id int64 1.61k 3.1k | question stringlengths 49 4.42k | output stringlengths 19 12.3k | answer stringlengths 5 224 | subject stringclasses 1
value | subfield stringclasses 4
values | difficulty stringclasses 4
values |
|---|---|---|---|---|---|---|
1,606 | Xenia and Sergey play the following game. Xenia thinks of a positive integer $N$ not exceeding 5000. Then she fixes 20 distinct positive integers $a_{1}, a_{2}, \ldots, a_{20}$ such that, for each $k=1,2, \ldots, 20$, the numbers $N$ and $a_{k}$ are congruent modulo $k$. By a move, Sergey tells Xenia a set $S$ of posit... | ["Sergey can determine Xenia's number in 2 but not fewer moves.\n\n\n\nWe first show that 2 moves are sufficient. Let Sergey provide the set $\\{17,18\\}$ on his first move, and the set $\\{18,19\\}$ on the second move. In Xenia's two responses, exactly one number occurs twice, namely, $a_{18}$. Thus, Sergey is able to... | ['2'] | Math | Combinatorics | Advanced |
1,610 | Given a positive integer $n$, determine the largest real number $\mu$ satisfying the following condition: for every $4 n$-point configuration $C$ in an open unit square $U$, there exists an open rectangle in $U$, whose sides are parallel to those of $U$, which contains exactly one point of $C$, and has an area greater ... | ['The required maximum is $\\frac{1}{2 n+2}$. To show that the condition in the statement is not met if $\\mu>\\frac{1}{2 n+2}$, let $U=(0,1) \\times(0,1)$, choose a small enough positive $\\epsilon$, and consider the configuration $C$ consisting of the $n$ four-element clusters of points $\\left(\\frac{i}{n+1} \\pm \\... | ['$\\frac{1}{2 n+2}$'] | Math | Algebra | Advanced |
1,612 |
Find (in closed form) the difference between the number of positive integers at most $2^{2017}$ with even weight and the number of positive integers at most $2^{2017}$ with odd weight. | ['For every integer $M \\geq 0$, let $A_{M}=\\sum_{n=-2^{M}+1}^{0}(-1)^{w(n)}$ and let $B_{M}=$ $\\sum_{n=1}^{2^{M}}(-1)^{w(n)}$; thus, $B_{M}$ evaluates the difference of the number of even weight integers in the range 1 through $2^{M}$ and the number of odd weight integers in that range.\n\n\n\nNotice that\n\n\n\n$$\... | ['$2^{1009}$'] | Math | Number Theory | Advanced |
1,613 | Determine all positive integers $n$ satisfying the following condition: for every monic polynomial $P$ of degree at most $n$ with integer coefficients, there exists a positive integer $k \leq n$, and $k+1$ distinct integers $x_{1}, x_{2}, \ldots, x_{k+1}$ such that
$$
P\left(x_{1}\right)+P\left(x_{2}\right)+\cdots+... | ['There is only one such integer, namely, $n=2$. In this case, if $P$ is a constant polynomial, the required condition is clearly satisfied; if $P=X+c$, then $P(c-1)+P(c+1)=$ $P(3 c)$; and if $P=X^{2}+q X+r$, then $P(X)=P(-X-q)$.\n\n\n\nTo rule out all other values of $n$, it is sufficient to exhibit a monic polynomial... | ['2'] | Math | Algebra | Advanced |
1,614 | Let $n$ be an integer greater than 1 and let $X$ be an $n$-element set. A non-empty collection of subsets $A_{1}, \ldots, A_{k}$ of $X$ is tight if the union $A_{1} \cup \cdots \cup A_{k}$ is a proper subset of $X$ and no element of $X$ lies in exactly one of the $A_{i}$ s. Find the largest cardinality of a collection ... | ["The required maximum is $2 n-2$. To describe a $(2 n-2)$-element collection satisfying the required conditions, write $X=\\{1,2, \\ldots, n\\}$ and set $B_{k}=\\{1,2, \\ldots, k\\}$, $k=1,2, \\ldots, n-1$, and $B_{k}=\\{k-n+2, k-n+3, \\ldots, n\\}, k=n, n+1, \\ldots, 2 n-2$. To show that no subcollection of the $B_{k... | ['$2n-2$'] | Math | Combinatorics | Advanced |
1,618 | Determine all prime numbers $p$ and all positive integers $x$ and $y$ satisfying $x^{3}+y^{3}=$ $p(x y+p)$. | ['Up to a swap of the first two entries, the only solutions are $(x, y, p)=(1,8,19)$, $(x, y, p)=(2,7,13)$ and $(x, y, p)=(4,5,7)$. The verification is routine.\n\n\n\nSet $s=x+y$. Rewrite the equation in the form $s\\left(s^{2}-3 x y\\right)=p(p+x y)$, and express $x y$ :\n\n\n\n$$\n\nx y=\\frac{s^{3}-p^{2}}{3 s+p} \\... | ['$(1,8,19), (2,7,13), (4,5,7)$'] | Math | Number Theory | Advanced |
1,620 | Let $n \geqslant 2$ be an integer, and let $f$ be a $4 n$-variable polynomial with real coefficients. Assume that, for any $2 n$ points $\left(x_{1}, y_{1}\right), \ldots,\left(x_{2 n}, y_{2 n}\right)$ in the plane, $f\left(x_{1}, y_{1}, \ldots, x_{2 n}, y_{2 n}\right)=0$ if and only if the points form the vertices of ... | ["The smallest possible degree is $2 n$. In what follows, we will frequently write $A_{i}=$ $\\left(x_{i}, y_{i}\\right)$, and abbreviate $P\\left(x_{1}, y_{1}, \\ldots, x_{2 n}, y_{2 n}\\right)$ to $P\\left(A_{1}, \\ldots, A_{2 n}\\right)$ or as a function of any $2 n$ points.\n\n\n\nSuppose that $f$ is valid. First, ... | ['$2n$'] | Math | Algebra | Advanced |
1,631 | For a positive integer $a$, define a sequence of integers $x_{1}, x_{2}, \ldots$ by letting $x_{1}=a$ and $x_{n+1}=2 x_{n}+1$ for $n \geq 1$. Let $y_{n}=2^{x_{n}}-1$. Determine the largest possible $k$ such that, for some positive integer $a$, the numbers $y_{1}, \ldots, y_{k}$ are all prime. | ['The largest such is $k=2$. Notice first that if $y_{i}$ is prime, then $x_{i}$ is prime as well. Actually, if $x_{i}=1$ then $y_{i}=1$ which is not prime, and if $x_{i}=m n$ for integer $m, n>1$ then $2^{m}-1 \\mid 2^{x_{i}}-1=y_{i}$, so $y_{i}$ is composite. In particular, if $y_{1}, y_{2}, \\ldots, y_{k}$ are prime... | ['2'] | Math | Algebra | Advanced |
1,641 | Let $n$ be a positive integer and fix $2 n$ distinct points on a circumference. Split these points into $n$ pairs and join the points in each pair by an arrow (i.e., an oriented line segment). The resulting configuration is good if no two arrows cross, and there are no arrows $\overrightarrow{A B}$ and $\overrightarrow... | ['The required number is $\\left(\\begin{array}{c}2 n \\\\ n\\end{array}\\right)$. To prove this, trace the circumference counterclockwise to label the points $a_{1}, a_{2}, \\ldots, a_{2 n}$.\n\nLet $\\mathcal{C}$ be any good configuration and let $O(\\mathcal{C})$ be the set of all points from which arrows emerge. We... | ['$\\binom{2n}{n}$'] | Math | Combinatorics | Advanced |
1,644 | Given positive integers $m$ and $n \geq m$, determine the largest number of dominoes $(1 \times 2$ or $2 \times 1$ rectangles) that can be placed on a rectangular board with $m$ rows and $2 n$ columns consisting of cells $(1 \times 1$ squares $)$ so that:
(i) each domino covers exactly two adjacent cells of the boar... | ['The required maximum is $m n-\\lfloor m / 2\\rfloor$ and is achieved by the brick-like vertically symmetric arrangement of blocks of $n$ and $n-1$ horizontal dominoes placed on alternate rows, so that the bottom row of the board is completely covered by $n$ dominoes.\n\n\n\nTo show that the number of dominoes in an a... | ['$m n-\\lfloor m / 2\\rfloor$'] | Math | Combinatorics | Advanced |
1,645 | A cubic sequence is a sequence of integers given by $a_{n}=n^{3}+b n^{2}+c n+d$, where $b, c$ and $d$ are integer constants and $n$ ranges over all integers, including negative integers.
Determine the possible values of $a_{2015} \cdot a_{2016}$ for a cubic sequence satisfying the condition in part (a). | ['The only possible value of $a_{2015} \\cdot a_{2016}$ is 0 . For simplicity, by performing a translation of the sequence (which may change the defining constants $b, c$ and $d$ ), we may instead concern ourselves with the values $a_{0}$ and $a_{1}$, rather than $a_{2015}$ and $a_{2016}$.\n\n\n\nSuppose now that we ha... | ['0'] | Math | Algebra | Advanced |
1,659 | Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that
$$
f(x+f(y))=f(x+y)+f(y)\tag{1}
$$
for all $x, y \in \mathbb{R}^{+}$. (Symbol $\mathbb{R}^{+}$denotes the set of all positive real numbers.) | ['First we show that $f(y)>y$ for all $y \\in \\mathbb{R}^{+}$. Functional equation (1) yields $f(x+f(y))>f(x+y)$ and hence $f(y) \\neq y$ immediately. If $f(y)<y$ for some $y$, then setting $x=y-f(y)$ we get\n\n$$\nf(y)=f((y-f(y))+f(y))=f((y-f(y))+y)+f(y)>f(y),\n$$\n\ncontradiction. Therefore $f(y)>y$ for all $y \\in ... | ['$f(x)=2 x$'] | Math | Algebra | Advanced |
1,662 | Let $n>1$ be an integer. In the space, consider the set
$$
S=\{(x, y, z) \mid x, y, z \in\{0,1, \ldots, n\}, x+y+z>0\}
$$
Find the smallest number of planes that jointly contain all $(n+1)^{3}-1$ points of $S$ but none of them passes through the origin. | ['It is easy to find $3 n$ such planes. For example, planes $x=i, y=i$ or $z=i$ $(i=1,2, \\ldots, n)$ cover the set $S$ but none of them contains the origin. Another such collection consists of all planes $x+y+z=k$ for $k=1,2, \\ldots, 3 n$.\n\nWe show that $3 n$ is the smallest possible number.\n\nLemma 1. Consider a ... | ['$3 n$'] | Math | Combinatorics | Advanced |
1,664 | Find all positive integers $n$, for which the numbers in the set $S=\{1,2, \ldots, n\}$ can be colored red and blue, with the following condition being satisfied: the set $S \times S \times S$ contains exactly 2007 ordered triples $(x, y, z)$ such that (i) $x, y, z$ are of the same color and (ii) $x+y+z$ is divisible b... | ['Suppose that the numbers $1,2, \\ldots, n$ are colored red and blue. Denote by $R$ and $B$ the sets of red and blue numbers, respectively; let $|R|=r$ and $|B|=b=n-r$. Call a triple $(x, y, z) \\in S \\times S \\times S$ monochromatic if $x, y, z$ have the same color, and bichromatic otherwise. Call a triple $(x, y, ... | ['$69$,$84$'] | Math | Combinatorics | Advanced |
1,675 | Determine the smallest positive real number $k$ with the following property.
Let $A B C D$ be a convex quadrilateral, and let points $A_{1}, B_{1}, C_{1}$ and $D_{1}$ lie on sides $A B, B C$, $C D$ and $D A$, respectively. Consider the areas of triangles $A A_{1} D_{1}, B B_{1} A_{1}, C C_{1} B_{1}$, and $D D_{1} C_{1... | ['Throughout the solution, triangles $A A_{1} D_{1}, B B_{1} A_{1}, C C_{1} B_{1}$, and $D D_{1} C_{1}$ will be referred to as border triangles. We will denote by $[\\mathcal{R}]$ the area of a region $\\mathcal{R}$.\n\nFirst, we show that $k \\geq 1$. Consider a triangle $A B C$ with unit area; let $A_{1}, B_{1}, K$ b... | ['$k=1$'] | Math | Geometry | Advanced |
1,678 | Find all pairs $(k, n)$ of positive integers for which $7^{k}-3^{n}$ divides $k^{4}+n^{2}$. | ['Suppose that a pair $(k, n)$ satisfies the condition of the problem. Since $7^{k}-3^{n}$ is even, $k^{4}+n^{2}$ is also even, hence $k$ and $n$ have the same parity. If $k$ and $n$ are odd, then $k^{4}+n^{2} \\equiv 1+1=2(\\bmod 4)$, while $7^{k}-3^{n} \\equiv 7-3 \\equiv 0(\\bmod 4)$, so $k^{4}+n^{2}$ cannot be divi... | ['$(2,4)$'] | Math | Number Theory | Advanced |
1,681 | Find all surjective functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for every $m, n \in \mathbb{N}$ and every prime $p$, the number $f(m+n)$ is divisible by $p$ if and only if $f(m)+f(n)$ is divisible by $p$.
( $\mathbb{N}$ is the set of all positive integers.) | ["Suppose that function $f: \\mathbb{N} \\rightarrow \\mathbb{N}$ satisfies the problem conditions.\n\nLemma. For any prime $p$ and any $x, y \\in \\mathbb{N}$, we have $x \\equiv y(\\bmod p)$ if and only if $f(x) \\equiv f(y)$ $(\\bmod p)$. Moreover, $p \\mid f(x)$ if and only if $p \\mid x$.\n\nProof. Consider an arb... | ['$f(n)=n$'] | Math | Number Theory | Advanced |
1,687 | Determine all pairs $(f, g)$ of functions from the set of positive integers to itself that satisfy
$$
f^{g(n)+1}(n)+g^{f(n)}(n)=f(n+1)-g(n+1)+1
$$
for every positive integer $n$. Here, $f^{k}(n)$ means $\underbrace{f(f(\ldots f}_{k}(n) \ldots))$. | ['The given relation implies\n\n$$\nf\\left(f^{g(n)}(n)\\right)<f(n+1) \\quad \\text { for all } n\n\\tag{1}\n$$\n\nwhich will turn out to be sufficient to determine $f$.\n\nLet $y_{1}<y_{2}<\\ldots$ be all the values attained by $f$ (this sequence might be either finite or infinite). We will prove that for every posit... | ['$f(n)=n$, $g(n)=1$'] | Math | Algebra | Advanced |
1,694 | Determine the greatest positive integer $k$ that satisfies the following property: The set of positive integers can be partitioned into $k$ subsets $A_{1}, A_{2}, \ldots, A_{k}$ such that for all integers $n \geq 15$ and all $i \in\{1,2, \ldots, k\}$ there exist two distinct elements of $A_{i}$ whose sum is $n$. | ['There are various examples showing that $k=3$ does indeed have the property under consideration. E.g. one can take\n\n$$\n\\begin{gathered}\nA_{1}=\\{1,2,3\\} \\cup\\{3 m \\mid m \\geq 4\\} \\\\\nA_{2}=\\{4,5,6\\} \\cup\\{3 m-1 \\mid m \\geq 4\\} \\\\\nA_{3}=\\{7,8,9\\} \\cup\\{3 m-2 \\mid m \\geq 4\\}\n\\end{gathere... | ['3'] | Math | Combinatorics | Advanced |
1,695 | Let $m$ be a positive integer and consider a checkerboard consisting of $m$ by $m$ unit squares. At the midpoints of some of these unit squares there is an ant. At time 0, each ant starts moving with speed 1 parallel to some edge of the checkerboard. When two ants moving in opposite directions meet, they both turn $90^... | ['For $m=1$ the answer is clearly correct, so assume $m>1$. In the sequel, the word collision will be used to denote meeting of exactly two ants, moving in opposite directions.\n\nIf at the beginning we place an ant on the southwest corner square facing east and an ant on the southeast corner square facing west, then t... | ['$\\frac{3 m}{2}-1$'] | Math | Combinatorics | Advanced |
1,697 | On a square table of 2011 by 2011 cells we place a finite number of napkins that each cover a square of 52 by 52 cells. In each cell we write the number of napkins covering it, and we record the maximal number $k$ of cells that all contain the same nonzero number. Considering all possible napkin configurations, what is... | ['Let $m=39$, then $2011=52 m-17$. We begin with an example showing that there can exist 3986729 cells carrying the same positive number.\n\n<img_3188>\n\nTo describe it, we number the columns from the left to the right and the rows from the bottom to the top by $1,2, \\ldots, 2011$. We will denote each napkin by the c... | ['3986729'] | Math | Combinatorics | Advanced |
1,709 | For each positive integer $k$, let $t(k)$ be the largest odd divisor of $k$. Determine all positive integers $a$ for which there exists a positive integer $n$ such that all the differences
$$
t(n+a)-t(n), \quad t(n+a+1)-t(n+1), \quad \ldots, \quad t(n+2 a-1)-t(n+a-1)
$$
are divisible by 4 . | ['A pair $(a, n)$ satisfying the condition of the problem will be called a winning pair. It is straightforward to check that the pairs $(1,1),(3,1)$, and $(5,4)$ are winning pairs.\n\nNow suppose that $a$ is a positive integer not equal to 1,3 , and 5 . We will show that there are no winning pairs $(a, n)$ by distingui... | ['1,3,5'] | Math | Number Theory | Advanced |
1,716 | Let $x_{1}, \ldots, x_{100}$ be nonnegative real numbers such that $x_{i}+x_{i+1}+x_{i+2} \leq 1$ for all $i=1, \ldots, 100$ (we put $x_{101}=x_{1}, x_{102}=x_{2}$ ). Find the maximal possible value of the sum
$$
S=\sum_{i=1}^{100} x_{i} x_{i+2}
$$ | ["Let $x_{2 i}=0, x_{2 i-1}=\\frac{1}{2}$ for all $i=1, \\ldots, 50$. Then we have $S=50 \\cdot\\left(\\frac{1}{2}\\right)^{2}=\\frac{25}{2}$. So, we are left to show that $S \\leq \\frac{25}{2}$ for all values of $x_{i}$ 's satisfying the problem conditions.\n\nConsider any $1 \\leq i \\leq 50$. By the problem conditi... | ['$\\frac{25}{2}$'] | Math | Algebra | Advanced |
1,718 | Denote by $\mathbb{Q}^{+}$the set of all positive rational numbers. Determine all functions $f: \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}$ which satisfy the following equation for all $x, y \in \mathbb{Q}^{+}$:
$$
f\left(f(x)^{2} y\right)=x^{3} f(x y)
\tag{1}
$$ | ['By substituting $y=1$, we get\n\n$$\nf\\left(f(x)^{2}\\right)=x^{3} f(x)\\tag{2}\n$$\n\nThen, whenever $f(x)=f(y)$, we have\n\n$$\nx^{3}=\\frac{f\\left(f(x)^{2}\\right)}{f(x)}=\\frac{f\\left(f(y)^{2}\\right)}{f(y)}=y^{3}\n$$\n\nwhich implies $x=y$, so the function $f$ is injective.\n\nNow replace $x$ by $x y$ in (2),... | ['$f(x)=\\frac{1}{x}$'] | Math | Algebra | Advanced |
1,723 | On some planet, there are $2^{N}$ countries $(N \geq 4)$. Each country has a flag $N$ units wide and one unit high composed of $N$ fields of size $1 \times 1$, each field being either yellow or blue. No two countries have the same flag.
We say that a set of $N$ flags is diverse if these flags can be arranged into an $... | ['When speaking about the diagonal of a square, we will always mean the main diagonal.\n\nLet $M_{N}$ be the smallest positive integer satisfying the problem condition. First, we show that $M_{N}>2^{N-2}$. Consider the collection of all $2^{N-2}$ flags having yellow first squares and blue second ones. Obviously, both c... | ['$M=2^{N-2}+1$'] | Math | Combinatorics | Advanced |
1,724 | 2500 chess kings have to be placed on a $100 \times 100$ chessboard so that
(i) no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex);
(ii) each row and each column contains exactly 25 kings.
Find the number of such arrangements. (Two arrangements differing by rotati... | ["Suppose that we have an arrangement satisfying the problem conditions. Divide the board into $2 \\times 2$ pieces; we call these pieces blocks. Each block can contain not more than one king (otherwise these two kings would attack each other); hence, by the pigeonhole principle each block must contain exactly one king... | ['2'] | Math | Combinatorics | Advanced |
1,736 | Find the least positive integer $n$ for which there exists a set $\left\{s_{1}, s_{2}, \ldots, s_{n}\right\}$ consisting of $n$ distinct positive integers such that
$$
\left(1-\frac{1}{s_{1}}\right)\left(1-\frac{1}{s_{2}}\right) \ldots\left(1-\frac{1}{s_{n}}\right)=\frac{51}{2010}
$$ | ['Suppose that for some $n$ there exist the desired numbers; we may assume that $s_{1}<s_{2}<\\cdots<s_{n}$. Surely $s_{1}>1$ since otherwise $1-\\frac{1}{s_{1}}=0$. So we have $2 \\leq s_{1} \\leq s_{2}-1 \\leq \\cdots \\leq s_{n}-(n-1)$, hence $s_{i} \\geq i+1$ for each $i=1, \\ldots, n$. Therefore\n\n$$\n\\begin{ali... | ['39'] | Math | Number Theory | Advanced |
1,737 | Find all pairs $(m, n)$ of nonnegative integers for which
$$
m^{2}+2 \cdot 3^{n}=m\left(2^{n+1}-1\right)
\tag{1}
$$ | ['For fixed values of $n$, the equation (1) is a simple quadratic equation in $m$. For $n \\leq 5$ the solutions are listed in the following table.\n\n| case | equation | discriminant | integer roots |\n| :--- | :--- | :--- | :--- |\n| $n=0$ | $m^{2}-m+2=0$ | -7 | none |\n| $n=1$ | $m^{2}-3 m+6=0$ | -15 | none |\n| $n=... | ['$(6,3),(9,3),(9,5),(54,5)$'] | Math | Number Theory | Advanced |
1,738 | Find the smallest number $n$ such that there exist polynomials $f_{1}, f_{2}, \ldots, f_{n}$ with rational coefficients satisfying
$$
x^{2}+7=f_{1}(x)^{2}+f_{2}(x)^{2}+\cdots+f_{n}(x)^{2} .
$$ | ['The equality $x^{2}+7=x^{2}+2^{2}+1^{2}+1^{2}+1^{2}$ shows that $n \\leq 5$. It remains to show that $x^{2}+7$ is not a sum of four (or less) squares of polynomials with rational coefficients.\n\nSuppose by way of contradiction that $x^{2}+7=f_{1}(x)^{2}+f_{2}(x)^{2}+f_{3}(x)^{2}+f_{4}(x)^{2}$, where the coefficients... | ['5'] | Math | Number Theory | Advanced |
1,747 | Determine the smallest number $M$ such that the inequality
$$
\left|a b\left(a^{2}-b^{2}\right)+b c\left(b^{2}-c^{2}\right)+c a\left(c^{2}-a^{2}\right)\right| \leq M\left(a^{2}+b^{2}+c^{2}\right)^{2}
$$
holds for all real numbers $a, b, c$. | ['We first consider the cubic polynomial\n\n$$\nP(t)=t b\\left(t^{2}-b^{2}\\right)+b c\\left(b^{2}-c^{2}\\right)+c t\\left(c^{2}-t^{2}\\right) .\n$$\n\nIt is easy to check that $P(b)=P(c)=P(-b-c)=0$, and therefore\n\n$$\nP(t)=(b-c)(t-b)(t-c)(t+b+c)\n$$\n\nsince the cubic coefficient is $b-c$. The left-hand side of the ... | ['$M=\\frac{9}{32} \\sqrt{2}$'] | Math | Algebra | Advanced |
1,750 | A diagonal of a regular 2006-gon is called odd if its endpoints divide the boundary into two parts, each composed of an odd number of sides. Sides are also regarded as odd diagonals.
Suppose the 2006-gon has been dissected into triangles by 2003 nonintersecting diagonals. Find the maximum possible number of isosceles ... | ['Call an isosceles triangle odd if it has two odd sides. Suppose we are given a dissection as in the problem statement. A triangle in the dissection which is odd and isosceles will be called iso-odd for brevity.\n\nLemma. Let $A B$ be one of dissecting diagonals and let $\\mathcal{L}$ be the shorter part of the bounda... | ['$1003$'] | Math | Combinatorics | Advanced |
1,760 | In triangle $A B C$, let $J$ be the centre of the excircle tangent to side $B C$ at $A_{1}$ and to the extensions of sides $A C$ and $A B$ at $B_{1}$ and $C_{1}$, respectively. Suppose that the lines $A_{1} B_{1}$ and $A B$ are perpendicular and intersect at $D$. Let $E$ be the foot of the perpendicular from $C_{1}$ to... | ['Let $K$ be the intersection point of lines $J C$ and $A_{1} B_{1}$. Obviously $J C \\perp A_{1} B_{1}$ and since $A_{1} B_{1} \\perp A B$, the lines $J K$ and $C_{1} D$ are parallel and equal. From the right triangle $B_{1} C J$ we obtain $J C_{1}^{2}=J B_{1}^{2}=J C \\cdot J K=J C \\cdot C_{1} D$ from which we infer... | ['$\\angle B E A_{1}=90$,$\\angle A E B_{1}=90$'] | Math | Geometry | Advanced |
1,766 | Determine all pairs $(x, y)$ of integers satisfying the equation
$$
1+2^{x}+2^{2 x+1}=y^{2}
$$ | ['If $(x, y)$ is a solution then obviously $x \\geq 0$ and $(x,-y)$ is a solution too. For $x=0$ we get the two solutions $(0,2)$ and $(0,-2)$.\n\nNow let $(x, y)$ be a solution with $x>0$; without loss of generality confine attention to $y>0$. The equation rewritten as\n\n$$\n2^{x}\\left(1+2^{x+1}\\right)=(y-1)(y+1)\n... | ['$(0,2),(0,-2),(4,23),(4,-23)$'] | Math | Number Theory | Advanced |
1,775 | Given a positive integer $n$, find the smallest value of $\left\lfloor\frac{a_{1}}{1}\right\rfloor+\left\lfloor\frac{a_{2}}{2}\right\rfloor+\cdots+\left\lfloor\frac{a_{n}}{n}\right\rfloor$ over all permutations $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ of $(1,2, \ldots, n)$. | ['Suppose that $2^{k} \\leqslant n<2^{k+1}$ with some nonnegative integer $k$. First we show a permutation $\\left(a_{1}, a_{2}, \\ldots, a_{n}\\right)$ such that $\\left\\lfloor\\frac{a_{1}}{1}\\right\\rfloor+\\left\\lfloor\\frac{a_{2}}{2}\\right\\rfloor+\\cdots+\\left\\lfloor\\frac{a_{n}}{n}\\right\\rfloor=k+1$; then... | ['$\\left\\lfloor\\log _{2} n\\right\\rfloor+1$'] | Math | Algebra | Advanced |
1,782 | Let $n \geqslant 3$ be an integer. An integer $m \geqslant n+1$ is called $n$-colourful if, given infinitely many marbles in each of $n$ colours $C_{1}, C_{2}, \ldots, C_{n}$, it is possible to place $m$ of them around a circle so that in any group of $n+1$ consecutive marbles there is at least one marble of colour $C_... | ['First suppose that there are $n(n-1)-1$ marbles. Then for one of the colours, say blue, there are at most $n-2$ marbles, which partition the non-blue marbles into at most $n-2$ groups with at least $(n-1)^{2}>n(n-2)$ marbles in total. Thus one of these groups contains at least $n+1$ marbles and this group does not co... | ['$m_{\\max }=n^{2}-n-1$'] | Math | Combinatorics | Advanced |
1,789 | Determine the largest $N$ for which there exists a table $T$ of integers with $N$ rows and 100 columns that has the following properties:
(i) Every row contains the numbers 1,2, ., 100 in some order.
(ii) For any two distinct rows $r$ and $s$, there is a column $c$ such that $|T(r, c)-T(s, c)| \geqslant 2$.
Here $T(... | ['Non-existence of a larger table. Let us consider some fixed row in the table, and let us replace (for $k=1,2, \\ldots, 50$ ) each of two numbers $2 k-1$ and $2 k$ respectively by the symbol $x_{k}$. The resulting pattern is an arrangement of 50 symbols $x_{1}, x_{2}, \\ldots, x_{50}$, where every symbol occurs exactl... | ['$\\frac{100!}{2^{50}}$'] | Math | Combinatorics | Advanced |
1,798 | Determine all integers $n \geqslant 1$ for which there exists a pair of positive integers $(a, b)$ such that no cube of a prime divides $a^{2}+b+3$ and
$$
\frac{a b+3 b+8}{a^{2}+b+3}=n
$$ | ['As $b \\equiv-a^{2}-3\\left(\\bmod a^{2}+b+3\\right)$, the numerator of the given fraction satisfies\n\n$$\na b+3 b+8 \\equiv a\\left(-a^{2}-3\\right)+3\\left(-a^{2}-3\\right)+8 \\equiv-(a+1)^{3} \\quad\\left(\\bmod a^{2}+b+3\\right)\n$$\n\nAs $a^{2}+b+3$ is not divisible by $p^{3}$ for any prime $p$, if $a^{2}+b+3$ ... | ['2'] | Math | Number Theory | Advanced |
1,800 | Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $\left(d_{1}, d_{2}, \ldots, d_{k}\right)$ such that for every $i=1,2, \ldots, k$, the number $d_{1}+\cdots+d_{i}$ is a perfect square. | ['For $i=1,2, \\ldots, k$ let $d_{1}+\\ldots+d_{i}=s_{i}^{2}$, and define $s_{0}=0$ as well. Obviously $0=s_{0}<s_{1}<s_{2}<\\ldots<s_{k}$, so\n\n$$\ns_{i} \\geqslant i \\quad \\text { and } \\quad d_{i}=s_{i}^{2}-s_{i-1}^{2}=\\left(s_{i}+s_{i-1}\\right)\\left(s_{i}-s_{i-1}\\right) \\geqslant s_{i}+s_{i-1} \\geqslant 2... | ['1,3'] | Math | Number Theory | Advanced |
1,807 | Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard:
- In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers on his napkin.
- In the... | ["Call a number $q$ good if every number in the second line appears in the third line unconditionally. We first show that the numbers 0 and \\pm 2 are good. The third line necessarily contains 0 , so 0 is good. For any two numbers $a, b$ in the first line, write $a=x-y$ and $b=u-v$, where $x, y, u, v$ are (not necessar... | ['$-2,0,2$'] | Math | Algebra | Advanced |
1,810 | An integer $n \geqslant 3$ is given. We call an $n$-tuple of real numbers $\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ Shiny if for each permutation $y_{1}, y_{2}, \ldots, y_{n}$ of these numbers we have
$$
\sum_{i=1}^{n-1} y_{i} y_{i+1}=y_{1} y_{2}+y_{2} y_{3}+y_{3} y_{4}+\cdots+y_{n-1} y_{n} \geqslant-1
$$
Find the l... | ['First of all, we show that we may not take a larger constant $K$. Let $t$ be a positive number, and take $x_{2}=x_{3}=\\cdots=t$ and $x_{1}=-1 /(2 t)$. Then, every product $x_{i} x_{j}(i \\neq j)$ is equal to either $t^{2}$ or $-1 / 2$. Hence, for every permutation $y_{i}$ of the $x_{i}$, we have\n\n$$\ny_{1} y_{2}+\... | ['$-(n-1) / 2$'] | Math | Algebra | Advanced |
1,818 | Let $n>1$ be an integer. An $n \times n \times n$ cube is composed of $n^{3}$ unit cubes. Each unit cube is painted with one color. For each $n \times n \times 1$ box consisting of $n^{2}$ unit cubes (of any of the three possible orientations), we consider the set of the colors present in that box (each color is listed... | ['Call a $n \\times n \\times 1$ box an $x$-box, a $y$-box, or a $z$-box, according to the direction of its short side. Let $C$ be the number of colors in a valid configuration. We start with the upper bound for $C$.\n\nLet $\\mathcal{C}_{1}, \\mathcal{C}_{2}$, and $\\mathcal{C}_{3}$ be the sets of colors which appear ... | ['$\\frac{n(n+1)(2 n+1)}{6}$'] | Math | Combinatorics | Advanced |
End of preview. Expand in Data Studio
- Downloads last month
- 6