MathDB

3

Part of 2022 Saint Petersburg Mathematical Olympiad

Problems(3)

Weird polynomial game

Source: St Petersburg 2022 11.3

9/28/2022
Ivan and Kolya play a game, Ivan starts. Initially, the polynomial x−1x-1 is written of the blackboard. On one move, the player deletes the current polynomial f(x)f(x) and replaces it with axn+1−f(−x)−2ax^{n+1}-f(-x)-2, where deg⁡(f)=n\deg(f)=n and aa is a real root of ff. The player who writes a polynomial which does not have real roots loses. Can Ivan beat Kolya?
algebra
Isogonal lines wrt angle BIC

Source: St Petersburg 2022 10.3

9/28/2022
Given is a triangle ABCABC with altitude AHAH, diameter of the circumcircle ADAD and incenter II. Prove that ∠BIH=∠DIC\angle BIH = \angle DIC.
geometry
Angle bisectors in a trapezoid and cyclic quads

Source: St Petersburg 2022 9.3

9/28/2022
Given is a trapezoid ABCDABCD, AD∥BCAD \parallel BC. The angle bisectors of the two pairs of opposite angles meet at X,YX, Y. Prove that AXYDAXYD and BXYCBXYC are cyclic.
geometry