NL-NWO 1976

Prove that there is no natural $n$ such that $8n + 7$ is the sum of three squares.

Given $\vartriangle ABC$ and a point $P$ inside that triangle. The parallelograms $CPBL$ , $APCM$ and $BPAN$ are constructed. Prove that $AL$ , $BM$ and $CN$ pass through one point $S$ , and that $S$ is the midpoint of $AL$ , $BM$ and $CN$ .

In how many ways can the king in the chessboard reach the eighth rank in $7$ moves from its original square on the first row?

For $a,b, x \in R$ holds: $x^2 - (2a^2 + 4)x + a^2 + 2a + b = 0$ . For which $b$ does this equation have at least one root between $0$ and $1$ for all $a$ ?

$f(k) = k + \left[ \frac{n}{k}\right ]$ , $k \in \{1,2,..., n\}$ , $k_0 =\left[ \sqrt{n} \right] + 1$ .

Prove that $f(k_0) < f(k)$ if $k \in \{1,2,..., n\}$