MathDB

2

Part of 2009 Poland - Second Round

Problems(2)

Some divisibilities imply a/b is less than root 3

Source: Polish Second Round 2009

8/4/2011
Given are two integers a>b>1a>b>1 such that a+b∣ab+1a+b \mid ab+1 and a−b∣ab−1a-b \mid ab-1. Prove that a<3ba<\sqrt{3}b.
number theory proposednumber theory
Intersection of 2 subsets amongst n has one element

Source: Polish Second Round 2009

8/4/2011
Find all integer numbers n≥4n\ge 4 which satisfy the following condition: from every nn different 33-element subsets of nn-element set it is possible to choose 22 subsets, which have exactly one element in common.
combinatorics proposedcombinatorics