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Part of 2018 Puerto Rico Team Selection Test

Problems(2)

(a_{i1} -a_1)... (a_{i5} -a_5) always even Puerto Rico OMCC TST 2018.2

Source:

9/16/2021
Let A={a1,a2,a3,a4,a5}A = \{a_1, a_2, a_3, a_4, a_5\} be a set of 55 positive integers. Show that for any rearrangement of AA, ai1a_{i1}, ai2a_{i2}, ai3a_{i3}, ai4a_{i4}, ai5a_{i5}, the product (ai1−a1)(ai2−a2)(ai3−a3)(ai4−a4)(ai5−a5)(a_{i1} -a_1) (a_{i2} -a_2) (a_{i3} -a_3) (a_{i4} -a_4) (a_{i5} -a_5) is always even.
combinatoricspermutationspermutation
cyclic wanted, <QAC=<ABC,<PAB=<ACB Puerto Rico Ibero IMO TST 2018.2

Source:

9/16/2021
Let ABCABC be an acute triangle and let P,QP,Q be points on BCBC such that ∠QAC=∠ABC\angle QAC =\angle ABC and ∠PAB=∠ACB\angle PAB = \angle ACB. We extend APAP to MM so that P P is the midpoint of AMAM and we extend AQAQ to NN so that QQ is the midpoint of ANAN. If T is the intersection point of BMBM and CNCN, show that quadrilateral ABTCABTC is cyclic.
geometrycyclic quadrilateralequal anglesConcyclic