KURSCHAK 1990

Let $p>2$ be a prime number and $n$ a positive integer. Prove that $pn^2$ has at most one positive divisor $d$ for which $n^2+d$ is a square number.

The incenter of $\triangle A_1A_2A_3$ is $I$ , and the center of the $A_i$ -excircle is $J_i$ ( $i=1,2,3$ ). Let $B_i$ be the intersection point of side $A_{i+1}A_{i+2}$ and the bisector of $\angle A_{i+1}IA_{i+2}$ ( $A_{i+3}:=A_i$ $\forall i$ ). Prove that the three lines $B_iJ_i$ are concurrent.

We would like to give a present to one of $100$ children. We do this by throwing a biased coin $k$ times, after predetermining who wins in each possible outcome of this lottery.

Prove that we can choose the probability $p$ of throwing heads, and the value of $k$ such that, by distributing the $2^k$ different outcomes between the children in the right way, we can guarantee that each child has the same probability of winning.