ITOT 45 Spring 2024 · THCS · O (Basic)

Whenever Tom cuts a pie or a piece into two pieces, their weights are equal. Whenever he cuts one piece into more than two pieces, all resulting weights are pairwise different. After several cuts he has17 pieces. Can all17 have equal weight? Pieces cannot be joined together.

An $8\times8$ chessboard is colored, one color per square. Any two diagonally adjacent squares, or any two squares a knight move apart, have different colors. What is the minimum number of colors? The Vietnamese source includes the following illustration of knight moves.

Five equilateral triangles are arranged as shown. The three large triangles are congruent, and the two small triangles are congruent. Find the angles of triangle$ABC$.

Twenty-five gold coins of different values occupy a$5\times5$ grid, one per cell. Two pirates alternately take one coin. A coin may be taken if no remaining coin lies to its left, or none lies to its right, or none lies above it, or none lies below it in the same row or column. Can the first pirate always guarantee at least half the total value?

There are$N$ boa snakes with distinct jaw sizes1,2,...,$N$ cm. A snake swallows every apple assigned to it whose diameter is at most its jaw size, and cannot swallow a larger apple. Jaw sizes cannot be recognized by appearance. In one evening, the keeper assigns any number of apples of chosen sizes to each snake; after the night he observes what each has eaten. What is the minimum total number of apples needed to determine every snake’s jaw size with certainty?