NL-NWO 1996

How many different (non similar) triangles are there whose angles have an integer number of degrees?

Investigate whether for two positive integers $m$ and $n$ the numbers $m^2 + n$ and $n^2 + m$ can be both squares of integers.

What is the largest number of horses that you can put on a chessboard without there being two horses that can beat each other? a. Describe an arrangement with that maximum number. b. Prove that a larger number is not possible.

(A chessboard consists of $8 \times 8$ spaces and a horse jumps from one field to another field according to the line "two squares vertically and one squared horizontally" or "one square vertically and two squares horizontally")

$a$ $b$ $c$ $d$ $e$ $f$ $g$ $h$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $P$

A line $l$ intersects the segment $AB$ perpendicular to $C$ . Three circles are drawn successively with $AB, AC$ and $BC$ as the diameter. The largest circle intersects $l$ in $D$ . The segments $DA$ and $DB$ still intersect the two smaller circles in $E$ and $F$ . a. Prove that quadrilateral $CFDE$ is a rectangle. b. Prove that the line through $E$ and $F$ touches the circles with diameters $AC$ and $BC$ in $E$ and $F$ .

$A$ $B$ $C$ $D$ $E$ $F$

For the positive integers $x , y$ and $z$ apply $\frac{1}{x}+\frac{1}{y}=\frac{1}{z}$ . Prove that if the three numbers $x , y,$ and $z$ have no common divisor greater than $1$ , $x + y$ is the square of an integer.