MathDB

2

Part of 2009 Iran MO (3rd Round)

Problems(2)

Stable Permutation

Source: Iran 3rd round 2009 - final exam problem 2

1/2/2015
Permutation π\pi of {1,…,n}\{1,\dots,n\} is called stable if the set {π(k)−k∣k=1,…,n}\{\pi (k)-k|k=1,\dots,n\} is consisted of exactly two different elements. Prove that the number of stable permutation of {1,…,n}\{1,\dots,n\} equals to σ(n)−τ(n)\sigma (n)-\tau (n) in which σ(n)\sigma (n) is the sum of positive divisors of nn and τ(n)\tau (n) is the number of positive divisors of nn.
Time allowed for this problem was 75 minutes.
combinatorics unsolvedcombinatorics
Iran(3rd round)2009

Source: Problem 2 Geometry

9/13/2009
2-There is given a trapezoid ABCD ABCD.We have the following properties: AD\parallel{}BC,DA \equal{} DB \equal{} DC,\angle BCD \equal{} 72^\circ. A point K K is taken on BD BD such that AD \equal{} AK,K \neq D.Let M M be the midpoint of CD CD.AM AM intersects BD BD at N N.PROVE BK \equal{} ND.
geometrytrapezoidsymmetrygeometry unsolved