1

Part of 1999 Bundeswettbewerb Mathematik

Problems(2)

Monkeys and Coconuts

Source: https://www.mathe-wettbewerbe.de/bwm/aufgaben/aufgaben-1999/aufgaben_99_1.pdf

Exactly 1600 Coconuts are distributed on exactly 100 monkeys, where some monkeys also can have 0 coconuts. Prove that, no matter how you distribute the coconuts, at least 4 monkeys will always have the same amount of coconuts.
(The original problem is written in German. So, I apologize when I've changed the original problem or something has become unclear while translating.)

number theoryalgebracombinatoricspigeonhole principle

vertices of a 2n-gon are labelled with the numbers 1,2,...,2n

Source: 1999 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p1

The vertices of a regular

2n

n > 2

an integer) are labelled with the numbers

1,2,...,2n

in some order. Assume that the sum of the labels at any two adjacent vertices equals the sum of the labels at the two diametrically opposite vertices. Prove that this is possible if and only if

n

combinatorial geometrycombinatorics