IMO 1974

Three players $A,B$ and $C$ play a game with three cards and on each of these $3$ cards it is written a positive integer, all $3$ numbers are different. A game consists of shuffling the cards, giving each player a card and each player is attributed a number of points equal to the number written on the card and then they give the cards back. After a number $(\geq 2)$ of games we find out that A has $20$ points, $B$ has $10$ points and $C$ has $9$ points. We also know that in the last game B had the card with the biggest number. Who had in the first game the card with the second value (this means the middle card concerning its value).

Let $ABC$ be a triangle. Prove that there exists a point $D$ on the side $AB$ of the triangle $ABC$ , such that $CD$ is the geometric mean of $AD$ and $DB$ , iff the triangle $ABC$ satisfies the inequality $\sin A\sin B\le\sin^2\frac{C}{2}$ .

Prove that for any n natural, the number $$\sum \limits_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k}$$ cannot be divided by $5$ .

Consider decompositions of an $8\times 8$ chessboard into $p$ non-overlapping rectangles subject to the following conditions: (i) Each rectangle has as many white squares as black squares. (ii) If $a_i$ is the number of white squares in the $i$ -th rectangle, then $a_1<a_2<\ldots <a_p$ . Find the maximum value of $p$ for which such a decomposition is possible. For this value of $p$ , determine all possible sequences $a_1,a_2,\ldots ,a_p$ .

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 $n(P)- \deg(P) \leq 2.$