MathDB

6

Part of 2013 Saint Petersburg Mathematical Olympiad

Problems(2)

St.Peterburg, P6 Grade 11, 2013

Source:

4/17/2014
Let (Ib)(I_b), (Ic)(I_c) are excircles of a triangle ABCABC. Given a circle ω \omega passes through AA and externally tangents to the circles (Ib)(I_b) and (Ic)(I_c) such that it intersects with BCBC at points MM, NN. Prove that ∠BAM=∠CAN \angle BAM=\angle CAN .
A. Smirnov
geometric transformationgeometrycircumcirclegeometry proposed
Soldiers color grass

Source: St Petersburg Olympiad 2013, Grade 10, P6

10/13/2017
There are 8585 soldiers with different heigth and age. Every day commander chooses random soldier and send him and also all soldiers that are taller and older than this soldier, or all soldiers that are lower and younger than this soldier to color grass. Prove that after 1010 days we can find two soldiers, that color grass at same days.
combinatorics