MathDB

2

Part of 2019 Iran Team Selection Test

Problems(3)

Iranian TST 2019, first exam, day1, problem 2

Source:

4/7/2019
a,a1,a2,…,ana, a_1,a_2,\dots ,a_n are natural numbers. We know that for any natural number kk which ak+1ak+1 is square, at least one of a1k+1,…,ank+1a_1k+1,\dots ,a_n k+1 is also square. Prove aa is one of a1,…,ana_1,\dots ,a_n Proposed by Mohsen Jamali
Iranian TSTTSTIran
Iranian TST 2019, second exam, day 1, problem 2

Source: Iranian TST

4/11/2019
In a triangle ABCABC, ∠A\angle A is 60∘60^\circ. On sides ABAB and ACAC we make two equilateral triangles (outside the triangle ABCABC) ABKABK and ACLACL. CKCK and ABAB intersect at SS , ACAC and BLBL intersect at RR , BLBL and CKCK intersect at TT. Prove the radical centre of circumcircle of triangles BSK,CLRBSK, CLR and BTCBTC is on the median of vertex AA in triangle ABCABC. Proposed by Ali Zamani
TSTIranian TSTgeometryIran
Iran combinatorial number thoery

Source: Iranian TST 2019, third exam day 1, problem 2

4/25/2019
Hesam chose 1010 distinct positive integers and he gave all pairwise gcd⁡\gcd's and pairwise lcm{\text lcm}'s (a total of 9090 numbers) to Masoud. Can Masoud always find the first 1010 numbers, just by knowing these 9090 numbers?
Proposed by Morteza Saghafian
Combinatorial Number Theorynumber theorycombinatorics