SE-SKOLORNAS 1971

Show that $$\left(1 + a + a^2\right)^2 < 3\left(1 + a^2 + a^4\right)$$ for real $a \neq 1$ .

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.

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.

Find $$\frac{65533^3 + 65534^3 + 65535^3 + 65536^3 + 65537^3 + 65538^3+ 65539^3}{32765\cdot 32766 + 32767\cdot 32768 + 32768\cdot 32769 + 32770\cdot 32771}$$

Show that $$\max\limits_{|x|\leq t} |1 - a \cos x| \geq \tan^2 \frac{t}{2}$$ for $a$ positive and $t \in (0, \frac{\pi}{2})$ .

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