NL-NWO 1994

A unit square is divided into two rectangles in such a way that the smaller rectangle can be put on the greater rectangle with every vertex of the smaller on exactly one of the edges of the greater. Calculate the dimensions of the smaller rectangle.

A sequence of integers $a_1,a_2,a_3,...$ is such that $a_1=2, a_2=3$ , and

$a_{n+1}=2a_{n-1}$ or $3a_n-2a_{n-1}$ for all $n \ge 2$ .

Prove that no number between $1600$ and $2000$ can be an element of the sequence.

$(a)$ Prove that every multiple of $6$ can be written as a sum of four cubes. $(b)$ Prove that every integer can be written as a sum of five cubes.

Let $P$ be a point on the diagonal $BD$ of a rectangle $ABCD$ , $F$ be the projection of $P$ on $BC$ , and $H \not= B$ be the point on $BC$ such that $BF=FH$ . If lines $PC$ and $AH$ intersect at $Q$ , prove that the areas of triangles $APQ$ and $CHQ$ are equal.

Three real numbers $a,b,c$ satisfy the inequality $|ax^2+bx+c| \le 1$ for all $x \in [-1,1]$ . Prove that $|cx^2+bx+a| \le 2$ for all $x \in [-1,1]$ .