MathDB

1

Part of 2014 Turkey Team Selection Test

Problems(2)

Number of Permutations

Source: Turkey TST 2014 Day 1 Problem 1

3/12/2014
Find the number of (a1,a2,...,a2014)(a_1,a_2, ... ,a_{2014}) permutations of the (1,2,...,2014)(1,2, . . . ,2014) such that, for all 1≤i<j≤20141\leq i<j\leq2014, i+ai≤j+aji+a_i \leq j+a_j.
inductionanalytic geometrygraphing linesslopecombinatorics proposedcombinatorics
Angle Equality

Source: Turkey TST 2014 Day 3 Problem 7

3/12/2014
Let PP be a point inside the acute triangle ABCABC with m(PAC^)=m(PCB^)m(\widehat{PAC})=m(\widehat{PCB}). DD is the midpoint of the segment PCPC. APAP and BCBC intersect at EE, and BPBP and DEDE intersect at QQ. Prove that sin⁡BCQ^=sin⁡BAP^\sin\widehat{BCQ}=\sin\widehat{BAP}.
trigonometrygeometry proposedgeometry