DE-MO 1998

Find all possible numbers of lines in a plane which intersect in exactly $37$ points.

Two pupils $A$ and $B$ play the following game. They begin with a pile of $1998$ matches and $A$ plays first. A player who is on turn must take a nonzero square number of matches from the pile. The winner is the one who makes the last move. Decide who has the winning strategy and give one such strategy.

For each nonnegative integer $k$ find all nonnegative integers $x,y,z$ such that $x^2 +y^2 +z^2 = 8^k$

Let $a$ be a positive real number. Then prove that the polynomial $$p(x)=a^3x^3+a^2x^2+ax+a$$ has integer roots if and only if $a=1$ and determine those roots.

A sequence ( $a_n$ ) is given by $a_0 = 0, a_1 = 1$ and $a_{k+2} = a_{k+1} +a_k$ for all integers $k \ge 0$ . Prove that the inequality $\sum_{k=0}^n \frac{a_k}{2^k}< 2$ holds for all positive integers $n$ .