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"content": "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 positive integers not exceeding 20 , and she tells him back the set $\\left\\{a_{k}: k \\in S\\right\\}$ without spelling out which number corresponds to which index. How many moves does Sergey need to determine for sure the number Xenia thought of?",
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2
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[
{
"content": "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 than or equal to $\\mu$.",
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$\frac{1}{2 n+2}$
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[
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"content": "\nFind (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.",
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$2^{1009}$
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[
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"content": "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\n\n\n\n$$\n\nP\\left(x_{1}\\right)+P\\left(x_{2}\\right)+\\cdots+P\\left(x_{k}\\right)=P\\left(x_{k+1}\\right) .\n\n$$\n\n\nNote. A polynomial is monic if the coefficient of the highest power is one.",
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2
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[
{
"content": "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 of proper non-empty subsets of $X$, no non-empty subcollection of which is tight.\n\n\n\nNote. A subset $A$ of $X$ is proper if $A \\neq X$. The sets in a collection are assumed to be distinct. The whole collection is assumed to be a subcollection.",
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$2n-2$
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[
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"content": "Determine all prime numbers $p$ and all positive integers $x$ and $y$ satisfying $x^{3}+y^{3}=$ $p(x y+p)$.",
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$(1,8,19), (2,7,13), (4,5,7)$
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[
{
"content": "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 a regular $2 n$-gon in some order, or are all equal.\n\n\n\nDetermine the smallest possible degree of $f$.",
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$2n$
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[
{
"content": "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.",
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2
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[
{
"content": "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 $\\overrightarrow{C D}$ such that $A B C D$ is a convex quadrangle oriented clockwise. Determine the number of good configurations.",
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$\binom{2n}{n}$
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[
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"content": "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:\n\n\n\n(i) each domino covers exactly two adjacent cells of the board;\n\n\n\n(ii) no two dominoes overlap;\n\n\n\n(iii) no two form a $2 \\times 2$ square; and\n\n\n\n(iv) the bottom row of the board is completely covered by $n$ dominoes.",
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$m n-\lfloor m / 2\rfloor$
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[
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"content": "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.\n\nDetermine the possible values of $a_{2015} \\cdot a_{2016}$ for a cubic sequence satisfying the condition in part (a).",
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0
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[
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"content": "Find all functions $f: \\mathbb{R}^{+} \\rightarrow \\mathbb{R}^{+}$ such that\n\n$$\nf(x+f(y))=f(x+y)+f(y)\\tag{1}\n$$\n\nfor all $x, y \\in \\mathbb{R}^{+}$. (Symbol $\\mathbb{R}^{+}$denotes the set of all positive real numbers.)",
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$f(x)=2 x$
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[
{
"content": "Let $n>1$ be an integer. In the space, consider the set\n$$\nS=\\{(x, y, z) \\mid x, y, z \\in\\{0,1, \\ldots, n\\}, x+y+z>0\\}\n$$\nFind the smallest number of planes that jointly contain all $(n+1)^{3}-1$ points of $S$ but none of them passes through the origin.",
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$3 n$
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"content": "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 by $n$.",
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$69$,$84$
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[
{
"content": "Determine the smallest positive real number $k$ with the following property.\n\nLet $A B C D$ 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 $D D_{1} C_{1}$; let $S$ be the sum of the two smallest ones, and let $S_{1}$ be the area of quadrilateral $A_{1} B_{1} C_{1} D_{1}$. Then we always have $k S_{1} \\geq S$.",
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$k=1$
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[
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"content": "Find all pairs $(k, n)$ of positive integers for which $7^{k}-3^{n}$ divides $k^{4}+n^{2}$.",
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$(2,4)$
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[
{
"content": "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 $p$.\n\n( $\\mathbb{N}$ is the set of all positive integers.)",
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$f(n)=n$
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{
"content": "Determine all pairs $(f, g)$ of functions from the set of positive integers to itself that satisfy\n\n$$\nf^{g(n)+1}(n)+g^{f(n)}(n)=f(n+1)-g(n+1)+1\n$$\n\nfor every positive integer $n$. Here, $f^{k}(n)$ means $\\underbrace{f(f(\\ldots f}_{k}(n) \\ldots))$.",
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$f(n)=n$, $g(n)=1$
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[
{
"content": "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 $n$.",
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3
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[
{
"content": "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 $90^{\\circ}$ clockwise and continue moving with speed 1 . When more than two ants meet, or when two ants moving in perpendicular directions meet, the ants continue moving in the same direction as before they met. When an ant reaches one of the edges of the checkerboard, it falls off and will not re-appear.\n\nConsidering all possible starting positions, determine the latest possible moment at which the last ant falls off the checkerboard or prove that such a moment does not necessarily exist.",
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] |
$\frac{3 m}{2}-1$
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