11th Swedish 1971


1. Show that (1 + a + a²)² < 3(1 + a² + a⁴) 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 (65533³ + 65534³ + 65535³ + 65536³ + 65537³ + 65538³+ 65539³)/(32765·32766 + 32767·32768 + 32768·32769 + 32770·32771).
5. Show that max_\|x\|≤t \|1 - a cos x\| ≥ tan²(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 (non-empty) subset of the cards has labels with total divisible by 100. Show that the labels must all be equal.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.