3

Part of 1972 Bundeswettbewerb Mathematik

Problems(2)

2^(n-1) subsets, every three have a common element

Source: Bundeswettbewerb Mathematik 1972, round 1, problem 3

2^{n-1}

subsets are choosen from a set with

n

elements, such that every three of these subsets have an element in common. Show that all subsets have an element in common.

inductioncombinatorics proposedcombinatorics

Arithmetic mean + interchange digits = geometric mean

Source: Bundeswettbewerb Mathematik 1972, round 2, problem 3

The arithmetic mean of two different positive integers

x,y

is a two digit integer. If one interchanges the digits, the geometric mean of these numbers is archieved. a) Find

x,y

. b) Show that a)'s solution is unique up to permutation if we work in base

g=10

, but that there is no solution in base

g=12

. c) Give more numbers

g

such that a) can be solved; give more of them such that a) can't be solved, too.

number theory proposednumber theory