MathDB

2

Part of 2018 Caucasus Mathematical Olympiad

Problems(2)

Easy combinatorial problem

Source: III Caucasus Mathematical Olympiad

3/17/2018
On a chessboard 8×88\times 8, n>6n>6 Knights are placed so that for any 6 Knights there are two Knights that attack each other. Find the greatest possible value of nn.
Chessboardcombinatorics
Just angle chasing

Source: III Caucasus Mathematical Olympiad

3/17/2018
Let II be the incenter of an acute-angled triangle ABCABC. Let PP, QQ, RR be points on sides ABAB, BCBC, CACA respectively, such that AP=ARAP=AR, BP=BQBP=BQ and ∠PIQ=∠BAC\angle PIQ = \angle BAC. Prove that QR⊥ACQR \perp AC.
geometry