MathDB

1

Part of 2018 Iran Team Selection Test

Problems(3)

Iran combinatorial number theory

Source: Iranian TST 2018, first exam day 1, problem 1

4/8/2018
Let A1,A2,...,AkA_1, A_2, ... , A_k be the subsets of {1,2,3,...,n}\left\{1,2,3,...,n\right\} such that for all 1≤i,j≤k1\leq i,j\leq k:Ai∩Aj≠∅A_i\cap A_j \neq \varnothing. Prove that there are nn distinct positive integers x1,x2,...,xnx_1,x_2,...,x_n such that for each 1≤j≤k1\leq j\leq k: lcmi∈Aj{xi}>lcmi∉Aj{xi}lcm_{i \in A_j}\left\{x_i\right\}>lcm_{i \notin A_j}\left\{x_i\right\} Proposed by Morteza Saghafian, Mahyar Sefidgaran
Iranian TSTcombinatoricsnumber theoryCombinatorial Number Theory
Functional equation

Source: Iranian TST 2018, second exam day 1, problem 1

4/15/2018
Find all functions f:R→Rf:\mathbb{R}\rightarrow \mathbb{R} that satisfy the following conditions: a. x+f(y+f(x))=y+f(x+f(y))   \forall x,y \in \mathbb{R} b. The set I={f(x)−f(y)x−y∣x,y∈R,x≠y}I=\left\{\frac{f(x)-f(y)}{x-y}\mid x,y\in \mathbb{R},x\neq y \right\} is an interval.
Proposed by Navid Safaei
functionalgebrafunctional equation
an easy geometry from iran tst

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

4/18/2018
Two circles ω1(O)\omega_1(O) and ω2\omega_2 intersect each other at A,BA,B ,and OO lies on ω2\omega_2. Let SS be a point on ABAB such that OS⊥ABOS\perp AB. Line OSOS intersects ω2\omega_2  at PP (other than OO). The bisector of ASP^\hat{ASP} intersects  ω1\omega_1 at LL (AA and LL are on the same side of the line OPOP). Let KK be a point on ω2\omega_2 such that PS=PKPS=PK (AA and KK are on the same side of the line OPOP). Prove that SL=KLSL=KL.
Proposed by Ali Zamani
geometryIranIranian TST