MathDB

3

Part of 1974 Bundeswettbewerb Mathematik

Problems(2)

Bundeswettbewerb Mathematik 1974 Problem 1.3

Source: Bundeswettbewerb Mathematik Round 1

10/16/2022
Let MM be a set with nn elements. How many pairs (A,B)(A, B) of subsets of MM are there such that AA is a subset of B?B?
combinatoricscountingSubsets
Bundeswettbewerb Mathematik 1974 Problem 2.3

Source: Bundeswettbewerb Mathematik 1974 Round 2

10/13/2022
A circle K1K_1 of radius r_1 = 1\slash 2 is inscribed in a semi-circle HH with diameter ABAB and radius 1.1. A sequence of different circles K2,K3,…K_2, K_3, \ldots with radii r2,r3,…r_2, r_3, \ldots respectively are drawn so that for each n≥1n\geq 1, the circle Kn+1K_{n+1} is tangent to HH, KnK_n and AB.AB. Prove that a_n = 1\slash r_n is an integer for each nn, and that it is a perfect square for nn even and twice a perfect square for nn odd.
geometrysemicirclecircleSequencesquare