IR-MO 1993

Prove that there exist infinitely many positive integers which can't be represented as sum of less than $10$ odd positive integers' perfect squares.

In the figure below, area of triangles $AOD, DOC,$ and $AOB$ is given. Find the area of triangle $OEF$ in terms of area of these three triangles.

$A$ $F$ $E$ $B$ $C$ $D$ $O$

Prove that there exists a subset $S$ of positive integers such that we can represent each positive integer as difference of two elements of $S$ in exactly one way.

In a convex quadrilateral $ABCD$ , diagonals $AC$ and $BD$ are equal. We construct four equilateral triangles with centers $O_1,O_2,O_3,O_4$ on the sides sides $AB, BC, CD, DA$ outside of this quadrilateral, respectively. Show that $O_1O_3 \perp O_2O_4$ .

Let $x_1, x_2, \ldots, x_{12}$ be twelve real numbers such that for each $1 \leq i \leq 12$ , we have $|x_i| \geq 1$ . Let $I=[a,b]$ be an interval such that $b-a \leq 2$ . Prove that number of the numbers of the form $t= \sum_{i=1}^{12} r_ix_i$ , where $r_i=\pm 1$ , which lie inside the interval $I$ , is less than $1000$ .