MathDB

4

Part of 2020 Iran Team Selection Test

Problems(2)

Wired-looking FE

Source: Iran TST1 Day2 P4

2/23/2020
Given a function g:[0,1]→Rg:[0,1] \to \mathbb{R} satisfying the property that for every non empty dissection of the trivial [0,1][0,1] to subsets A,BA,B we have either ∃x∈A;g(x)∈B\exists x \in A; g(x) \in B or ∃x∈B;g(x)∈A\exists x \in B; g(x) \in A and we have furthermore g(x)>xg(x)>x for x∈[0,1]x \in [0,1]. Prove that there exist infinite x∈[0,1]x \in [0,1] with g(x)=1g(x)=1.
Proposed by Ali Zamani
functional equationIranian TSTalgebra
Concurrency

Source: Iranian TST 2020, second exam day 2, problem 4

3/12/2020
Let ABCABC be an isosceles triangle (AB=ACAB=AC) with incenter II. Circle ω\omega passes through CC and II and is tangent to AIAI. ω\omega intersects ACAC and circumcircle of ABCABC at QQ and DD, respectively. Let MM be the midpoint of ABAB and NN be the midpoint of CQCQ. Prove that ADAD, MNMN and BCBC are concurrent.
Proposed by Alireza Dadgarnia
geometryincentercircumcircleHarmonicshomothety