NL-NWO 1968

On the base $AB$ of the isosceles triangle $ABC$ , lies the point $P$ such that $AP : PB = 1 : 2$ . Determine the minimum of $\angle ACP$ .

It holds: $N,a > 0$ . Prove that $\frac12 \left(\frac{N}{a}+a \right) \ge \sqrt{N}$ , and if $N \ge 1$ and $a = [\sqrt{N}]$ . Prove that if $a \ne \sqrt{N}: \frac12 \left(\frac{N}{a}+a \right)$ is a better approximation for $\sqrt{N}$ than $a$ .

$\vartriangle ABC \sim \vartriangle A'B'C'$ . $\vartriangle ABC$ has sides $a,b,c$ and $\vartriangle A'B'C'$ has sides $a',b',c'$ . Two sides of $\vartriangle ABC$ are equal to sides of $\vartriangle A'B'C'$ . Furthermore, $a < a'$ , $a < b < c$ , $a = 8$ . Prove that there is exactly one pair of such triangles with all sides integers.

Given is a triangle $ABC$ . A line $\ell$ passes through reflection wrt $BC$ changes into the line $\ell'$ , $\ell'$ changes into $\ell''$ through reflection wrt $AC$ and $\ell''$ through reflection wrt $AB$ changes into $\ell'''$ . Construct the line $\ell$ given that $\ell'''$ coincides with $\ell$ .

A square of side $n$ ( $n$ natural) is divided into $n^2$ squares of side $1$ . Each pair of "horizontal" boundary lines and each pair of "vertical" boundary lines enclose a rectangle (a square is also considered a rectangle). A rectangle has a length and a width; the width is less than or equal to the length. (a) Prove that there are $8$ rectangles of width $n - 1$ . (b) Determine the number of rectangles with width $n -k$ ( $0\le k \le n -1,k$ integer). (c) Determine a formula for $1^3 + 2^3 +...+ n^3$ .