OME 1992

Determine the smallest number N, multiple of 83, such that N^2 has 63 positive divisors.

Given two circles of radii $r$ and $r'$ exterior to each other, construct a line parallel to a given line and intersecting the two circles in chords with the sum of lengths $\ell$ .

Prove that if $a,b,c,d$ are nonnegative integers satisfying $(a+b)^2+2a+b= (c+d)^2+2c+d$ , then $a = c$ and $b = d$ . Show that the same is true if $a,b,c,d$ satisfy $(a+b)^2+3a+b=(c+d)^2+3c+d$ , but show that there exist $a,b,c,d$ with $a \ne c$ and $b \ne d$ satisfying $(a+b)^2+4a+b = (c+d)^2+4c+d$ .

Prove that the arithmetic progression $3,7,11,15,...$ . contains infinitely many prime numbers.

Given a triangle $ABC$ , show how to construct the point $P$ such that $\angle PAB= \angle PBC= \angle PCA$ . Express this angle in terms of $\angle A,\angle B,\angle C$ using trigonometric functions.

For a positive integer $n$ , let $S(n)$ be the set of complex numbers $z = x+iy$ ( $x,y \in R$ ) with $|z| = 1$ satisfying $(x+iy)^n+(x-iy)^n = 2x^n$ . (a) Determine $S(n)$ for $n = 2,3,4$ . (b) Find an upper bound (depending on $n$ ) of the number of elements of $S(n)$ for $n > 5$ .