MathDB

1

Part of 2024 Czech-Polish-Slovak Junior Match

Problems(2)

Parallelity with barycenter

Source: Czech-Polish-Slovak Junior Match 2024, T-1

5/29/2024
Let GG be the barycenter of triangle ABCABC. Let DD be a point such that AGDBAGDB is a parallelogram. Show that BG∥CDBG \parallel CD.
geometry proposedgeometrybarycenterparallelogram
Can we move from (1,2) to (2,3) on the blackboard?

Source: Czech-Polish-Slovak Junior Match 2024, I-1

5/29/2024
Initially, the numbers 11 and 22 are written on the board. A move consists of choosing a positive real number xx and replacing (a,b)(a,b) on the board by (a+xb,b+xa)\left(a+\frac{x}{b},b+\frac{x}{a}\right). Is it possible to create in finitely many moves a situation where the numbers on the board are 22 and 33?
combinatorics