MathDB

2

Part of 2023 JBMO TST - Turkey

Problems(2)

Concurrent lines with midpoint

Source: Turkey JBMO TST 2023 Day 1 P2

4/30/2023
Let ABCABC is acute angled triangle and K,LK,L is points on AC,BCAC,BC respectively such that ∠AKB=∠ALB\angle{AKB}=\angle{ALB}. PP is intersection of ALAL and BKBK and QQ is the midpoint of segment KLKL. Let T,ST,S are the intersection AL,BKAL,BK with (ABC)(ABC) respectively. Prove that TK,SL,PQTK,SL,PQ are concurrent.
geometryconcurrency
marbles placed chessboard

Source: Turkey JBMO TST 2023 Day 2 P2

4/30/2023
A marble is placed on each 3333 unit square of a 10∗1010*10 chessboard. After that, the number of marbles in the same row or column with that square is written on each of the remaining empty unit squares. What is the maximum sum of the numbers written on the board?
Chessboardcombinatorics