MathDB

4

Part of 2018 Iran Team Selection Test

Problems(3)

Iran geometry

Source: Iranian TST 2018, first exam day 2, problem 4

4/8/2018
Let ABCABC be a triangle (∠A≠90∘\angle A\neq 90^\circ). BE,CFBE,CF are the altitudes of the triangle. The bisector of ∠A\angle A intersects EF,BCEF,BC at M,NM,N. Let PP be a point such that MP⊥EFMP\perp EF and NP⊥BCNP\perp BC. Prove that APAP passes through the midpoint of BCBC.
Proposed by Iman Maghsoudi, Hooman Fattahi
iran tst 2018 number theory

Source: Iranian TST 2018, second exam day 2, problem 4

4/17/2018
Call a positive integer "useful but not optimized " (!), if it can be written as a sum of distinct powers of 33 and powers of 55. Prove that there exist infinitely many positive integers which they are not "useful but not optimized". (e.g. 37=(30+31+33)+(50+51)37=(3^0+3^1+3^3)+(5^0+5^1) is a " useful but not optimized" number)
Proposed by Mohsen Jamali
number theoryIranIranian TSTTST
A number theory problem from Iran TST

Source: Iranian TST 2018, third exam day 2, problem 4

4/19/2018
We say distinct positive integers  a1,a2,…,ana_1,a_2,\ldots ,a_n are "good" if their sum is equal to the sum of all pairwise gcd⁡\gcd 's among them. Prove that there are infinitely many nn s such that nn good numbers exist.
Proposed by Morteza Saghafian
number theoryIranIranian TST