Found problems: 396
2006 China Team Selection Test, 1
Let $k$ be an odd number that is greater than or equal to $3$. Prove that there exists a $k^{th}$-degree integer-valued polynomial with non-integer-coefficients that has the following properties:
(1) $f(0)=0$ and $f(1)=1$; and.
(2) There exist infinitely many positive integers $n$ so that if the following equation: \[ n= f(x_1)+\cdots+f(x_s), \] has integer solutions $x_1, x_2, \dots, x_s$, then $s \geq 2^k-1$.
2021 IMO Shortlist, C2
Let $n\ge 3$ be a fixed integer. There are $m\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\emph{not}$ possible.
[i]Carl Schildkraut, USA[/i]
1998 Poland - Second Round, 5
Let $a_1,a_2,\ldots,a_7, b_1,b_2,\ldots,b_7\geq 0$ be real numbers satisfying $a_i+b_i\le 2$ for all $i=\overline{1,7}$.
Prove that there exist $k\ne m$ such that $|a_k-a_m|+|b_k-b_m|\le 1$.
Thanks for show me the mistake typing
1993 India National Olympiad, 7
Let $A = \{ 1,2, 3 , \ldots, 100 \}$ and $B$ be a subset of $A$ having $53$ elements. Show that $B$ has 2 distinct elements $x$ and $y$ whose sum is divisible by $11$.
2006 China Team Selection Test, 3
$k$ and $n$ are positive integers that are greater than $1$. $N$ is the set of positive integers. $A_1, A_2, \cdots A_k$ are pairwise not-intersecting subsets of $N$ and $A_1 \cup A_2 \cup \cdots \cup A_k = N$.
Prove that for some $i \in \{ 1,2,\cdots,k \}$, there exsits infinity many non-factorable n-th degree polynomials so that coefficients of one polynomial are pairwise distinct and all the coeficients are in $A_i$.
2012 China Team Selection Test, 2
Prove that there exists a positive real number $C$ with the following property: for any integer $n\ge 2$ and any subset $X$ of the set $\{1,2,\ldots,n\}$ such that $|X|\ge 2$, there exist $x,y,z,w \in X$(not necessarily distinct) such that
\[0<|xy-zw|<C\alpha ^{-4}\]
where $\alpha =\frac{|X|}{n}$.
2012 Math Prize for Girls Olympiad, 4
Let $f$ be a function from the set of rational numbers to the set of real numbers. Suppose that for all rational numbers $r$ and $s$, the expression $f(r + s) - f(r) - f(s)$ is an integer. Prove that there is a positive integer $q$ and an integer $p$ such that
\[
\Bigl\lvert f\Bigl(\frac{1}{q}\Bigr) - p \Bigr\rvert \le \frac{1}{2012} \, .
\]
1985 IMO Longlists, 49
Given a set $M$ of $1985$ positive integers, none of which has a prime divisor larger than $26$, prove that the set has four distinct elements whose geometric mean is an integer.
2024 Auckland Mathematical Olympiad, 10
Prove that circles constructed on the sides of a convex quadrilateral as diameters completely cover this quadrilateral.
2000 Kurschak Competition, 3
Let $k\ge 0$ be an integer and suppose the integers $a_1,a_2,\dots,a_n$ give at least $2k$ different residues upon division by $(n+k)$. Show that there are some $a_i$ whose sum is divisible by $n+k$.
2013 USAMTS Problems, 4
An infinite sequence $(a_0,a_1,a_2,\dots)$ of positive integers is called a $\emph{ribbon}$ if the sum of any eight consecutive terms is at most $16$; that is, for all $i\ge0$,
\[a_i+a_{i+1}+\dots+a_{i+7}\le16.\]A positive integer $m$ is called a $\emph{cut size}$ if every ribbon contains a set of consecutive elements that sum to $m$; that is, given any ribbon $(a_0,a_1,a_2,\dots)$, there exist nonnegative integers $k\le l$ such that
\[a_k+a_{k+1}+\dots+a_l=m.\]Find, with proof, all cut sizes, or prove that none exist.
1993 All-Russian Olympiad, 3
What is the maximum number of checkers it is possible to put on a $ n \times n$ chessboard such that in every row and in every column there is an even number of checkers?
2016 Bundeswettbewerb Mathematik, 4
There are $33$ children in a given class. Each child writes a number on the blackboard, which indicates how many other children possess the same forename as oneself. Afterwards, each child does the same thing with their surname. After they've finished, each of the numbers $0,1,2,\dots,10$ appear at least once on the blackboard.
Prove that there are at least two children in this class that have the same forename and surname.
2010 Romania Team Selection Test, 1
Given an integer number $n \geq 3$, consider $n$ distinct points on a circle, labelled $1$ through $n$.
Determine the maximum number of closed chords $[ij]$, $i \neq j$, having pairwise non-empty intersections.
[i]János Pach[/i]
2009 China Western Mathematical Olympiad, 3
A total of $n$ people compete in a mathematical match which contains $15$ problems where $n>12$. For each problem, $1$ point is given for a right answer and $0$ is given for a wrong answer. Analysing each possible situation, we find that if the sum of points every group of $12$ people get is no less than $36$, then there are at least $3$ people that got the right answer of a certain problem, among the $n$ people. Find the least possible $n$.
2013 IMO Shortlist, A2
Prove that in any set of $2000$ distinct real numbers there exist two pairs $a>b$ and $c>d$ with $a \neq c$ or $b \neq d $, such that \[ \left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}. \]
2000 Cono Sur Olympiad, 2
The numbers $1,2,\ldots,64$ are written in the squares of an $8\times 8$ chessboard, one number to each square. Then $2\times 2$ tiles are placed on the chessboard (without overlapping) so that each tile covers exactly four squares whose numbers sum to less than $100$. Find, with proof, the maximum number of tiles that can be placed on the chessboard, and give an example of a distribution of the numbers $1,2,\ldots,64$ into the squares of the chessboard that admits this maximum number of tiles.
2003 Romania Team Selection Test, 10
Let $\mathcal{P}$ be the set of all primes, and let $M$ be a subset of $\mathcal{P}$, having at least three elements, and such that for any proper subset $A$ of $M$ all of the prime factors of the number $ -1+\prod_{p\in A}p$ are found in $M$. Prove that $M= \mathcal{P}$.
[i]Valentin Vornicu[/i]
2019 Bangladesh Mathematical Olympiad, 8
The set of natural numbers $\mathbb{N}$ are partitioned into a finite number of subsets.Prove that there exists a subset of $S$ so that for any natural numbers $n$,there are infinitely many multiples of $n$ in $S$.
2004 Romania National Olympiad, 4
Let $\mathcal U = \left\{ \left( x,y \right) | x,y \in \mathbb Z, \ 0 \leq x,y < 4 \right\}$.
(a) Prove that we can choose $6$ points from $\mathcal U$ such that there are no isosceles triangles with the vertices among the chosen points.
(b) Prove that no matter how we choose $7$ points from $\mathcal U$, there are always three which form an isosceles triangle.
[i]Radu Gologan, Dinu Serbanescu[/i]
2015 Kosovo Team Selection Test, 4
Let $P_1,P_2,...,P_{2556}$ be distinct points inside a regular hexagon $ABCDEF$ of side $1$. If any three points from the set $S=\{A,B,C,D,E,F,P_1,P_2...,P_{2556}\}$ aren't collinear, prove that there exists a triangle with area smaller than $\frac{1}{1700}$, with vertices from the set $S$.
2013 Putnam, 4
A finite collection of digits $0$ and $1$ is written around a circle. An [i]arc[/i] of length $L\ge 0$ consists of $L$ consecutive digits around the circle. For each arc $w,$ let $Z(w)$ and $N(w)$ denote the number of $0$'s in $w$ and the number of $1$'s in $w,$ respectively. Assume that $|Z(w)-Z(w')|\le 1$ for any two arcs $w,w'$ of the same length. Suppose that some arcs $w_1,\dots,w_k$ have the property that \[Z=\frac1k\sum_{j=1}^kZ(w_j)\text{ and }N=\frac1k\sum_{j=1}^k N(w_j)\] are both integers. Prove that there exists an arc $w$ with $Z(w)=Z$ and $N(w)=N.$
2003 USA Team Selection Test, 1
For a pair of integers $a$ and $b$, with $0 < a < b < 1000$, set $S\subseteq \{ 1, 2, \dots , 2003\}$ is called a [i]skipping set[/i] for $(a, b)$ if for any pair of elements $s_1, s_2 \in S$, $|s_1 - s_2|\not\in \{ a, b\}$. Let $f(a, b)$ be the maximum size of a skipping set for $(a, b)$. Determine the maximum and minimum values of $f$.
2004 Baltic Way, 9
A set $S$ of $n-1$ natural numbers is given ($n\ge 3$). There exist at least at least two elements in this set whose difference is not divisible by $n$. Prove that it is possible to choose a non-empty subset of $S$ so that the sum of its elements is divisible by $n$.
2014 China National Olympiad, 3
For non-empty number sets $S, T$, define the sets $S+T=\{s+t\mid s\in S, t\in T\}$ and $2S=\{2s\mid s\in S\}$.
Let $n$ be a positive integer, and $A, B$ be two non-empty subsets of $\{1,2\ldots,n\}$. Show that there exists a subset $D$ of $A+B$ such that
1) $D+D\subseteq 2(A+B)$,
2) $|D|\geq\frac{|A|\cdot|B|}{2n}$,
where $|X|$ is the number of elements of the finite set $X$.