MathDB

3

Part of 2020 Iran Team Selection Test

Problems(2)

Polynomial permutation

Source: Iran TST1 Day1 P3

2/22/2020
We call a number nn interesting if for each permutation σ\sigma of 1,2,…,n1,2,\ldots,n there exist polynomials P1,P2,…,PnP_1,P_2,\ldots ,P_n and ϵ>0\epsilon > 0 such that: i)i) P1(0)=P2(0)=…=Pn(0)P_1(0)=P_2(0)=\ldots =P_n(0) ii)ii) P1(x)>P2(x)>…>Pn(x)P_1(x)>P_2(x)>\ldots >P_n(x) for −ϵ<x<0-\epsilon<x<0 iii)iii) Pσ(1)(x)>Pσ(2)(x)>…>Pσ(n)(x)P_{\sigma (1)} (x)>P_{\sigma (2)}(x)> \ldots >P_{\sigma (n)} (x) for 0<x<ϵ0<x<\epsilon Find all interesting nn.
Proposed by Mojtaba Zare Bidaki
algebrapolynomialIranian TST
Iran geometry

Source: Iranian TST 2020, second exam day 1, problem 3

3/11/2020
Given a triangle ABCABC with circumcircle Γ\Gamma. Points EE and FF are the foot of angle bisectors of BB and CC, II is incenter and KK is the intersection of AIAI and EFEF. Suppose that TT be the midpoint of arc BACBAC. Circle Γ\Gamma intersects the AA-median and circumcircle of AEFAEF for the second time at XX and SS. Let S′S' be the reflection of SS across AIAI and JJ be the second intersection of circumcircle of AS′KAS'K and AXAX. Prove that quadrilateral TJIXTJIX is cyclic.
Proposed by Alireza Dadgarnia and Amir Parsa Hosseini
geometrycircumcircleincenterIranian TST