

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


2. 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.


3. 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.


4. 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) ≤ p'(x)^{2} for all x.


5. A 3 x 1 paper rectangle is folded twice to give a square side 1. The square is folded along a diagonal to give a rightangled 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?


6. Show that (n  m)!/m! ≤ (n/2 + 1/2)^{n2m} for positive integers m, n with 2m ≤ n.

