MathDB

4

Part of 2012 Dutch IMO TST

Problems(2)

a_i + j = n + 1 where i = a_j, permutations of 1,2,...,n , when 4 divides n

Source: Dutch IMO TST1 2012 p4

1/10/2020
Let nn be a positive integer divisible by 44. We consider the permutations (a1,a2,...,an)(a_1, a_2,...,a_n) of (1,2,...,n)(1,2,..., n) having the following property: for each j we have ai+j=n+1a_i + j = n + 1 where i=aji = a_j . Prove that there are exactly (12n)!(14n)!\frac{ (\frac12 n)!}{(\frac14 n)!} such permutations.
combinatoricspermutations
angle chasing candidate, right angle wanted, equal angles given

Source: Dutch IMO TST2 2012 p4

8/6/2019
Let △ABC\vartriangle ABC be a triangle. The angle bisector of ∠CAB\angle CAB intersectsBC BC at LL. On the interior of line segments ACAC and ABAB, two points, MM and NN, respectively, are chosen in such a way that the lines AL,BMAL, BM and CNCN are concurrent, and such that ∠AMN=∠ALB\angle AMN = \angle ALB. Prove that ∠NML=90o\angle NML = 90^o.
geometryright angleequal anglesAngle Chasing