Školjka

%V0 Prove that if $n$ is a positive integer such that the equation $$x^3-3xy^2+y^3=n$$ has a solution in integers $x,y$, then it has at least three such solutions. Show that the equation has no solutions in integers for $n=2891$.

%V0 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.

%V0 Does there exist a function $f : \mathbb N \to \mathbb N$, such that $f(f(n)) =n + 1987$ for every natural number $n$? (IMO Problem 4) Proposed by Vietnam.

%V0 An $n \times n, n \geq 2$ chessboard is numbered by the numbers $1, 2, \ldots, n^2$ (and every number occurs). Prove that there exist two neighbouring (with common edge) squares such that their numbers differ by at least $n.$

%V0 Prove that $\sum_{k = 0}^{995} \frac {( - 1)^k}{1991 - k} {1991 - k \choose k} = \frac {1}{1991}$

%V0 An $n \times n$ matrix whose entries come from the set $S = \{1, 2, \ldots , 2n - 1\}$ is called a silver matrix if, for each $i = 1, 2, \ldots , n$, the $i$-th row and the $i$-th column together contain all elements of $S$. Show that: (a) there is no silver matrix for $n = 1997$; (b) silver matrices exist for infinitely many values of $n$.