\{1,3,6\}$, $\{2,3,6\... | Let $H(n)$ denote the number of sets of positive integers such that the least common multiple of the integers in the set equals $n$.
E.g.:
The integers in the following ten sets all have a least common multiple of $6$:
$\{2,3\}$, $\{1,2,3\}$, $\{6\}$, $\{1,6\}$, $\{2,6\}$, $\{1,2,6\}$, $\{3,6\}$, $\{1,3,6\}$, $\{2,3,6\... | <p>
Let $H(n)$ denote the number of sets of positive integers such that the <strong>least common multiple</strong> of the integers in the set equals $n$.<br/>
E.g.:<br/>
The integers in the following ten sets all have a least common multiple of $6$:<br/>
$\{2,3\}$, $\{1,2,3\}$, $\{6\}$, $\{1,6\}$, $\{2,6\}$, $\{1,2,6\}... | 834171904 | Sunday, 12th February 2017, 10:00 am | 290 | 60% | hard |
858 | LCM | Define $G(N) = \sum_S \operatorname{lcm}(S)$ where $S$ ranges through all subsets of $\{1, \dots, N\}$ and $\operatorname{lcm}$ denotes the lowest common multiple. Note that the $\operatorname{lcm}$ of the empty set is $1$.
You are given $G(5) = 528$ and $G(20) = 8463108648960$.
Find $G(800)$. Give your answer modulo... | Define $G(N) = \sum_S \operatorname{lcm}(S)$ where $S$ ranges through all subsets of $\{1, \dots, N\}$ and $\operatorname{lcm}$ denotes the lowest common multiple. Note that the $\operatorname{lcm}$ of the empty set is $1$.
You are given $G(5) = 528$ and $G(20) = 8463108648960$.
Find $G(800)$. Give your answer modulo... | <p>
Define $G(N) = \sum_S \operatorname{lcm}(S)$ where $S$ ranges through all subsets of $\{1, \dots, N\}$ and $\operatorname{lcm}$ denotes the lowest common multiple. Note that the $\operatorname{lcm}$ of the empty set is $1$.</p>
<p>
You are given $G(5) = 528$ and $G(20) = 8463108648960$.</p>
<p>
Find $G(800)$. Give ... | 973077199 | Sunday, 8th October 2023, 05:00 am | 237 | 50% | medium |
125 | Palindromic Sums | The palindromic number $595$ is interesting because it can be written as the sum of consecutive squares: $6^2 + 7^2 + 8^2 + 9^2 + 10^2 + 11^2 + 12^2$.
There are exactly eleven palindromes below one-thousand that can be written as consecutive square sums, and the sum of these palindromes is $4164$. Note that $1 = 0^2 + ... | The palindromic number $595$ is interesting because it can be written as the sum of consecutive squares: $6^2 + 7^2 + 8^2 + 9^2 + 10^2 + 11^2 + 12^2$.
There are exactly eleven palindromes below one-thousand that can be written as consecutive square sums, and the sum of these palindromes is $4164$. Note that $1 = 0^2 + ... | <p>The palindromic number $595$ is interesting because it can be written as the sum of consecutive squares: $6^2 + 7^2 + 8^2 + 9^2 + 10^2 + 11^2 + 12^2$.</p>
<p>There are exactly eleven palindromes below one-thousand that can be written as consecutive square sums, and the sum of these palindromes is $4164$. Note that $... | 2906969179 | Friday, 4th August 2006, 06:00 pm | 14974 | 25% | easy |
878 | XOR-Equation B | We use $x\oplus y$ for the bitwise XOR of $x$ and $y$.
Define the XOR-product of $x$ and $y$, denoted by $x \otimes y$, similar to a long multiplication in base $2$, except that the intermediate results are XORed instead of the usual integer addition.
For example, $7 \otimes 3 = 9$, or in base $2$, $111_2 \otimes 11_2... | We use $x\oplus y$ for the bitwise XOR of $x$ and $y$.
Define the XOR-product of $x$ and $y$, denoted by $x \otimes y$, similar to a long multiplication in base $2$, except that the intermediate results are XORed instead of the usual integer addition.
For example, $7 \otimes 3 = 9$, or in base $2$, $111_2 \otimes 11_2... | We use $x\oplus y$ for the bitwise XOR of $x$ and $y$.<br/>
Define the <dfn>XOR-product</dfn> of $x$ and $y$, denoted by $x \otimes y$, similar to a long multiplication in base $2$, except that the intermediate results are XORed instead of the usual integer addition.<br/>
For example, $7 \otimes 3 = 9$, or in base $2$... | 23707109 | Saturday, 17th February 2024, 01:00 pm | 258 | 40% | medium |
60 | Prime Pair Sets | The primes $3$, $7$, $109$, and $673$, are quite remarkable. By taking any two primes and concatenating them in any order the result will always be prime. For example, taking $7$ and $109$, both $7109$ and $1097$ are prime. The sum of these four primes, $792$, represents the lowest sum for a set of four primes with thi... | The primes $3$, $7$, $109$, and $673$, are quite remarkable. By taking any two primes and concatenating them in any order the result will always be prime. For example, taking $7$ and $109$, both $7109$ and $1097$ are prime. The sum of these four primes, $792$, represents the lowest sum for a set of four primes with thi... | <p>The primes $3$, $7$, $109$, and $673$, are quite remarkable. By taking any two primes and concatenating them in any order the result will always be prime. For example, taking $7$ and $109$, both $7109$ and $1097$ are prime. The sum of these four primes, $792$, represents the lowest sum for a set of four primes with ... | 26033 | Friday, 2nd January 2004, 06:00 pm | 30207 | 20% | easy |
531 | Chinese Leftovers | Let $g(a, n, b, m)$ be the smallest non-negative solution $x$ to the system:
$x = a \bmod n$
$x = b \bmod m$
if such a solution exists, otherwise $0$.
E.g. $g(2,4,4,6)=10$, but $g(3,4,4,6)=0$.
Let $\phi(n)$ be Euler's totient function.
Let $f(n,m)=g(\phi(n),n,\phi(m),m)$
Find $\sum f(n,m)$ for $1000000 \le n \l... | Let $g(a, n, b, m)$ be the smallest non-negative solution $x$ to the system:
$x = a \bmod n$
$x = b \bmod m$
if such a solution exists, otherwise $0$.
E.g. $g(2,4,4,6)=10$, but $g(3,4,4,6)=0$.
Let $\phi(n)$ be Euler's totient function.
Let $f(n,m)=g(\phi(n),n,\phi(m),m)$
Find $\sum f(n,m)$ for $1000000 \le n \l... | <p>
Let $g(a, n, b, m)$ be the smallest non-negative solution $x$ to the system:<br/>
$x = a \bmod n$<br/>
$x = b \bmod m$<br/>
if such a solution exists, otherwise $0$.
</p>
<p>
E.g. $g(2,4,4,6)=10$, but $g(3,4,4,6)=0$.
</p>
<p>
Let $\phi(n)$ be Euler's totient function.
</p>
<p>
Let $f(n,m)=g(\phi(n),n,\phi(m),m)$
</... | 4515432351156203105 | Sunday, 25th October 2015, 04:00 am | 1290 | 25% | easy |
613 | Pythagorean Ant | Dave is doing his homework on the balcony and, preparing a presentation about Pythagorean triangles, has just cut out a triangle with side lengths 30cm, 40cm and 50cm from some cardboard, when a gust of wind blows the triangle down into the garden.
Another gust blows a small ant straight onto this triangle. The poor an... | Dave is doing his homework on the balcony and, preparing a presentation about Pythagorean triangles, has just cut out a triangle with side lengths 30cm, 40cm and 50cm from some cardboard, when a gust of wind blows the triangle down into the garden.
Another gust blows a small ant straight onto this triangle. The poor an... | <p>Dave is doing his homework on the balcony and, preparing a presentation about Pythagorean triangles, has just cut out a triangle with side lengths 30cm, 40cm and 50cm from some cardboard, when a gust of wind blows the triangle down into the garden.<br>
Another gust blows a small ant straight onto this triangle. The ... | 0.3916721504 | Sunday, 5th November 2017, 04:00 am | 1990 | 20% | easy |
625 | Gcd Sum | $G(N)=\sum_{j=1}^N\sum_{i=1}^j \gcd(i,j)$.
You are given: $G(10)=122$.
Find $G(10^{11})$. Give your answer modulo $998244353$. | $G(N)=\sum_{j=1}^N\sum_{i=1}^j \gcd(i,j)$.
You are given: $G(10)=122$.
Find $G(10^{11})$. Give your answer modulo $998244353$. | <p>
$G(N)=\sum_{j=1}^N\sum_{i=1}^j \gcd(i,j)$. <br/>
You are given: $G(10)=122$.</p>
<p>
Find $G(10^{11})$. Give your answer modulo $998244353$.
</p> | 551614306 | Saturday, 21st April 2018, 04:00 pm | 713 | 35% | medium |
494 | Collatz Prefix Families | The Collatz sequence is defined as:
$a_{i+1} = \left\{ \large{\frac {a_i} 2 \atop 3 a_i+1} {\text{if }a_i\text{ is even} \atop \text{if }a_i\text{ is odd}} \right.$.
The Collatz conjecture states that starting from any positive integer, the sequence eventually reaches the cycle $1,4,2,1, \dots$.
We shall define the ... | The Collatz sequence is defined as:
$a_{i+1} = \left\{ \large{\frac {a_i} 2 \atop 3 a_i+1} {\text{if }a_i\text{ is even} \atop \text{if }a_i\text{ is odd}} \right.$.
The Collatz conjecture states that starting from any positive integer, the sequence eventually reaches the cycle $1,4,2,1, \dots$.
We shall define the ... | <p>
The Collatz sequence is defined as:
$a_{i+1} = \left\{ \large{\frac {a_i} 2 \atop 3 a_i+1} {\text{if }a_i\text{ is even} \atop \text{if }a_i\text{ is odd}} \right.$.
</p>
<p>
The Collatz conjecture states that starting from any positive integer, the sequence eventually reaches the cycle $1,4,2,1, \dots$.<br/>
We s... | 2880067194446832666 | Saturday, 20th December 2014, 07:00 pm | 245 | 100% | hard |
793 | Median of Products | Let $S_i$ be an integer sequence produced with the following pseudo-random number generator:
$S_0 = 290797$
$S_{i+1} = S_i ^2 \bmod 50515093$
Let $M(n)$ be the median of the pairwise products $ S_i S_j $ for $0 \le i \lt j \lt n$.
You are given $M(3) = 3878983057768$ and $M(103) = 492700616748525$.
Find $M(1\,... | Let $S_i$ be an integer sequence produced with the following pseudo-random number generator:
$S_0 = 290797$
$S_{i+1} = S_i ^2 \bmod 50515093$
Let $M(n)$ be the median of the pairwise products $ S_i S_j $ for $0 \le i \lt j \lt n$.
You are given $M(3) = 3878983057768$ and $M(103) = 492700616748525$.
Find $M(1\,... | <p>
Let $S_i$ be an integer sequence produced with the following pseudo-random number generator:
</p>
<ul>
<li>$S_0 = 290797$</li>
<li> $S_{i+1} = S_i ^2 \bmod 50515093$</li>
</ul>
<p>
Let $M(n)$ be the median of the pairwise products $ S_i S_j $ for $0 \le i \lt j \lt n$.
</p>
<p>
You are given $M(3) = 3878983057768$ ... | 475808650131120 | Sunday, 10th April 2022, 08:00 am | 740 | 20% | easy |
65 | Convergents of $e$ | The square root of $2$ can be written as an infinite continued fraction.
$\sqrt{2} = 1 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2 + ...}}}}$
The infinite continued fraction can be written, $\sqrt{2} = [1; (2)]$, $(2)$ indicates that $2$ repeats ad infinitum. In a similar way, $\sqrt{23} = [4; (1, 3, 1, 8)]... | The square root of $2$ can be written as an infinite continued fraction.
$\sqrt{2} = 1 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2 + ...}}}}$
The infinite continued fraction can be written, $\sqrt{2} = [1; (2)]$, $(2)$ indicates that $2$ repeats ad infinitum. In a similar way, $\sqrt{23} = [4; (1, 3, 1, 8)]... | <p>The square root of $2$ can be written as an infinite continued fraction.</p>
<p>$\sqrt{2} = 1 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2 + ...}}}}$</p>
<p>The infinite continued fraction can be written, $\sqrt{2} = [1; (2)]$, $(2)$ indicates that $2$ repeats <i>ad infinitum</i>. In a similar way, $\sqrt... | 272 | Friday, 12th March 2004, 06:00 pm | 33122 | 15% | easy |
235 | An Arithmetic Geometric Sequence | Given is the arithmetic-geometric sequence $u(k) = (900-3k)r^{k - 1}$.
Let $s(n) = \sum_{k = 1}^n u(k)$.
Find the value of $r$ for which $s(5000) = -600\,000\,000\,000$.
Give your answer rounded to $12$ places behind the decimal point. | Given is the arithmetic-geometric sequence $u(k) = (900-3k)r^{k - 1}$.
Let $s(n) = \sum_{k = 1}^n u(k)$.
Find the value of $r$ for which $s(5000) = -600\,000\,000\,000$.
Give your answer rounded to $12$ places behind the decimal point. | <p>
Given is the arithmetic-geometric sequence $u(k) = (900-3k)r^{k - 1}$.<br/>
Let $s(n) = \sum_{k = 1}^n u(k)$.
</p>
<p>
Find the value of $r$ for which $s(5000) = -600\,000\,000\,000$.
</p>
<p>
Give your answer rounded to $12$ places behind the decimal point.
</p> | 1.002322108633 | Saturday, 7th March 2009, 05:00 am | 5406 | 40% | medium |
593 | Fleeting Medians | We define two sequences $S = \{S(1), S(2), ..., S(n)\}$ and $S_2 = \{S_2(1), S_2(2), ..., S_2(n)\}$:
$S(k) = (p_k)^k \bmod 10007$ where $p_k$ is the $k$th prime number.
$S_2(k) = S(k) + S(\lfloor\frac{k}{10000}\rfloor + 1)$ where $\lfloor \cdot \rfloor$ denotes the floor function.
Then let $M(i, j)$ be the median of el... | We define two sequences $S = \{S(1), S(2), ..., S(n)\}$ and $S_2 = \{S_2(1), S_2(2), ..., S_2(n)\}$:
$S(k) = (p_k)^k \bmod 10007$ where $p_k$ is the $k$th prime number.
$S_2(k) = S(k) + S(\lfloor\frac{k}{10000}\rfloor + 1)$ where $\lfloor \cdot \rfloor$ denotes the floor function.
Then let $M(i, j)$ be the median of el... | <p>We define two sequences $S = \{S(1), S(2), ..., S(n)\}$ and $S_2 = \{S_2(1), S_2(2), ..., S_2(n)\}$:</p>
<p>$S(k) = (p_k)^k \bmod 10007$ where $p_k$ is the $k$th prime number.</p>
<p>$S_2(k) = S(k) + S(\lfloor\frac{k}{10000}\rfloor + 1)$ where $\lfloor \cdot \rfloor$ denotes the floor function.</p>
<p>Then let $M(i,... | 96632320042.0 | Saturday, 4th March 2017, 07:00 pm | 601 | 35% | medium |
599 | Distinct Colourings of a Rubik's Cube | The well-known Rubik's Cube puzzle has many fascinating mathematical properties. The 2×2×2 variant has 8 cubelets with a total of 24 visible faces, each with a coloured sticker. Successively turning faces will rearrange the cubelets, although not all arrangements of cubelets are reachable without dismantling the puzzle... | The well-known Rubik's Cube puzzle has many fascinating mathematical properties. The 2×2×2 variant has 8 cubelets with a total of 24 visible faces, each with a coloured sticker. Successively turning faces will rearrange the cubelets, although not all arrangements of cubelets are reachable without dismantling the puzzle... | <p>
The well-known <strong>Rubik's Cube</strong> puzzle has many fascinating mathematical properties. The 2×2×2 variant has 8 cubelets with a total of 24 visible faces, each with a coloured sticker. Successively turning faces will rearrange the cubelets, although not all arrangements of cubelets are reachable without d... | 12395526079546335 | Saturday, 15th April 2017, 01:00 pm | 342 | 60% | hard |
502 | Counting Castles | We define a block to be a rectangle with a height of $1$ and an integer-valued length. Let a castle be a configuration of stacked blocks.
Given a game grid that is $w$ units wide and $h$ units tall, a castle is generated according to the following rules:
Blocks can be placed on top of other blocks as long as nothing st... | We define a block to be a rectangle with a height of $1$ and an integer-valued length. Let a castle be a configuration of stacked blocks.
Given a game grid that is $w$ units wide and $h$ units tall, a castle is generated according to the following rules:
Blocks can be placed on top of other blocks as long as nothing st... | <p>We define a <dfn>block</dfn> to be a rectangle with a height of $1$ and an integer-valued length. Let a <dfn>castle</dfn> be a configuration of stacked blocks.</p>
<p>Given a game grid that is $w$ units wide and $h$ units tall, a castle is generated according to the following rules:</p>
<ol><li>Blocks can be placed ... | 749485217 | Saturday, 7th February 2015, 04:00 pm | 332 | 100% | hard |
41 | Pandigital Prime | We shall say that an $n$-digit number is pandigital if it makes use of all the digits $1$ to $n$ exactly once. For example, $2143$ is a $4$-digit pandigital and is also prime.
What is the largest $n$-digit pandigital prime that exists? | We shall say that an $n$-digit number is pandigital if it makes use of all the digits $1$ to $n$ exactly once. For example, $2143$ is a $4$-digit pandigital and is also prime.
What is the largest $n$-digit pandigital prime that exists? | <p>We shall say that an $n$-digit number is pandigital if it makes use of all the digits $1$ to $n$ exactly once. For example, $2143$ is a $4$-digit pandigital and is also prime.</p>
<p>What is the largest $n$-digit pandigital prime that exists?</p> | 7652413 | Friday, 11th April 2003, 06:00 pm | 74695 | 5% | easy |
543 | Prime-Sum Numbers | Define function $P(n, k) = 1$ if $n$ can be written as the sum of $k$ prime numbers (with repetitions allowed), and $P(n, k) = 0$ otherwise.
For example, $P(10,2) = 1$ because $10$ can be written as either $3 + 7$ or $5 + 5$, but $P(11,2) = 0$ because no two primes can sum to $11$.
Let $S(n)$ be the sum of all $P(i,k)$... | Define function $P(n, k) = 1$ if $n$ can be written as the sum of $k$ prime numbers (with repetitions allowed), and $P(n, k) = 0$ otherwise.
For example, $P(10,2) = 1$ because $10$ can be written as either $3 + 7$ or $5 + 5$, but $P(11,2) = 0$ because no two primes can sum to $11$.
Let $S(n)$ be the sum of all $P(i,k)$... | <p>Define function $P(n, k) = 1$ if $n$ can be written as the sum of $k$ prime numbers (with repetitions allowed), and $P(n, k) = 0$ otherwise.</p>
<p>For example, $P(10,2) = 1$ because $10$ can be written as either $3 + 7$ or $5 + 5$, but $P(11,2) = 0$ because no two primes can sum to $11$.</p>
<p>Let $S(n)$ be the su... | 199007746081234640 | Saturday, 16th January 2016, 04:00 pm | 766 | 30% | easy |
862 | Larger Digit Permutation | For a positive integer $n$ define $T(n)$ to be the number of strictly larger integers which can be formed by permuting the digits of $n$.
Leading zeros are not allowed and so for $n = 2302$ the total list of permutations would be:
$2023,2032,2203,2230,\mathbf{2302},2320,3022,32 02,3220$
giving $T(2302)=4$.
Further ... | For a positive integer $n$ define $T(n)$ to be the number of strictly larger integers which can be formed by permuting the digits of $n$.
Leading zeros are not allowed and so for $n = 2302$ the total list of permutations would be:
$2023,2032,2203,2230,\mathbf{2302},2320,3022,32 02,3220$
giving $T(2302)=4$.
Further ... | <p>
For a positive integer $n$ define $T(n)$ to be the number of strictly larger integers which can be formed by permuting the digits of $n$.</p>
<p>
Leading zeros are not allowed and so for $n = 2302$ the total list of permutations would be:</p>
<div style="text-align:center;">
$2023,2032,2203,2230,\mathbf{2302},2320,... | 6111397420935766740 | Saturday, 4th November 2023, 04:00 pm | 802 | 10% | easy |
388 | Distinct Lines | Consider all lattice points $(a,b,c)$ with $0 \le a,b,c \le N$.
From the origin $O(0,0,0)$ all lines are drawn to the other lattice points.
Let $D(N)$ be the number of distinct such lines.
You are given that $D(1\,000\,000) = 831909254469114121$.
Find $D(10^{10})$. Give as your answer the first nine digits followe... | Consider all lattice points $(a,b,c)$ with $0 \le a,b,c \le N$.
From the origin $O(0,0,0)$ all lines are drawn to the other lattice points.
Let $D(N)$ be the number of distinct such lines.
You are given that $D(1\,000\,000) = 831909254469114121$.
Find $D(10^{10})$. Give as your answer the first nine digits followe... | <p>
Consider all lattice points $(a,b,c)$ with $0 \le a,b,c \le N$.
</p>
<p>
From the origin $O(0,0,0)$ all lines are drawn to the other lattice points.<br/>
Let $D(N)$ be the number of <i>distinct</i> such lines.
</p>
<p>
You are given that $D(1\,000\,000) = 831909254469114121$.
</p><p>
Find $D(10^{10})$. Give as your... | 831907372805129931 | Saturday, 9th June 2012, 02:00 pm | 653 | 55% | medium |
533 | Minimum Values of the Carmichael Function | The Carmichael function $\lambda(n)$ is defined as the smallest positive integer $m$ such that $a^m = 1$ modulo $n$ for all integers $a$ coprime with $n$.
For example $\lambda(8) = 2$ and $\lambda(240) = 4$.
Define $L(n)$ as the smallest positive integer $m$ such that $\lambda(k) \ge n$ for all $k \ge m$.
For example, ... | The Carmichael function $\lambda(n)$ is defined as the smallest positive integer $m$ such that $a^m = 1$ modulo $n$ for all integers $a$ coprime with $n$.
For example $\lambda(8) = 2$ and $\lambda(240) = 4$.
Define $L(n)$ as the smallest positive integer $m$ such that $\lambda(k) \ge n$ for all $k \ge m$.
For example, ... | <p>The <strong>Carmichael function</strong> $\lambda(n)$ is defined as the smallest positive integer $m$ such that $a^m = 1$ modulo $n$ for all integers $a$ coprime with $n$.<br/>
For example $\lambda(8) = 2$ and $\lambda(240) = 4$.</p>
<p>Define $L(n)$ as the smallest positive integer $m$ such that $\lambda(k) \ge n$ ... | 789453601 | Sunday, 8th November 2015, 10:00 am | 367 | 50% | medium |
432 | Totient Sum | Let $S(n,m) = \sum\phi(n \times i)$ for $1 \leq i \leq m$. ($\phi$ is Euler's totient function)
You are given that $S(510510,10^6)= 45480596821125120$.
Find $S(510510,10^{11})$.
Give the last $9$ digits of your answer. | Let $S(n,m) = \sum\phi(n \times i)$ for $1 \leq i \leq m$. ($\phi$ is Euler's totient function)
You are given that $S(510510,10^6)= 45480596821125120$.
Find $S(510510,10^{11})$.
Give the last $9$ digits of your answer. | <p>
Let $S(n,m) = \sum\phi(n \times i)$ for $1 \leq i \leq m$. ($\phi$ is Euler's totient function)<br/>
You are given that $S(510510,10^6)= 45480596821125120$.
</p>
<p>
Find $S(510510,10^{11})$.<br/>
Give the last $9$ digits of your answer.
</p> | 754862080 | Saturday, 15th June 2013, 01:00 pm | 559 | 70% | hard |
585 | Nested Square Roots | Consider the term $\small \sqrt{x+\sqrt{y}+\sqrt{z}}$ that is representing a nested square root. $x$, $y$ and $z$ are positive integers and $y$ and $z$ are not allowed to be perfect squares, so the number below the outer square root is irrational. Still it can be shown that for some combinations of $x$, $y$ and $z$ the... | Consider the term $\small \sqrt{x+\sqrt{y}+\sqrt{z}}$ that is representing a nested square root. $x$, $y$ and $z$ are positive integers and $y$ and $z$ are not allowed to be perfect squares, so the number below the outer square root is irrational. Still it can be shown that for some combinations of $x$, $y$ and $z$ the... | <p>Consider the term $\small \sqrt{x+\sqrt{y}+\sqrt{z}}$ that is representing a <strong>nested square root</strong>. $x$, $y$ and $z$ are positive integers and $y$ and $z$ are not allowed to be perfect squares, so the number below the outer square root is irrational. Still it can be shown that for some combinations of ... | 17714439395932 | Saturday, 7th January 2017, 07:00 pm | 201 | 100% | hard |
133 | Repunit Nonfactors | A number consisting entirely of ones is called a repunit. We shall define $R(k)$ to be a repunit of length $k$; for example, $R(6) = 111111$.
Let us consider repunits of the form $R(10^n)$.
Although $R(10)$, $R(100)$, or $R(1000)$ are not divisible by $17$, $R(10000)$ is divisible by $17$. Yet there is no value of $n$ ... | A number consisting entirely of ones is called a repunit. We shall define $R(k)$ to be a repunit of length $k$; for example, $R(6) = 111111$.
Let us consider repunits of the form $R(10^n)$.
Although $R(10)$, $R(100)$, or $R(1000)$ are not divisible by $17$, $R(10000)$ is divisible by $17$. Yet there is no value of $n$ ... | <p>A number consisting entirely of ones is called a repunit. We shall define $R(k)$ to be a repunit of length $k$; for example, $R(6) = 111111$.</p>
<p>Let us consider repunits of the form $R(10^n)$.</p>
<p>Although $R(10)$, $R(100)$, or $R(1000)$ are not divisible by $17$, $R(10000)$ is divisible by $17$. Yet there is... | 453647705 | Friday, 1st December 2006, 06:00 pm | 6222 | 50% | medium |
428 | Necklace of Circles | Let $a$, $b$ and $c$ be positive numbers.
Let $W, X, Y, Z$ be four collinear points where $|WX| = a$, $|XY| = b$, $|YZ| = c$ and $|WZ| = a + b + c$.
Let $C_{in}$ be the circle having the diameter $XY$.
Let $C_{out}$ be the circle having the diameter $WZ$.
The triplet $(a, b, c)$ is called a necklace triplet if you can... | Let $a$, $b$ and $c$ be positive numbers.
Let $W, X, Y, Z$ be four collinear points where $|WX| = a$, $|XY| = b$, $|YZ| = c$ and $|WZ| = a + b + c$.
Let $C_{in}$ be the circle having the diameter $XY$.
Let $C_{out}$ be the circle having the diameter $WZ$.
The triplet $(a, b, c)$ is called a necklace triplet if you can... | <p>Let $a$, $b$ and $c$ be positive numbers.<br/>
Let $W, X, Y, Z$ be four collinear points where $|WX| = a$, $|XY| = b$, $|YZ| = c$ and $|WZ| = a + b + c$.<br/>
Let $C_{in}$ be the circle having the diameter $XY$.<br/>
Let $C_{out}$ be the circle having the diameter $WZ$.<br/></p>
<p>
The triplet $(a, b, c)$ is called... | 747215561862 | Sunday, 19th May 2013, 01:00 am | 274 | 90% | hard |
139 | Pythagorean Tiles | Let $(a, b, c)$ represent the three sides of a right angle triangle with integral length sides. It is possible to place four such triangles together to form a square with length $c$.
For example, $(3, 4, 5)$ triangles can be placed together to form a $5$ by $5$ square with a $1$ by $1$ hole in the middle and it can be ... | Let $(a, b, c)$ represent the three sides of a right angle triangle with integral length sides. It is possible to place four such triangles together to form a square with length $c$.
For example, $(3, 4, 5)$ triangles can be placed together to form a $5$ by $5$ square with a $1$ by $1$ hole in the middle and it can be ... | <p>Let $(a, b, c)$ represent the three sides of a right angle triangle with integral length sides. It is possible to place four such triangles together to form a square with length $c$.</p>
<p>For example, $(3, 4, 5)$ triangles can be placed together to form a $5$ by $5$ square with a $1$ by $1$ hole in the middle and ... | 10057761 | Saturday, 27th January 2007, 09:00 am | 6359 | 50% | medium |
122 | Efficient Exponentiation | The most naive way of computing $n^{15}$ requires fourteen multiplications:
$$n \times n \times \cdots \times n = n^{15}.$$
But using a "binary" method you can compute it in six multiplications:
\begin{align}
n \times n &= n^2\\
n^2 \times n^2 &= n^4\\
n^4 \times n^4 &= n^8\\
n^8 \times n^4 &= n^{12}\\
n^{12} \times n^... | The most naive way of computing $n^{15}$ requires fourteen multiplications:
$$n \times n \times \cdots \times n = n^{15}.$$
But using a "binary" method you can compute it in six multiplications:
\begin{align}
n \times n &= n^2\\
n^2 \times n^2 &= n^4\\
n^4 \times n^4 &= n^8\\
n^8 \times n^4 &= n^{12}\\
n^{12} \times n^... | <p>The most naive way of computing $n^{15}$ requires fourteen multiplications:
$$n \times n \times \cdots \times n = n^{15}.$$</p>
<p>But using a "binary" method you can compute it in six multiplications:</p>
\begin{align}
n \times n &= n^2\\
n^2 \times n^2 &= n^4\\
n^4 \times n^4 &= n^8\\
n^8 \times n^4 &= n^{12}\\
n^... | 1582 | Friday, 2nd June 2006, 06:00 pm | 8782 | 40% | medium |
47 | Distinct Primes Factors | The first two consecutive numbers to have two distinct prime factors are:
\begin{align}
14 &= 2 \times 7\\
15 &= 3 \times 5.
\end{align}
The first three consecutive numbers to have three distinct prime factors are:
\begin{align}
644 &= 2^2 \times 7 \times 23\\
645 &= 3 \times 5 \times 43\\
646 &= 2 \times 17 \times 19.... | The first two consecutive numbers to have two distinct prime factors are:
\begin{align}
14 &= 2 \times 7\\
15 &= 3 \times 5.
\end{align}
The first three consecutive numbers to have three distinct prime factors are:
\begin{align}
644 &= 2^2 \times 7 \times 23\\
645 &= 3 \times 5 \times 43\\
646 &= 2 \times 17 \times 19.... | <p>The first two consecutive numbers to have two distinct prime factors are:</p>
\begin{align}
14 &= 2 \times 7\\
15 &= 3 \times 5.
\end{align}
<p>The first three consecutive numbers to have three distinct prime factors are:</p>
\begin{align}
644 &= 2^2 \times 7 \times 23\\
645 &= 3 \times 5 \times 43\\
646 &= 2 \times... | 134043 | Friday, 4th July 2003, 06:00 pm | 63771 | 5% | easy |
654 | Neighbourly Constraints | Let $T(n, m)$ be the number of $m$-tuples of positive integers such that the sum of any two neighbouring elements of the tuple is $\le n$.
For example, $T(3, 4)=8$, via the following eight $4$-tuples:
$(1, 1, 1, 1)$
$(1, 1, 1, 2)$
$(1, 1, 2, 1)$
$(1, 2, 1, 1)$
$(1, 2, 1, 2)$
$(2, 1, 1, 1)$
$(2, 1, 1, 2)$
$(2, 1, 2, 1... | Let $T(n, m)$ be the number of $m$-tuples of positive integers such that the sum of any two neighbouring elements of the tuple is $\le n$.
For example, $T(3, 4)=8$, via the following eight $4$-tuples:
$(1, 1, 1, 1)$
$(1, 1, 1, 2)$
$(1, 1, 2, 1)$
$(1, 2, 1, 1)$
$(1, 2, 1, 2)$
$(2, 1, 1, 1)$
$(2, 1, 1, 2)$
$(2, 1, 2, 1... | <p>
Let $T(n, m)$ be the number of $m$-tuples of positive integers such that the sum of any two neighbouring elements of the tuple is $\le n$.
</p>
<p>
For example, $T(3, 4)=8$, via the following eight $4$-tuples:<br>
$(1, 1, 1, 1)$<br/>
$(1, 1, 1, 2)$<br/>
$(1, 1, 2, 1)$<br/>
$(1, 2, 1, 1)$<br/>
$(1, 2, 1, 2)$<br/>
$(... | 815868280 | Sunday, 3rd February 2019, 04:00 am | 385 | 45% | medium |
185 | Number Mind | The game Number Mind is a variant of the well known game Master Mind.
Instead of coloured pegs, you have to guess a secret sequence of digits. After each guess you're only told in how many places you've guessed the correct digit. So, if the sequence was 1234 and you guessed 2036, you'd be told that you have one correct... | The game Number Mind is a variant of the well known game Master Mind.
Instead of coloured pegs, you have to guess a secret sequence of digits. After each guess you're only told in how many places you've guessed the correct digit. So, if the sequence was 1234 and you guessed 2036, you'd be told that you have one correct... | <p>The game Number Mind is a variant of the well known game Master Mind.</p>
<p>Instead of coloured pegs, you have to guess a secret sequence of digits. After each guess you're only told in how many places you've guessed the correct digit. So, if the sequence was 1234 and you guessed 2036, you'd be told that you have o... | 4640261571849533 | Saturday, 8th March 2008, 01:00 am | 3494 | 55% | medium |
54 | Poker Hands | In the card game poker, a hand consists of five cards and are ranked, from lowest to highest, in the following way:
High Card: Highest value card.
One Pair: Two cards of the same value.
Two Pairs: Two different pairs.
Three of a Kind: Three cards of the same value.
Straight: All cards are consecutive values.
Flush: All... | In the card game poker, a hand consists of five cards and are ranked, from lowest to highest, in the following way:
High Card: Highest value card.
One Pair: Two cards of the same value.
Two Pairs: Two different pairs.
Three of a Kind: Three cards of the same value.
Straight: All cards are consecutive values.
Flush: All... | <p>In the card game poker, a hand consists of five cards and are ranked, from lowest to highest, in the following way:</p>
<ul><li><b>High Card</b>: Highest value card.</li>
<li><b>One Pair</b>: Two cards of the same value.</li>
<li><b>Two Pairs</b>: Two different pairs.</li>
<li><b>Three of a Kind</b>: Three cards of ... | 376 | Friday, 10th October 2003, 06:00 pm | 39978 | 10% | easy |
660 | Pandigital Triangles | We call an integer sided triangle $n$-pandigital if it contains one angle of $120$ degrees and, when the sides of the triangle are written in base $n$, together they use all $n$ digits of that base exactly once.
For example, the triangle $(217, 248, 403)$ is $9$-pandigital because it contains one angle of $120$ degree... | We call an integer sided triangle $n$-pandigital if it contains one angle of $120$ degrees and, when the sides of the triangle are written in base $n$, together they use all $n$ digits of that base exactly once.
For example, the triangle $(217, 248, 403)$ is $9$-pandigital because it contains one angle of $120$ degree... | <p>We call an integer sided triangle <dfn>$n$-pandigital</dfn> if it contains one angle of $120$ degrees and, when the sides of the triangle are written in base $n$, together they use all $n$ digits of that base exactly once.</p>
<p>
For example, the triangle $(217, 248, 403)$ is $9$-pandigital because it contains one ... | 474766783 | Saturday, 9th March 2019, 07:00 pm | 335 | 40% | medium |
537 | Counting Tuples | Let $\pi(x)$ be the prime counting function, i.e. the number of prime numbers less than or equal to $x$.
For example,$\pi(1)=0$, $\pi(2)=1$, $\pi(100)=25$.
Let $T(n, k)$ be the number of $k$-tuples $(x_1, \dots, x_k)$ which satisfy:
1. every $x_i$ is a positive integer;
2. $\displaystyle \sum_{i=1}^k \pi(x_i)=n$
Fo... | Let $\pi(x)$ be the prime counting function, i.e. the number of prime numbers less than or equal to $x$.
For example,$\pi(1)=0$, $\pi(2)=1$, $\pi(100)=25$.
Let $T(n, k)$ be the number of $k$-tuples $(x_1, \dots, x_k)$ which satisfy:
1. every $x_i$ is a positive integer;
2. $\displaystyle \sum_{i=1}^k \pi(x_i)=n$
Fo... | <p>
Let $\pi(x)$ be the prime counting function, i.e. the number of prime numbers less than or equal to $x$.<br/>
For example,$\pi(1)=0$, $\pi(2)=1$, $\pi(100)=25$.
</p>
<p>
Let $T(n, k)$ be the number of $k$-tuples $(x_1, \dots, x_k)$ which satisfy:<br/>
1. every $x_i$ is a positive integer;<br/>
2. $\displaystyle \su... | 779429131 | Saturday, 5th December 2015, 10:00 pm | 709 | 35% | medium |
632 | Square Prime Factors | For an integer $n$, we define the square prime factors of $n$ to be the primes whose square divides $n$. For example, the square prime factors of $1500=2^2 \times 3 \times 5^3$ are $2$ and $5$.
Let $C_k(N)$ be the number of integers between $1$ and $N$ inclusive with exactly $k$ square prime factors. You are given some... | For an integer $n$, we define the square prime factors of $n$ to be the primes whose square divides $n$. For example, the square prime factors of $1500=2^2 \times 3 \times 5^3$ are $2$ and $5$.
Let $C_k(N)$ be the number of integers between $1$ and $N$ inclusive with exactly $k$ square prime factors. You are given some... | <p>For an integer $n$, we define the <dfn>square prime factors</dfn> of $n$ to be the primes whose square divides $n$. For example, the square prime factors of $1500=2^2 \times 3 \times 5^3$ are $2$ and $5$.</p>
<p>Let $C_k(N)$ be the number of integers between $1$ and $N$ inclusive with exactly $k$ square prime factor... | 728378714 | Saturday, 28th July 2018, 01:00 pm | 548 | 35% | medium |
786 | Billiard | The following diagram shows a billiard table of a special quadrilateral shape.
The four angles $A, B, C, D$ are $120^\circ, 90^\circ, 60^\circ, 90^\circ$ respectively, and the lengths $AB$ and $AD$ are equal.
The diagram on the left shows the trace of an infinitesimally small billiard ball, departing from point $A... | The following diagram shows a billiard table of a special quadrilateral shape.
The four angles $A, B, C, D$ are $120^\circ, 90^\circ, 60^\circ, 90^\circ$ respectively, and the lengths $AB$ and $AD$ are equal.
The diagram on the left shows the trace of an infinitesimally small billiard ball, departing from point $A... | <p>
The following diagram shows a billiard table of a special quadrilateral shape.
The four angles $A, B, C, D$ are $120^\circ, 90^\circ, 60^\circ, 90^\circ$ respectively, and the lengths $AB$ and $AD$ are equal.
</p>
<div style="text-align:center;">
<img alt="" class="dark_img" height="160" src="project/images/p786_bi... | 45594532839912702 | Sunday, 20th February 2022, 10:00 am | 139 | 100% | hard |
685 | Inverse Digit Sum II | Writing down the numbers which have a digit sum of 10 in ascending order, we get:
$19, 28, 37, 46,55,64,73,82,91,109, 118,\dots$
Let $f(n,m)$ be the $m^{\text{th}}$ occurrence of the digit sum $n$. For example, $f(10,1)=19$, $f(10,10)=109$ and $f(10,100)=1423$.
Let $\displaystyle S(k)=\sum_{n=1}^k f(n^3,n^4)$. For ex... | Writing down the numbers which have a digit sum of 10 in ascending order, we get:
$19, 28, 37, 46,55,64,73,82,91,109, 118,\dots$
Let $f(n,m)$ be the $m^{\text{th}}$ occurrence of the digit sum $n$. For example, $f(10,1)=19$, $f(10,10)=109$ and $f(10,100)=1423$.
Let $\displaystyle S(k)=\sum_{n=1}^k f(n^3,n^4)$. For ex... | <p>Writing down the numbers which have a digit sum of 10 in ascending order, we get:
$19, 28, 37, 46,55,64,73,82,91,109, 118,\dots$</p>
<p>
Let $f(n,m)$ be the $m^{\text{th}}$ occurrence of the digit sum $n$. For example, $f(10,1)=19$, $f(10,10)=109$ and $f(10,100)=1423$.</p>
<p>
Let $\displaystyle S(k)=\sum_{n=1}^k f(... | 662878999 | Saturday, 19th October 2019, 04:00 pm | 224 | 65% | hard |
886 | Coprime Permutations | A permutation of $\{2,3,\ldots,n\}$ is a rearrangement of these numbers. A coprime permutation is a rearrangement such that all pairs of adjacent numbers are coprime.
Let $P(n)$ be the number of coprime permutations of $\{2,3,\ldots,n\}$.
For example, $P(4)=2$ as there are two coprime permutations, $(2,3,4)$ and $(4,3,... | A permutation of $\{2,3,\ldots,n\}$ is a rearrangement of these numbers. A coprime permutation is a rearrangement such that all pairs of adjacent numbers are coprime.
Let $P(n)$ be the number of coprime permutations of $\{2,3,\ldots,n\}$.
For example, $P(4)=2$ as there are two coprime permutations, $(2,3,4)$ and $(4,3,... | <p>A permutation of $\{2,3,\ldots,n\}$ is a rearrangement of these numbers. A <i>coprime permutation</i> is a rearrangement such that all pairs of adjacent numbers are coprime.</p>
<p>Let $P(n)$ be the number of coprime permutations of $\{2,3,\ldots,n\}$.</p>
<p>For example, $P(4)=2$ as there are two coprime permutatio... | 5570163 | Saturday, 13th April 2024, 02:00 pm | 269 | 40% | medium |
674 | Solving $\mathcal{I}$-equations | We define the $\mathcal{I}$ operator as the function
\[\mathcal{I}(x,y) = (1+x+y)^2+y-x\]
and $\mathcal{I}$-expressions as arithmetic expressions built only from variable names and applications of $\mathcal{I}$. A variable name may consist of one or more letters. For example, the three expressions $x$, $\mathcal{I}(x,y... | We define the $\mathcal{I}$ operator as the function
\[\mathcal{I}(x,y) = (1+x+y)^2+y-x\]
and $\mathcal{I}$-expressions as arithmetic expressions built only from variable names and applications of $\mathcal{I}$. A variable name may consist of one or more letters. For example, the three expressions $x$, $\mathcal{I}(x,y... | <p>We define the $\mathcal{I}$ operator as the function
\[\mathcal{I}(x,y) = (1+x+y)^2+y-x\]
and $\mathcal{I}$-expressions as arithmetic expressions built only from variable names and applications of $\mathcal{I}$. A variable name may consist of one or more letters. For example, the three expressions $x$, $\mathcal{I}(... | 416678753 | Sunday, 9th June 2019, 10:00 am | 184 | 75% | hard |
169 | Sums of Powers of Two | Define $f(0)=1$ and $f(n)$ to be the number of different ways $n$ can be expressed as a sum of integer powers of $2$ using each power no more than twice.
For example, $f(10)=5$ since there are five different ways to express $10$:
\begin{align}
& 1 + 1 + 8\\
& 1 + 1 + 4 + 4\\
& 1 + 1 + 2 + 2 + 4\\
& 2 + 4 + 4\\
& 2 + 8
... | Define $f(0)=1$ and $f(n)$ to be the number of different ways $n$ can be expressed as a sum of integer powers of $2$ using each power no more than twice.
For example, $f(10)=5$ since there are five different ways to express $10$:
\begin{align}
& 1 + 1 + 8\\
& 1 + 1 + 4 + 4\\
& 1 + 1 + 2 + 2 + 4\\
& 2 + 4 + 4\\
& 2 + 8
... | <p>Define $f(0)=1$ and $f(n)$ to be the number of different ways $n$ can be expressed as a sum of integer powers of $2$ using each power no more than twice.</p>
<p>For example, $f(10)=5$ since there are five different ways to express $10$:</p>
\begin{align}
& 1 + 1 + 8\\
& 1 + 1 + 4 + 4\\
& 1 + 1 + 2 + 2 + 4\\
& 2 + 4 ... | 178653872807 | Friday, 23rd November 2007, 09:00 pm | 5688 | 50% | medium |
448 | Average Least Common Multiple | The function $\operatorname{\mathbf{lcm}}(a,b)$ denotes the least common multiple of $a$ and $b$.
Let $A(n)$ be the average of the values of $\operatorname{lcm}(n,i)$ for $1 \le i \le n$.
E.g: $A(2)=(2+2)/2=2$ and $A(10)=(10+10+30+20+10+30+70+40+90+10)/10=32$.
Let $S(n)=\sum A(k)$ for $1 \le k \le n$.
$S(100)=122726$... | The function $\operatorname{\mathbf{lcm}}(a,b)$ denotes the least common multiple of $a$ and $b$.
Let $A(n)$ be the average of the values of $\operatorname{lcm}(n,i)$ for $1 \le i \le n$.
E.g: $A(2)=(2+2)/2=2$ and $A(10)=(10+10+30+20+10+30+70+40+90+10)/10=32$.
Let $S(n)=\sum A(k)$ for $1 \le k \le n$.
$S(100)=122726$... | <p>
The function $\operatorname{\mathbf{lcm}}(a,b)$ denotes the least common multiple of $a$ and $b$.<br/>
Let $A(n)$ be the average of the values of $\operatorname{lcm}(n,i)$ for $1 \le i \le n$.<br/>
E.g: $A(2)=(2+2)/2=2$ and $A(10)=(10+10+30+20+10+30+70+40+90+10)/10=32$.
</p>
Let $S(n)=\sum A(k)$ for $1 \le k \le n... | 106467648 | Sunday, 24th November 2013, 01:00 am | 391 | 80% | hard |
383 | Divisibility Comparison Between Factorials | Let $f_5(n)$ be the largest integer $x$ for which $5^x$ divides $n$.
For example, $f_5(625000) = 7$.
Let $T_5(n)$ be the number of integers $i$ which satisfy $f_5((2 \cdot i - 1)!) \lt 2 \cdot f_5(i!)$ and $1 \le i \le n$.
It can be verified that $T_5(10^3) = 68$ and $T_5(10^9) = 2408210$.
Find $T_5(10^{18})$. | Let $f_5(n)$ be the largest integer $x$ for which $5^x$ divides $n$.
For example, $f_5(625000) = 7$.
Let $T_5(n)$ be the number of integers $i$ which satisfy $f_5((2 \cdot i - 1)!) \lt 2 \cdot f_5(i!)$ and $1 \le i \le n$.
It can be verified that $T_5(10^3) = 68$ and $T_5(10^9) = 2408210$.
Find $T_5(10^{18})$. | <p>
Let $f_5(n)$ be the largest integer $x$ for which $5^x$ divides $n$.<br/>
For example, $f_5(625000) = 7$.
</p>
<p>
Let $T_5(n)$ be the number of integers $i$ which satisfy $f_5((2 \cdot i - 1)!) \lt 2 \cdot f_5(i!)$ and $1 \le i \le n$.<br/>
It can be verified that $T_5(10^3) = 68$ and $T_5(10^9) = 2408210$.
</p>
<... | 22173624649806 | Saturday, 5th May 2012, 11:00 pm | 532 | 50% | medium |
173 | Hollow Square Laminae I | We shall define a square lamina to be a square outline with a square "hole" so that the shape possesses vertical and horizontal symmetry. For example, using exactly thirty-two square tiles we can form two different square laminae:
With one-hundred tiles, and not necessarily using all of the tiles at one time, it is p... | We shall define a square lamina to be a square outline with a square "hole" so that the shape possesses vertical and horizontal symmetry. For example, using exactly thirty-two square tiles we can form two different square laminae:
With one-hundred tiles, and not necessarily using all of the tiles at one time, it is p... | <p>We shall define a square lamina to be a square outline with a square "hole" so that the shape possesses vertical and horizontal symmetry. For example, using exactly thirty-two square tiles we can form two different square laminae:</p>
<div class="center">
<img alt="" src="resources/images/0173_square_laminas.gif?167... | 1572729 | Saturday, 22nd December 2007, 01:00 pm | 9962 | 30% | easy |
270 | Cutting Squares | A square piece of paper with integer dimensions $N \times N$ is placed with a corner at the origin and two of its sides along the $x$- and $y$-axes. Then, we cut it up respecting the following rules:
We only make straight cuts between two points lying on different sides of the square, and having integer coordinates.
Tw... | A square piece of paper with integer dimensions $N \times N$ is placed with a corner at the origin and two of its sides along the $x$- and $y$-axes. Then, we cut it up respecting the following rules:
We only make straight cuts between two points lying on different sides of the square, and having integer coordinates.
Tw... | <p>A square piece of paper with integer dimensions $N \times N$ is placed with a corner at the origin and two of its sides along the $x$- and $y$-axes. Then, we cut it up respecting the following rules:
</p><ul><li>We only make straight cuts between two points lying on different sides of the square, and having integer ... | 82282080 | Saturday, 26th December 2009, 01:00 am | 720 | 80% | hard |
512 | Sums of Totients of Powers | Let $\varphi(n)$ be Euler's totient function.
Let $f(n)=(\sum_{i=1}^{n}\varphi(n^i)) \bmod (n+1)$.
Let $g(n)=\sum_{i=1}^{n} f(i)$.
$g(100)=2007$.
Find $g(5 \times 10^8)$. | Let $\varphi(n)$ be Euler's totient function.
Let $f(n)=(\sum_{i=1}^{n}\varphi(n^i)) \bmod (n+1)$.
Let $g(n)=\sum_{i=1}^{n} f(i)$.
$g(100)=2007$.
Find $g(5 \times 10^8)$. | <p>Let $\varphi(n)$ be Euler's totient function.</p><p>
Let $f(n)=(\sum_{i=1}^{n}\varphi(n^i)) \bmod (n+1)$.</p><p>
Let $g(n)=\sum_{i=1}^{n} f(i)$.</p><p>
$g(100)=2007$.
</p>
<p>
Find $g(5 \times 10^8)$.
</p> | 50660591862310323 | Saturday, 18th April 2015, 10:00 pm | 1670 | 25% | easy |
807 | Loops of Ropes | Given a circle $C$ and an integer $n > 1$, we perform the following operations.
In step $0$, we choose two uniformly random points $R_0$ and $B_0$ on $C$.
In step $i$ ($1 \leq i < n$), we first choose a uniformly random point $R_i$ on $C$ and connect the points $R_{i - 1}$ and $R_i$ with a red rope; then choose a unifo... | Given a circle $C$ and an integer $n > 1$, we perform the following operations.
In step $0$, we choose two uniformly random points $R_0$ and $B_0$ on $C$.
In step $i$ ($1 \leq i < n$), we first choose a uniformly random point $R_i$ on $C$ and connect the points $R_{i - 1}$ and $R_i$ with a red rope; then choose a unifo... | <p>Given a circle $C$ and an integer $n > 1$, we perform the following operations.</p>
<p>In step $0$, we choose two uniformly random points $R_0$ and $B_0$ on $C$.<br>
In step $i$ ($1 \leq i < n$), we first choose a uniformly random point $R_i$ on $C$ and connect the points $R_{i - 1}$ and $R_i$ with a red rope;... | 0.1091523673 | Sunday, 11th September 2022, 02:00 am | 141 | 90% | hard |
804 | Counting Binary Quadratic Representations | Let $g(n)$ denote the number of ways a positive integer $n$ can be represented in the form: $$x^2+xy+41y^2$$ where $x$ and $y$ are integers. For example, $g(53)=4$ due to $(x,y) \in \{(-4,1),(-3,-1),(3,1),(4,-1)\}$.
Define $\displaystyle T(N)=\sum_{n=1}^{N}g(n)$. You are given $T(10^3)=474$ and $T(10^6)=492128$.
Find $... | Let $g(n)$ denote the number of ways a positive integer $n$ can be represented in the form: $$x^2+xy+41y^2$$ where $x$ and $y$ are integers. For example, $g(53)=4$ due to $(x,y) \in \{(-4,1),(-3,-1),(3,1),(4,-1)\}$.
Define $\displaystyle T(N)=\sum_{n=1}^{N}g(n)$. You are given $T(10^3)=474$ and $T(10^6)=492128$.
Find $... | <p>Let $g(n)$ denote the number of ways a positive integer $n$ can be represented in the form: $$x^2+xy+41y^2$$ where $x$ and $y$ are integers. For example, $g(53)=4$ due to $(x,y) \in \{(-4,1),(-3,-1),(3,1),(4,-1)\}$.</p>
<p>Define $\displaystyle T(N)=\sum_{n=1}^{N}g(n)$. You are given $T(10^3)=474$ and $T(10^6)=49212... | 4921370551019052 | Saturday, 25th June 2022, 05:00 pm | 725 | 20% | easy |
675 | $2^{\omega(n)}$ | Let $\omega(n)$ denote the number of distinct prime divisors of a positive integer $n$.
So $\omega(1) = 0$ and $\omega(360) = \omega(2^{3} \times 3^{2} \times 5) = 3$.
Let $S(n)$ be $ \sum_{d \mid n} 2^{\omega(d)} $.
E.g. $S(6) = 2^{\omega(1)}+2^{\omega(2)}+2^{\omega(3)}+2^{\omega(6)} = 2^0+2^1+2^1+2^2 = 9$.
Le... | Let $\omega(n)$ denote the number of distinct prime divisors of a positive integer $n$.
So $\omega(1) = 0$ and $\omega(360) = \omega(2^{3} \times 3^{2} \times 5) = 3$.
Let $S(n)$ be $ \sum_{d \mid n} 2^{\omega(d)} $.
E.g. $S(6) = 2^{\omega(1)}+2^{\omega(2)}+2^{\omega(3)}+2^{\omega(6)} = 2^0+2^1+2^1+2^2 = 9$.
Le... | <p>
Let $\omega(n)$ denote the number of distinct prime divisors of a positive integer $n$.<br/>
So $\omega(1) = 0$ and $\omega(360) = \omega(2^{3} \times 3^{2} \times 5) = 3$.
</p>
<p>
Let $S(n)$ be $ \sum_{d \mid n} 2^{\omega(d)} $.
<br/>
E.g. $S(6) = 2^{\omega(1)}+2^{\omega(2)}+2^{\omega(3)}+2^{\omega(6)} = 2^0+2... | 416146418 | Saturday, 15th June 2019, 01:00 pm | 844 | 25% | easy |
145 | Reversible Numbers | Some positive integers $n$ have the property that the sum $[n + \operatorname{reverse}(n)]$ consists entirely of odd (decimal) digits. For instance, $36 + 63 = 99$ and $409 + 904 = 1313$. We will call such numbers reversible; so $36$, $63$, $409$, and $904$ are reversible. Leading zeroes are not allowed in either $n$ o... | Some positive integers $n$ have the property that the sum $[n + \operatorname{reverse}(n)]$ consists entirely of odd (decimal) digits. For instance, $36 + 63 = 99$ and $409 + 904 = 1313$. We will call such numbers reversible; so $36$, $63$, $409$, and $904$ are reversible. Leading zeroes are not allowed in either $n$ o... | <p>Some positive integers $n$ have the property that the sum $[n + \operatorname{reverse}(n)]$ consists entirely of odd (decimal) digits. For instance, $36 + 63 = 99$ and $409 + 904 = 1313$. We will call such numbers <dfn>reversible</dfn>; so $36$, $63$, $409$, and $904$ are reversible. Leading zeroes are not allowed i... | 608720 | Friday, 16th March 2007, 01:00 pm | 18209 | 20% | easy |
539 | Odd Elimination | Start from an ordered list of all integers from $1$ to $n$. Going from left to right, remove the first number and every other number afterward until the end of the list. Repeat the procedure from right to left, removing the right most number and every other number from the numbers left. Continue removing every other nu... | Start from an ordered list of all integers from $1$ to $n$. Going from left to right, remove the first number and every other number afterward until the end of the list. Repeat the procedure from right to left, removing the right most number and every other number from the numbers left. Continue removing every other nu... | <p>
Start from an ordered list of all integers from $1$ to $n$. Going from left to right, remove the first number and every other number afterward until the end of the list. Repeat the procedure from right to left, removing the right most number and every other number from the numbers left. Continue removing every othe... | 426334056 | Sunday, 20th December 2015, 04:00 am | 919 | 35% | medium |
445 | Retractions A | For every integer $n>1$, the family of functions $f_{n,a,b}$ is defined
by
$f_{n,a,b}(x)\equiv a x + b \mod n\,\,\, $ for $a,b,x$ integer and $0< a <n, 0 \le b < n,0 \le x < n$.
We will call $f_{n,a,b}$ a retraction if $\,\,\, f_{n,a,b}(f_{n,a,b}(x)) \equiv f_{n,a,b}(x) \mod n \,\,\,$ for every $0 \le x < n$.
Let... | For every integer $n>1$, the family of functions $f_{n,a,b}$ is defined
by
$f_{n,a,b}(x)\equiv a x + b \mod n\,\,\, $ for $a,b,x$ integer and $0< a <n, 0 \le b < n,0 \le x < n$.
We will call $f_{n,a,b}$ a retraction if $\,\,\, f_{n,a,b}(f_{n,a,b}(x)) \equiv f_{n,a,b}(x) \mod n \,\,\,$ for every $0 \le x < n$.
Let... | <p>
For every integer $n>1$, the family of functions $f_{n,a,b}$ is defined
by <br>
$f_{n,a,b}(x)\equiv a x + b \mod n\,\,\, $ for $a,b,x$ integer and $0< a <n, 0 \le b < n,0 \le x < n$. </br></p>
<p>
We will call $f_{n,a,b}$ a <i>retraction</i> if $\,\,\, f_{n,a,b}(f_{n,a,b}(x)) \equiv f_{n,a,b}(x) \... | 659104042 | Saturday, 16th November 2013, 10:00 pm | 444 | 50% | medium |
86 | Cuboid Route | A spider, S, sits in one corner of a cuboid room, measuring $6$ by $5$ by $3$, and a fly, F, sits in the opposite corner. By travelling on the surfaces of the room the shortest "straight line" distance from S to F is $10$ and the path is shown on the diagram.
However, there are up to three "shortest" path candidates ... | A spider, S, sits in one corner of a cuboid room, measuring $6$ by $5$ by $3$, and a fly, F, sits in the opposite corner. By travelling on the surfaces of the room the shortest "straight line" distance from S to F is $10$ and the path is shown on the diagram.
However, there are up to three "shortest" path candidates ... | <p>A spider, S, sits in one corner of a cuboid room, measuring $6$ by $5$ by $3$, and a fly, F, sits in the opposite corner. By travelling on the surfaces of the room the shortest "straight line" distance from S to F is $10$ and the path is shown on the diagram.</p>
<div class="center">
<img alt="" class="dark_img" src... | 1818 | Friday, 7th January 2005, 06:00 pm | 14368 | 35% | medium |
440 | GCD and Tiling | We want to tile a board of length $n$ and height $1$ completely, with either $1 \times 2$ blocks or $1 \times 1$ blocks with a single decimal digit on top:
For example, here are some of the ways to tile a board of length $n = 8$:
Let $T(n)$ be the number of ways to tile a board of length $n$ as described above.
F... | We want to tile a board of length $n$ and height $1$ completely, with either $1 \times 2$ blocks or $1 \times 1$ blocks with a single decimal digit on top:
For example, here are some of the ways to tile a board of length $n = 8$:
Let $T(n)$ be the number of ways to tile a board of length $n$ as described above.
F... | <p>We want to tile a board of length $n$ and height $1$ completely, with either $1 \times 2$ blocks or $1 \times 1$ blocks with a single decimal digit on top:</p>
<div class="center">
<img alt="0440_tiles.png" src="resources/images/0440_tiles.png?1678992053"/>
</div>
<p>For example, here are some of the ways to tile a ... | 970746056 | Sunday, 13th October 2013, 07:00 am | 429 | 60% | hard |
639 | Summing a Multiplicative Function | A multiplicative function $f(x)$ is a function over positive integers satisfying $f(1)=1$ and $f(a b)=f(a) f(b)$ for any two coprime positive integers $a$ and $b$.
For integer $k$ let $f_k(n)$ be a multiplicative function additionally satisfying $f_k(p^e)=p^k$ for any prime $p$ and any integer $e>0$.
For example, $f_1... | A multiplicative function $f(x)$ is a function over positive integers satisfying $f(1)=1$ and $f(a b)=f(a) f(b)$ for any two coprime positive integers $a$ and $b$.
For integer $k$ let $f_k(n)$ be a multiplicative function additionally satisfying $f_k(p^e)=p^k$ for any prime $p$ and any integer $e>0$.
For example, $f_1... | <p>A <b>multiplicative function</b> $f(x)$ is a function over positive integers satisfying $f(1)=1$ and $f(a b)=f(a) f(b)$ for any two coprime positive integers $a$ and $b$.</p>
<p>For integer $k$ let $f_k(n)$ be a multiplicative function additionally satisfying $f_k(p^e)=p^k$ for any prime $p$ and any integer $e>0$... | 797866893 | Sunday, 21st October 2018, 07:00 am | 304 | 70% | hard |
852 | Coins in a Box | This game has a box of $N$ unfair coins and $N$ fair coins. Fair coins have probability 50% of landing heads while unfair coins have probability 75% of landing heads.
The player begins with a score of 0 which may become negative during play.
At each round the player randomly picks a coin from the box and guesses its ty... | This game has a box of $N$ unfair coins and $N$ fair coins. Fair coins have probability 50% of landing heads while unfair coins have probability 75% of landing heads.
The player begins with a score of 0 which may become negative during play.
At each round the player randomly picks a coin from the box and guesses its ty... | <p>This game has a box of $N$ unfair coins and $N$ fair coins. Fair coins have probability 50% of landing heads while unfair coins have probability 75% of landing heads.</p>
<p>The player begins with a score of 0 which may become negative during play.</p>
<p>At each round the player randomly picks a coin from the box a... | 130.313496 | Saturday, 2nd September 2023, 02:00 pm | 244 | 50% | medium |
888 | 1249 Nim | Two players play a game with a number of piles of stones, alternating turns. Each turn a player can choose to remove 1, 2, 4, or 9 stones from a single pile; or alternatively they can choose to split a pile containing two or more stones into two non-empty piles. The winner is the player who removes the last stone.
A c... | Two players play a game with a number of piles of stones, alternating turns. Each turn a player can choose to remove 1, 2, 4, or 9 stones from a single pile; or alternatively they can choose to split a pile containing two or more stones into two non-empty piles. The winner is the player who removes the last stone.
A c... | <p>
Two players play a game with a number of piles of stones, alternating turns. Each turn a player can choose to remove 1, 2, 4, or 9 stones from a single pile; or alternatively they can choose to split a pile containing two or more stones into two non-empty piles. The winner is the player who removes the last stone.<... | 227429102 | Saturday, 27th April 2024, 08:00 pm | 189 | 45% | medium |
227 | The Chase | The Chase is a game played with two dice and an even number of players.
The players sit around a table and the game begins with two opposite players having one die each. On each turn, the two players with a die roll it.
If the player rolls 1, then the die passes to the neighbour on the left.
If the player rolls 6, then... | The Chase is a game played with two dice and an even number of players.
The players sit around a table and the game begins with two opposite players having one die each. On each turn, the two players with a die roll it.
If the player rolls 1, then the die passes to the neighbour on the left.
If the player rolls 6, then... | <p><dfn>The Chase</dfn> is a game played with two dice and an even number of players.</p>
<p>The players sit around a table and the game begins with two opposite players having one die each. On each turn, the two players with a die roll it.</p>
<p>If the player rolls 1, then the die passes to the neighbour on the left.... | 3780.618622 | Saturday, 10th January 2009, 01:00 am | 2354 | 65% | hard |
111 | Primes with Runs | Considering $4$-digit primes containing repeated digits it is clear that they cannot all be the same: $1111$ is divisible by $11$, $2222$ is divisible by $22$, and so on. But there are nine $4$-digit primes containing three ones:
$$1117, 1151, 1171, 1181, 1511, 1811, 2111, 4111, 8111.$$
We shall say that $M(n, d)$ repr... | Considering $4$-digit primes containing repeated digits it is clear that they cannot all be the same: $1111$ is divisible by $11$, $2222$ is divisible by $22$, and so on. But there are nine $4$-digit primes containing three ones:
$$1117, 1151, 1171, 1181, 1511, 1811, 2111, 4111, 8111.$$
We shall say that $M(n, d)$ repr... | <p>Considering $4$-digit primes containing repeated digits it is clear that they cannot all be the same: $1111$ is divisible by $11$, $2222$ is divisible by $22$, and so on. But there are nine $4$-digit primes containing three ones:
$$1117, 1151, 1171, 1181, 1511, 1811, 2111, 4111, 8111.$$</p>
<p>We shall say that $M(n... | 612407567715 | Friday, 16th December 2005, 06:00 pm | 8210 | 45% | medium |
617 | Mirror Power Sequence | For two integers $n,e \gt 1$, we define an $(n,e)$-MPS (Mirror Power Sequence) to be an infinite sequence of integers $(a_i)_{i\ge 0}$ such that for all $i\ge 0$, $a_{i+1} = \min(a_i^e,n-a_i^e)$ and $a_i \gt 1$. Examples of such sequences are the two $(18,2)$-MPS sequences made of alternating $2$ and $4$.
Note that eve... | For two integers $n,e \gt 1$, we define an $(n,e)$-MPS (Mirror Power Sequence) to be an infinite sequence of integers $(a_i)_{i\ge 0}$ such that for all $i\ge 0$, $a_{i+1} = \min(a_i^e,n-a_i^e)$ and $a_i \gt 1$. Examples of such sequences are the two $(18,2)$-MPS sequences made of alternating $2$ and $4$.
Note that eve... | <p>For two integers $n,e \gt 1$, we define an <dfn>$(n,e)$-MPS (Mirror Power Sequence)</dfn> to be an infinite sequence of integers $(a_i)_{i\ge 0}$ such that for all $i\ge 0$, $a_{i+1} = \min(a_i^e,n-a_i^e)$ and $a_i \gt 1$.<br/> Examples of such sequences are the two $(18,2)$-MPS sequences made of alternating $2$ and... | 1001133757 | Saturday, 30th December 2017, 04:00 pm | 407 | 40% | medium |
614 | Special Partitions 2 | An integer partition of a number $n$ is a way of writing $n$ as a sum of positive integers. Partitions that differ only by the order of their summands are considered the same.
We call an integer partition special if 1) all its summands are distinct, and 2) all its even summands are also divisible by $4$.For example, th... | An integer partition of a number $n$ is a way of writing $n$ as a sum of positive integers. Partitions that differ only by the order of their summands are considered the same.
We call an integer partition special if 1) all its summands are distinct, and 2) all its even summands are also divisible by $4$.For example, th... | <p>An <strong>integer partition</strong> of a number $n$ is a way of writing $n$ as a sum of positive integers. Partitions that differ only by the order of their summands are considered the same.</p>
<p>We call an integer partition <dfn>special</dfn> if 1) all its summands are distinct, and 2) all its even summands are... | 130694090 | Sunday, 19th November 2017, 07:00 am | 331 | 55% | medium |
903 | Total Permutation Powers | A permutation $\pi$ of $\{1, \dots, n\}$ can be represented in one-line notation as $\pi(1),\ldots,\pi(n) $. If all $n!$ permutations are written in lexicographic order then $\textrm{rank}(\pi)$ is the position of $\pi$ in this 1-based list.
For example, $\text{rank}(2,1,3) = 3$ because the six permutations of $\{1, 2,... | A permutation $\pi$ of $\{1, \dots, n\}$ can be represented in one-line notation as $\pi(1),\ldots,\pi(n) $. If all $n!$ permutations are written in lexicographic order then $\textrm{rank}(\pi)$ is the position of $\pi$ in this 1-based list.
For example, $\text{rank}(2,1,3) = 3$ because the six permutations of $\{1, 2,... | <p>A permutation $\pi$ of $\{1, \dots, n\}$ can be represented in <b>one-line notation</b> as $\pi(1),\ldots,\pi(n) $. If all $n!$ permutations are written in lexicographic order then $\textrm{rank}(\pi)$ is the position of $\pi$ in this 1-based list.</p>
<p>For example, $\text{rank}(2,1,3) = 3$ because the six permuta... | 128553191 | Sunday, 28th July 2024, 11:00 am | 118 | 90% | hard |
664 | An Infinite Game | Peter is playing a solitaire game on an infinite checkerboard, each square of which can hold an unlimited number of tokens.
Each move of the game consists of the following steps:
Choose one token $T$ to move. This may be any token on the board, as long as not all of its four adjacent squares are empty.
Select and disc... | Peter is playing a solitaire game on an infinite checkerboard, each square of which can hold an unlimited number of tokens.
Each move of the game consists of the following steps:
Choose one token $T$ to move. This may be any token on the board, as long as not all of its four adjacent squares are empty.
Select and disc... | <p>Peter is playing a solitaire game on an infinite checkerboard, each square of which can hold an unlimited number of tokens.</p>
<p>Each move of the game consists of the following steps:</p>
<ol>
<li>Choose one token $T$ to move. This may be any token on the board, as long as not all of its four adjacent squares are ... | 35295862 | Sunday, 7th April 2019, 07:00 am | 212 | 65% | hard |
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