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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.

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IMO Shortlist 1971 problem 13

Let $A = (a_{ij})$ , where $i,j = 1,2,\ldots,n$ , be a square matrix with all $a_{ij}$ non-negative integers. For each $i,j$ such that $a_{ij} = 0$ , the sum of the elements in the $i$ th row and the $j$ th column is at least $n$ . Prove that the sum of all the elements in the matrix is at least $\frac {n^2}{2}$ .

IMO Shortlist 1974 problem 3

Let $P(x)$ be a polynomial with integer coefficients. We denote $\deg(P)$ its degree which is $\geq 1.$ Let $n(P)$ be the number of all the integers $k$ for which we have $(P(k))^{2}=1.$ Prove that $n(P)- \deg(P) \leq 2.$

IMO Shortlist 1984 problem 16

Let $a,b,c,d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$ . Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$ , then $a=1$ .

IMO Shortlist 1987 problem 15

Let $x_1,x_2,\ldots,x_n$ be real numbers satisfying $x_1^2+x_2^2+\ldots+x_n^2=1$ . Prove that for every integer $k\ge2$ there are integers $a_1,a_2,\ldots,a_n$ , not all zero, such that $|a_i|\le k-1$ for all $i$ , and $|a_1x_1+a_2x_2+\ldots+a_nx_n|\le{(k-1)\sqrt n\over k^n-1}$ . (IMO Problem 3)

Proposed by Germany, FR

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 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 $S(n)=n^{2}-14.$