MathDB

3

Part of 2015 Spain Mathematical Olympiad

Problems(2)

Game with subtract 1 or divide by 2.

Source: Spanish Mathematical Olympiad 2015, Problem 3

4/30/2015
On the board is written an integer N≥2N \geq 2. Two players AA and BB play in turn, starting with AA. Each player in turn replaces the existing number by the result of performing one of two operations: subtract 1 and divide by 2, provided that a positive integer is obtained. The player who reaches the number 1 wins. Determine the smallest even number NN requires you to play at least 20152015 times to win (BB shifts are not counted).
combinatorics
Simple but not easy. 2015 Spain Math Olympiad.

Source:

4/28/2015
Let ABCABC be a triangle. MM, and NN points on BCBC, such that BM=CNBM=CN, with MM in the interior of BNBN. Let PP and QQ be points in ANAN and AMAM respectively such that ∠PMC=∠MAB\angle PMC= \angle MAB, and ∠QNB=∠NAC\angle QNB= \angle NAC. Prove that ∠QBC=∠PCB \angle QBC= \angle PCB.
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