This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 2

1977 Bundeswettbewerb Mathematik, 2
Tags: pyramid , combinatorics , way , number , Sum
A beetle crawls along the edges of an $n$-lateral pyramid, starting and ending at the midpoint $A$ of a base edge and passing through each point at most once. How many ways are there for the beetle to do this (two ways are said to be equal if they go through the same vertices)? Show that the sum of the numbers of passed vertices (over all these ways) equals $1^2 +2^2 +\ldots +n^2. $

2003 German National Olympiad, 3
Tags: combinatorics , square , board , way , weights , grid
Consider a $N\times N$ square board where $N\geq 3$ is an odd integer. The caterpillar Carl sits at the center of the square; all other cells contain distinct positive integers. An integer $n$ weights $1\slash n$ kilograms. Carl wants to leave the board but can eat at most $2$ kilograms. Determine whether Carl can always find a way out when a) $N=2003.$ b) $N$ is an arbitrary odd integer.