MathDB

3

Part of 2018 Iran Team Selection Test

Problems(3)

Iran geometry

Source: Iranian TST 2018, first exam, day1, problem 3

4/7/2018
In triangle ABCABC let MM be the midpoint of BCBC. Let ω\omega be a circle inside of ABCABC and is tangent to AB,ACAB,AC at E,FE,F, respectively. The tangents from MM to ω\omega meet ω\omega at P,QP,Q such that PP and BB lie on the same side of AMAM. Let X≡PM∩BFX \equiv PM \cap BF and Y≡QM∩CEY \equiv QM \cap CE . If 2PM=BC2PM=BC prove that XYXY is tangent to ω\omega.
Proposed by Iman Maghsoudi
geometry
Number theory

Source: Iranian TST 2018, second exam, day1, problem 3

4/15/2018
Let a1,a2,a3,⋯a_1,a_2,a_3,\cdots be an infinite sequence of distinct integers. Prove that there are infinitely many primes pp that distinct positive integers i,j,ki,j,k can be found such that p∣aiajak−1p\mid a_ia_ja_k-1.
Proposed by Mohsen Jamali
number theory
a really nice polynomial problem

Source: Iranian TST 2018, third exam day 1, problem 3

4/18/2018
n>1n>1 and distinct positive integers a1,a2,…,an+1a_1,a_2,\ldots,a_{n+1} are  given. Does there exist a polynomial p(x)∈Z[x]p(x)\in\Bbb{Z}[x] of degree  ≤n\le n that satisfies the following conditions? a. ∀1≤i<j≤n+1:gcd⁡(p(ai),p(aj))>1\forall_{1\le i < j\le n+1}: \gcd(p(a_i),p(a_j))>1 b. ∀1≤i<j<k≤n+1:gcd⁡(p(ai),p(aj),p(ak))=1\forall_{1\le i < j < k\le n+1}: \gcd(p(a_i),p(a_j),p(a_k))=1
Proposed by Mojtaba Zare
polynomialInteger PolynomialIranian TSTnumber theoryIran