KURSCHAK 1993

Let $a$ and $b$ be positive integers. Prove that the numbers $an^2+b$ and $a(n+1)^2+b$ are both perfect squares only for finitely many integers $n$ .

Triangle $ABC$ is not isosceles. The incircle of $\triangle ABC$ touches the sides $BC$ , $CA$ , $AB$ in the points $K$ , $L$ , $M$ . The parallel with $LM$ through $B$ meets $KL$ at $D$ , the parallel with $LM$ through $C$ meets $KM$ at $E$ .

Prove that $DE$ passes through the midpoint of $\overline{LM}$ .

Let $n$ be a fixed positive integer. Compute over $\mathbb{R}$ the minimum of the following polynomial:

$$f(x)=\sum_{t=0}^{2n}(2n+1-t)x^t.$$