2

Part of 1977 Bundeswettbewerb Mathematik

Problems(2)

Bundeswettbewerb Mathematik 1977 Problem 2.2

Source: Bundeswettbewerb Mathematik 1977 Round 2

On a plane are given three non-collinear points

A, B, C

. We are given a disk of diameter different from that of the circle passing through

A, B, C

large enough to cover all three points. Construct the fourth vertex of the parallelogram

ABCD

using only this disk (The disk is to be used as a circular ruler, for constructing a circle passing through two given points).

geometryconstructionparallelogramdisks

Bundeswettbewerb Mathematik 1977 Problem 1.2

Source: Bundeswettbewerb Mathematik 1977 Round 1

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.

pyramidcombinatoricswaynumberSum