SE-SKOLORNAS 1970

Show that infinitely many positive integers cannot be written as a sum of three fourth powers of integers.

$6$ open disks in the plane are such that the center of no disk lies inside another. Show that no point lies inside all $6$ disks.

A polynomial with integer coefficients takes the value $5$ at five distinct integers. Show that it does not take the value $9$ at any integer.

Let $p(x) = (x- x_1)(x- x_2)(x- x_3)$ , where $x_1, x_2$ and $x_3$ are real. Show that $p(x) p''(x) \le p'(x)^2$ for all $x$ .

A $3\times 1$ paper rectangle is folded twice to give a square side $1$ . The square is folded along a diagonal to give a right-angled triangle. A needle is driven through an interior point of the triangle, making $6$ holes in the paper. The paper is then unfolded. Where should the point be in order to maximise the smallest distance between any two holes?

Show that $\frac{(n - m)!}{m!} \le \left(\frac{n}{2} + \frac{1}{2}\right)^{n-2m}$ for positive integers $m, n$ with $2m \le n$ .