

1. Show that (1 + a + a^{2})^{2} < 3(1 + a^{2} + a^{4}) for real a ≠ 1.


2. An arbitrary number of lines divide the plane into regions. Show that the regions can be colored red and blue so that neighboring regions have different colors.


3. A table is covered by 15 pieces of paper. Show that we can remove 7 pieces so that the remaining 8 cover at least 8/15 of the table.


4. Find (65533^{3} + 65534^{3} + 65535^{3} + 65536^{3} + 65537^{3} + 65538^{3}+ 65539^{3})/(32765·32766 + 32767·32768 + 32768·32769 + 32770·32771).


5. Show that max_{x≤t} 1  a cos x ≥ tan^{2}(t/2) for a positive and t ∈ (0, π/2).


6. 99 cards each have a label chosen from 1, 2, ... , 99, such that no (nonempty) subset of the cards has labels with total divisible by 100. Show that the labels must all be equal.

