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IMO Shortlist 1963 problem 6

Five students $A, B, C, D, E$ took part in a contest. One prediction was that the contestants would finish in the order $ABCDE$ . This prediction was very poor. In fact, no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second prediction had the contestants finishing in the order $DAECB$ . This prediction was better. Exactly two of the contestants finished in the places predicted, and two disjoint pairs of students predicted to finish consecutively actually did so. Determine the order in which the contestants finished.

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IMO Shortlist 1987 problem 20

Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$ , then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$ .(IMO Problem 6)

Original Formulation

Let $f(x) = x^2 + x + p$ , $p \in \mathbb N.$ Prove that if the numbers $f(0), f(1), \cdots , f(\sqrt{p\over 3} )$ are primes, then all the numbers $f(0), f(1), \cdots , f(p - 2)$ are primes.

Proposed by Soviet Union.

IMO Shortlist 1988 problem 9

Let $a$ and $b$ be two positive integers such that $a \cdot b + 1$ divides $a^{2} + b^{2}$ . Show that $\frac {a^{2} + b^{2}}{a \cdot b + 1}$ is a perfect square.

IMO Shortlist 1989 problem 23

A permutation $\{x_1, \ldots, x_{2n}\}$ of the set $\{1,2, \ldots, 2n\}$ where $n$ is a positive integer, is said to have property $T$ if $|x_i - x_{i + 1}| = n$ for at least one $i$ in $\{1,2, \ldots, 2n - 1\}.$ Show that, for each $n$ , there are more permutations with property $T$ than without.

IMO Shortlist 1991 problem 28

We call a set $S$ on the real line $\mathbb{R}$ superinvariant if for any stretching $A$ of the set by the transformation taking $x$ to $A(x) = x_0 + a(x - x_0), a > 0$ there exists a translation $B,$ $B(x) = x+b,$ such that the images of $S$ under $A$ and $B$ agree; i.e., for any $x \in S$ there is a $y \in S$ such that $A(x) = B(y)$ and for any $t \in S$ there is a $u \in S$ such that $B(t) = A(u).$ Determine all superinvariant sets.

IMO Shortlist 1992 problem 21

For each positive integer $\,n,\;S(n)\,$ is defined to be the greatest integer such that, for every positive integer $\,k\leq S(n),\;n^{2}\,$ can be written as the sum of $\,k\,$ positive squares.

a.) Prove that $\,S(n)\leq n^{2}-14\,$ for each $\,n\geq 4$ .
b.) Find an integer $\,n\,$ such that $\,S(n)=n^{2}-14$ .
c.) Prove that there are infintely many integers $\,n\,$ such that $S(n)=n^{2}-14.$

IMO Shortlist 1997 problem 24

For each positive integer $n$ , let $f(n)$ denote the number of ways of representing $n$ as a sum of powers of 2 with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For instance, $f(4) = 4$ , because the number 4 can be represented in the following four ways: 4; 2+2; 2+1+1; 1+1+1+1.

Prove that, for any integer $n \geq 3$ we have $2^{\frac {n^2}{4}} < f(2^n) < 2^{\frac {n^2}2}$ .