MathDB

3

Part of 2010 International Zhautykov Olympiad

Problems(2)

Problem by IZhO 2010

Source:

9/3/2010
Let ABCABC arbitrary triangle (AB≠BC≠AC≠ABAB \neq BC \neq AC \neq AB) And O,I,H it's circum-center, incenter and ortocenter (point of intersection altitudes). Prove, that 1) ∠OIH>900\angle OIH > 90^0(2 points) 2)∠OIH>1350\angle OIH >135^0(7 points)
balls for 1) and 2) not additive.
geometryincentertrigonometryinequalitiesEulergeometry proposed
IZhO 2010 1-st round-3

Source:

9/4/2010
A rectangle formed by the lines of checkered paper is divided into figures of three kinds: isosceles right triangles (1) with base of two units, squares (2) with unit side, and parallelograms (3) formed by two sides and two diagonals of unit squares (figures may be oriented in any way). Prove that the number of figures of the third kind is even.
http://up.iranblog.com/Files7/dda310bab8b6455f90ce.jpg
geometryrectangleparallelogramanalytic geometrygraphing linesslopecombinatorics proposed