MathDB

6

Part of 2023 Saint Petersburg Mathematical Olympiad

Problems(2)

Geo with reflections

Source: St. Petersburg 2023 9.7=11.6

8/12/2023
Given is a triangle ABCABC. Let XX be the reflection of BB in ACAC and YY is the reflection of CC in ABAB. The tangent to (XAY)(XAY) at AA meets XYXY and BCBC at E,FE, F. Show that AE=AFAE=AF.
geometrygeometric transformationreflection
Graph with club flavortext

Source: St Petersburg 2023 10.6

8/12/2023
There are several gentlemen in the meeting of the Diogenes Club, some of which are friends with each other (friendship is mutual). Let's name a participant unsociable if he has exactly one friend among those present at the meeting. By the club rules, the only friend of any unsociable member can leave the meeting (gentlemen leave the meeting one at a time). The purpose of the meeting is to achieve a situation in which that there are no friends left among the participants. Prove that if the goal is achievable, then the number of participants remaining at the meeting does not depend on who left and in what order.
combinatoricsSt. Petersburg MOgraph theory