MathDB

2

Part of 2014 Iran MO (2nd Round)

Problems(2)

Powerful sets: a, b in S, then a^b or b^a in S

Source: Iran National Olympiad - 2014 Second Round - D2P2

5/3/2014
A subset SS of positive real numbers is called powerful if for any two distinct elements a,ba, b of SS, at least one of aba^{b} or bab^{a} is also an element of SS. a) Give an example of a four elements powerful set. b) Prove that every finite powerful set has at most four elements.
combinatorics unsolvedcombinatorics
Angles and points on a square

Source: Iran National Olympiad - 2014 Second Round - D1P2

5/1/2014
Let ABCDABCD be a square. Let N,PN,P be two points on sides AB,ADAB, AD, respectively such that NP=NCNP=NC, and let QQ be a point on ANAN such that ∠QPN=∠NCB\angle QPN = \angle NCB. Prove that ∠BCQ=12∠AQP. \angle BCQ = \dfrac{1}{2} \angle AQP .
geometrygeometry proposed