International Mathematical Olympiad  – IMO 1975  Problems

Problem 1

Let $x_i, y_i~(i = 1, 2, \ldots, n)$ be real numbers such that\[x_1 \geq x_2 \geq \cdots \geq x_n \text{ and } y_1 \geq y_2 \geq \cdots \geq y_n.\]Prove that, if $z_1, z_2, \cdots, z_n$ is any permutation of $y_1, y_2, \cdots, y_n$, then\[\sum_{i=1}^n (x_i – y_i)^2 \leq \sum_{i=1}^n (x_i – z_i)^2.\]

Problem 2

Let $a_1, a_2, a_3, \cdots$ be an infinite increasing sequence of positive integers. Prove that for every $p \geq 1$ 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 $q > p$.

Problem 3

On 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 $QR = RP$.

Problem 4

When $4444^{4444}$ 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.)

Problem 5

Determine, with proof, whether or not 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.

Problem 6

Find all polynomials $P$, in two variables, with the following properties:

(i) for a positive integer $n$ and all real $t, x, y$\[P(tx, ty) = t^n P(x, y)\](that is, $P$ is homogeneous of degree $n$),

(ii) for all real $a, b, c,$\[P(b+c, a) + P(c+a, b) + P(a+b, c) = 0,\]
(iii) $P(1, 0) = 1.$