MathDB

3

Part of 2019 Slovenia Team Selection Test

Problems(2)

Slovenia 2019 TST1 P3

Source: 2019 Slovenia 1st TST Problem 3

2/19/2019
Let nn be any positive integer and MM a set that contains nn positive integers. A sequence with 2n2^n elements is christmassy if every element of the sequence is an element of MM. Prove that, in any christmassy sequence there exist some successive elements, the product of whom is a perfect square.
TSTcombinatorics
Slovenia 2019 TST2 P3

Source: 2019 Slovenia 2nd TST Problem 3

2/13/2019
Let ABCABC be a non-right triangle and let MM be the midpoint of BCBC. Let DD be a point on AMAM (D≠A, D≠M). Let ω1 be a circle through DD that intersects BCBC at BB and let ω2 be a circle through DD that intersects BCBC at CC. Let ABAB intersect ω1 at BB and EE, and let ACAC intersect ω2 at CC and FF. Prove, that the tangent on ω1 at EE and the tangent on ω2 at FF intersect on AMAM.
Team Selection Testgeometry