CMO 1991

Show that the equation $x^2+y^5=z^3$ has infinitely many solutions in integers $x, y,z$ for which $xyz \neq 0$ .

Let $n$ be a fixed positive integer. Find the sum of all positive integers with the property that in base $2$ each has exactly $2n$ digits, consisting of $n$ 1's and $n$ 0's. (The first digit cannot be $0$ .)

Let $C$ be a circle and $P$ a given point in the plane. Each line through $P$ which intersects $C$ determines a chord of $C$ . Show that the midpoints of these chords lie on a circle.

Can ten distinct numbers $a_1, a_2, b_1, b_2, b_3, c_1, c_2, d_1, d_2, d_3$ be chosen from $\{0, 1, 2, \ldots, 14\}$ , so that the $14$ differences $|a_1 - b_1|$ , $|a_1 - b_2|$ , $|a_1 - b_3|$ , $|a_2 - b_1|$ , $|a_2 - b_2|$ , $|a_2 - b_3|$ , $|c_1 - d_1|$ , $|c_1 - d_2|$ , $|c_1 - d_3|$ , $|c_2 - d_1|$ , $|c_2 - d_2|$ , $|c_2 - d_3|$ , $|a_1 - c_1|$ , and $|a_2 - c_2|$ are all distinct?

The sides of an equilateral triangle $ABC$ are divided into $n$ equal parts $(n \geq 2) .$ For each point on a side, we draw the lines parallel to other sides of the triangle $ABC,$ e.g. for $n=3$ we have the following diagram: [asy] unitsize(150); defaultpen(linewidth(0.7)); int n = 3; /*