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IMO Shortlist 1998 problem C7

A solitaire game is played on an $m\times n$ rectangular board, using $mn$ markers which are white on one side and black on the other. Initially, each square of the board contains a marker with its white side up, except for one corner square, which contains a marker with its black side up. In each move, one may take away one marker with its black side up, but must then turn over all markers which are in squares having an edge in common with the square of the removed marker. Determine all pairs $(m,n)$ of positive integers such that all markers can be removed from the board.

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IMO Shortlist 1998 problem C6

Ten points are marked in the plane so that no three of them lie on a line. Each pair of points is connected with a segment. Each of these segments is painted with one of $k$ colors, in such a way that for any $k$ of the ten points, there are $k$ segments each joining two of them and no two being painted with the same color. Determine all integers $k$ , $1\leq k\leq 10$ , for which this is possible.

IMO Shortlist 1999 problem C7

Let $p >3$ be a prime number. For each nonempty subset $T$ of $\{0,1,2,3, \ldots , p-1\}$ , let $E(T)$ be the set of all $(p-1)$ -tuples $(x_1, \ldots ,x_{p-1} )$ , where each $x_i \in T$ and $x_1+2x_2+ \ldots + (p-1)x_{p-1}$ is divisible by $p$ and let $|E(T)|$ denote the number of elements in $E(T)$ . Prove that

$|E(\{0,1,3\})| \geq |E(\{0,1,2\})|$

with equality if and only if $p = 5$ .

IMO Shortlist 2003 problem C6

Let $f(k)$ be the number of all non-negative integers $n$ satisfying the following conditions:

(1) The integer $n$ has exactly $k$ digits in the decimal representation (where the first digit is not necessarily non-zero!), i. e. we have $0 \leq n <10^k$ .

(2) These $k$ digits of n can be permuted in such a way that the resulting number is divisible by 11.

Show that for any positive integer number $m,$ we have $f\left(2m\right) = 10 f\left(2m - 1\right)$ .

IMO Shortlist 2005 problem C7

Suppose that $a_1$ , $a_2$ , $\ldots$ , $a_n$ are integers such that $n\mid a_1 + a_2 + \ldots + a_n$ .
Prove that there exist two permutations $\left(b_1,b_2,\ldots,b_n\right)$ and $\left(c_1,c_2,\ldots,c_n\right)$ of $\left(1,2,\ldots,n\right)$ such that for each integer $i$ with $1\leq i\leq n$ , we have
$n\mid a_i - b_i - c_i$

IMO Shortlist 2007 problem C7

Let $\alpha < \frac {3 - \sqrt {5}}{2}$ be a positive real number. Prove that there exist positive integers $n$ and $p > \alpha \cdot 2^n$ for which one can select $2 \cdot p$ pairwise distinct subsets $S_1, \ldots, S_p, T_1, \ldots, T_p$ of the set $\{1,2, \ldots, n\}$ such that $S_i \cap T_j \neq \emptyset$ for all $1 \leq i,j \leq p$

Author: Gerhard Wöginger, Austria

IMO Shortlist 2008 problem C5

Let $S = \{x_1, x_2, \ldots, x_{k + l}\}$ be a $(k + l)$ -element set of real numbers contained in the interval $[0, 1]$ ; $k$ and $l$ are positive integers. A $k$ -element subset $A\subset S$ is called nice if
$\left |\frac {1}{k}\sum_{x_i\in A} x_i - \frac {1}{l}\sum_{x_j\in S\setminus A} x_j\right |\le \frac {k + l}{2kl}$
Prove that the number of nice subsets is at least $\dfrac{2}{k + l}\dbinom{k + l}{k}$ .

Proposed by Andrey Badzyan, Russia