MathDB

2

Part of 2008 IMO Shortlist

Problems(3)

Permutations and divisibility

Source: Serbia TST 2009, IMO Shortlist 2008, Combinatorics problem 2

4/17/2009
Let n∈Nn \in \mathbb N and AnA_n set of all permutations (a1,…,an)(a_1, \ldots, a_n) of the set {1,2,…,n}\{1, 2, \ldots , n\} for which k∣2(a1+⋯+ak), for all 1≤k≤n.k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n. Find the number of elements of the set AnA_n.
Proposed by Vidan Govedarica, Serbia
combinatoricspermutationDivisibilityIMO Shortlist
IMO Shortlist 2008, Geometry problem 2

Source: IMO Shortlist 2008, Geometry problem 2, German TST 2, P1, 2009

7/9/2009
Given trapezoid ABCD ABCD with parallel sides AB AB and CD CD, assume that there exist points E E on line BC BC outside segment BC BC, and F F inside segment AD AD such that \angle DAE \equal{} \angle CBF. Denote by I I the point of intersection of CD CD and EF EF, and by J J the point of intersection of AB AB and EF EF. Let K K be the midpoint of segment EF EF, assume it does not lie on line AB AB. Prove that I I belongs to the circumcircle of ABK ABK if and only if K K belongs to the circumcircle of CDJ CDJ. Proposed by Charles Leytem, Luxembourg
geometrytrapezoidcircumcircleIMO ShortlistCharles Leytem
IMO ShortList 2008, Number Theory problem 2

Source: IMO ShortList 2008, Number Theory problem 2, German TST 2, P2, 2009

7/9/2009
Let a1 a_1, a2 a_2, … \ldots, an a_n be distinct positive integers, n≥3 n\ge 3. Prove that there exist distinct indices i i and j j such that a_i \plus{} a_j does not divide any of the numbers 3a1 3a_1, 3a2 3a_2, … \ldots, 3an 3a_n. Proposed by Mohsen Jamaali, Iran
number theorymodular arithmeticSequenceDivisibilityIMO Shortlist