MathDB

2

Part of 2000 Polish MO Finals

Problems(2)

Chessboard

Source:

1/25/2005
In the unit squre For the given natural number n≥2n \geq 2 find the smallest number kk that from each set of kk unit squares of the nnxnn chessboard one can achoose a subset such that the number of the unit squares contained in this subset an lying in a row or column of the chessboard is even
graph theorycombinatorics unsolvedcombinatorics
Midpoint of base in isosceles triangle [<APM + <BPC = 180°]

Source: Poland National Olympiad 2000, Day 1, Problem 2

1/25/2005
Let a triangle ABCABC satisfy AC=BCAC = BC; in other words, let ABCABC be an isosceles triangle with base ABAB. Let PP be a point inside the triangle ABCABC such that ∠PAB=∠PBC\angle PAB = \angle PBC. Denote by MM the midpoint of the segment ABAB. Show that ∠APM+∠BPC=180∘\angle APM + \angle BPC = 180^{\circ}.
geometrytrigonometrycircumcircleanalytic geometrygeometric transformationreflectioncyclic quadrilateral