CWMO 2003

Place the numbers $1, 2, 3, 4, 5, 6, 7, 8$ at the vertices of a cuboid such that the sum of any $3$ numbers on a side is not less than $10$ . Find the smallest possible sum of the 4 numbers on a side.

Let $a_1, a_2, \ldots, a_{2n}$ be $2n$ real numbers satisfying the condition $\sum_{i = 1}^{2n - 1} (a_{i + 1} - a_i)^2 = 1$ . Find the greatest possible value of $(a_{n + 1} + a_{n + 2} + \ldots + a_{2n}) - (a_1 + a_2 + \ldots + a_n)$ .

Let $n$ be a given positive integer. Find the smallest positive integer $u_n$ such that for any positive integer $d$ , in any $u_n$ consecutive odd positive integers, the number of them that can be divided by $d$ is not smaller than the number of odd integers among $1, 3, 5, \ldots, 2n - 1$ that can be divided by $d$ .

Given that the sum of the distances from point $P$ in the interior of a convex quadrilateral $ABCD$ to the sides $AB, BC, CD, DA$ is a constant, prove that $ABCD$ is a parallelogram.

The sequence $\{a_n\}$ satisfies $a_0 = 0, a_{n + 1} = ka_n + \sqrt {(k^2 - 1)a_n^2 + 1}, n = 0, 1, 2, \ldots$ , where $k$ is a fixed positive integer. Prove that all the terms of the sequence are integral and that $2k$ divides $a_{2n}, n = 0, 1, 2, \ldots$ .

A circle can be inscribed in the convex quadrilateral $ABCD$ . The incircle touches the sides $AB, BC, CD, DA$ at $A_1, B_1, C_1, D_1$ respectively. The points $E, F, G, H$ are the midpoints of $A_1B_1, B_1C_1, C_1D_1, D_1A_1$ respectively. Prove that the quadrilateral $EFGH$ is a rectangle if and only if $A, B, C, D$ are concyclic.

The non-negative numbers $x_1, x_2, \ldots, x_5$ satisfy $\sum_{i = 1}^5 \frac {1}{1 + x_i} = 1$ . Prove that $\sum_{i = 1}^5 \frac {x_i}{4 + x_i^2} \leq 1$ .

$1650$ students are arranged in $22$ rows and $75$ columns. It is known that in any two columns, the number of pairs of students in the same row and of the same sex is not greater than $11$ . Prove that the number of boys is not greater than $928$ .