ITOT 42 Autumn 2020 - THCS O - Additive practice

Can 100 distinct points be chosen on a circle so that exactly 1000 right triangles have their three vertices among the chosen points?

Eight tennis players enter one knockout tournament per year: four first-round matches, two semifinals and one final, with random pairings at each appropriate round. After several years every pair of players has met exactly once. Prove (a) each player has reached the semifinals more than once; (b) each player has reached at least one final. Original weights: 2+3.

A pile contains $n$ stones. Two players alternate taking either one stone or $p$ stones, where $p$ is a prime divisor of the current pile size. The player taking the last stone wins. For which positive integers $n$ can the first player force a win?

An equilateral triangle has side $d$. A point $P$ has positive distances $a,b,c$ from its vertices. Prove there is an equilateral triangle of side $a$ and a point $Q$ whose distances from its vertices are $b,c,d$.

Eight elephants are numbered 1–8. Their original weights satisfy $w_i=w_{i-1}+w_{i-2}$ for $i=3,\ldots,8$, but the weights are otherwise unknown. At most one elephant has lost weight and none has gained weight. Using elephants alone, how can two balance-scale weighings identify the lighter elephant or show that none has lost weight?