International Mathematical Olympiad · 1974

Problems

Problem 1Three players $A$ , $B$ , and $C$ play the following game. On each of three cards an integer is written; the three numbers $p, q, r$ satisfy $0 < p < q < r$ . The three cards are shuffled and one is dealt to each player, and each player receives as many counters as the number on the card they hold. The cards are shuffled and dealt again, and this is repeated for at least two rounds (counters from earlier rounds stay with the players). After the last round, $A$ has $20$ counters in all, $B$ has $10$ , and $C$ has $9$ . In the last round, $B$ received $r$ counters. Who received $q$ counters on the first round?Solutions: 1
Problem 2In the triangle $ABC$ , prove that there is a point $D$ on side $AB$ such that $CD$ is the geometric mean of $AD$ and $DB$ if and only if $\sin A \sin B \le \sin^2 \frac{C}{2}$ .Solutions: 2
Problem 3Prove that $\sum_{k=0}^{n}\binom{2n+1}{2k+1}2^{3k}$ is not divisible by $5$ for any integer $n\ge0$ .Solutions: 1
Problem 4Consider decompositions of an $8\times8$ chessboard into $p$ non-overlapping rectangles. Each rectangle has as many white squares as black squares. If $a_i$ is the number of white squares in the $i$ -th rectangle, then $a_1<a_2<\cdots<a_p$ . Find the maximum possible $p$ , and for this $p$ determine all possible sequences $a_1,\ldots,a_p$ .Solutions: 1
Problem 5Determine all possible values of $S=\dfrac{a}{a+b+d}+\dfrac{b}{a+b+c}+\dfrac{c}{b+c+d}+\dfrac{d}{a+c+d}$ , where $a,b,c,d$ are arbitrary positive real numbers.Solutions: 1
Problem 6Let $P$ be a non-constant polynomial with integer coefficients. If $n(P)$ is the number of distinct integers $k$ such that $P(k)^2=1$ , prove that $n(P)-\deg(P)\le2$ .Solutions: 1