OMA 2008

$101$ positive integers are written on a line. Prove that we can write signs $+$ , signs $\times$ and parenthesis between them, without changing the order of the numbers, in such a way that the resulting expression makes sense and the result is divisible by $16!$ .

In every cell of a $60 \times 60$ board is written a real number, whose absolute value is less or equal than $1$ . The sum of all numbers on the board equals $600$ . Prove that there is a $12 \times 12$ square in the board such that the absolute value of the sum of all numbers on it is less or equal than $24$ .

On a circle of center $O$ , let $A$ and $B$ be points on the circle such that $\angle AOB = 120^o$ . Point $C$ lies on the small arc $AB$ and point $D$ lies on the segment $AB$ . Let also $AD = 2, BD = 1$ and $CD = \sqrt2$ . Calculate the area of triangle $ABC$ .

Find all real numbers $x$ which satisfy the following equation: $[2x]+[3x]+[7x]=2008$ .

Note: $[x]$ means the greatest integer less or equal than $x$ .

Find all perfect powers whose last $4$ digits are $2,0,0,8$ , in that order.

Consider a board of $a \times b$ , with $a$ and $b$ integers greater than or equal to $2$ . Initially their squares are colored black and white like a chess board. The permitted operation consists of choosing two squares with a common side and recoloring them as follows: a white square becomes black; a black box turns green; a green box turns white. Determine for which values of $a$ and $b$ it is possible, by a succession of allowed operations, to make all the squares that were initially white end black and all the squares that were initially black end white.

Clarification: Initially there are no green squares, but they appear after the first operation.