International Mathematical Olympiad · 1975

Problems

Problem 1Let $x_1 \ge x_2 \ge \cdots \ge x_n$ and $y_1 \ge y_2 \ge \cdots \ge y_n$ be real numbers. Prove that if $z_1, z_2, \ldots, z_n$ is any permutation of $y_1, y_2, \ldots, y_n$ , then $\sum_{i=1}^{n} (x_i - y_i)^2 \le \sum_{i=1}^{n} (x_i - z_i)^2$ .Solutions: 1
Problem 2Let $a_1, a_2, a_3, \ldots$ be an infinite increasing sequence of positive integers. Prove that for every $p \ge 1$ there are infinitely many $a_m$ which can be written in the form $a_m = xa_p + ya_q$ with $x, y$ positive integers and $q > p$ .Solutions: 1
Problem 3On the sides of an arbitrary triangle $ABC$ , triangles $ABR$ , $BCP$ , $CAQ$ are constructed externally with $\angle CBP = \angle CAQ = 45^\circ$ , $\angle BCP = \angle ACQ = 30^\circ$ , $\angle ABR = \angle BAR = 15^\circ$ . Prove that $\angle QRP = 90^\circ$ and $QR = RP$ .Solutions: 1
Problem 4When $4444^{4444}$ is written in decimal notation, the sum of its digits is $A$ . Let $B$ be the sum of the digits of $A$ . Find the sum of the digits of $B$ . ( $A$ and $B$ are written in decimal notation.)Solutions: 1
Problem 5Determine, with proof, whether one can find $1975$ points on the circumference of a circle with unit radius such that the distance between any two of them is a rational number.Solutions: 1
Problem 6Find all polynomials $P$ in two variables such that, for a positive integer $n$ , $P(tx,ty)=t^nP(x,y)$ for all real $t,x,y$ ; for all real $a,b,c$ , $P(b+c,a)+P(c+a,b)+P(a+b,c)=0$ ; and $P(1,0)=1$ .Solutions: 1