

1. Show that in base n^{2}+1 the numbers n^{2}(n^{2}+2)^{2} and n^{4}(n^{2}+2)^{2} have the same digits but in opposite order.


2. Find all continuous functions f(x) such that f(x) + f(x^{2}) = 2 for all real x.


3. For which positive integers n is n^{3}  18n^{2} + 115n  391 a cube?


4. ABCD is a regular tetrahedron. Find a point P on the edge BD such that the sphere diameter AP touches the edge CD.


5. The positive reals x_{1}, x_{2}, x_{3}, x_{4}, x_{5} satisfy x_{1} < x_{2} and x_{2} ≤ each of x_{3}, x_{4}, x_{5}. Also a > 0. Show that 1/(x_{1}+x_{3})^{a} + 1/(x_{2}+x_{4})^{a} + 1/(x_{2}+x_{5})^{a} < 1/(x_{1}+x_{2})^{a} + 1/(x_{2}+x_{3})^{a} + 1/(x_{4}+x_{5})^{a}.


6. 4n points are arranged around a circle. The points are colored alternately yellow and blue. The yellow points are divided into pairs and each pair is joined by a yellow line segment. Similarly for the blue points. At most two segments meet at any point inside the circle. Show that there are at least n points of intersection between a yellow segment and a blue segment.

