MathDB

2

Part of 2022 Saint Petersburg Mathematical Olympiad

Problems(2)

Conditional geo with angle 45°

Source: St Petersburg 2022 11.2

9/28/2022
Given is a triangle ABCABC with ∠BAC=45\angle BAC=45; AD,BE,CFAD, BE, CF are altitudes and EF∩BC=XEF \cap BC=X. If AX∥DEAX \parallel DE, find the angles of the triangle.
geometry
Set Theory masked with children and songs

Source: St Petersburg 2022 9.2

9/28/2022
1212 schoolchildren are engaged in a circle of patriotic songs, each of them knows a few songs (maybe none). We will say that a group of schoolchildren can sing a song if at least one member of the group knows it. Supervisor the circle noticed that any group of 1010 circle members can sing exactly 2020 songs, and any group of 88 circle members - exactly 1616 songs. Prove that the group of all 1212 circle members can sing exactly 2424 songs.
combinatorics