MathDB

1

Part of 2022 China Team Selection Test

Problems(4)

Concurrency in a cyclic hexagon

Source: 2022 China TST, Test 1, P1 (posting for better LaTeX)

3/24/2022
In a cyclic convex hexagon ABCDEFABCDEF, ABAB and DCDC intersect at GG, AFAF and DEDE intersect at HH. Let M,NM, N be the circumcenters of BCGBCG and EFHEFH, respectively. Prove that the BEBE, CFCF and MNMN are concurrent.
geometryhexagonconcurrency
Eulerian cycle in a grid graph

Source: 2022 China TST, Test 2, P1

3/28/2022
Find all pairs of positive integers (m,n)(m, n), such that in a m×nm \times n table (with m+1m+1 horizontal lines and n+1n+1 vertical lines), a diagonal can be drawn in some unit squares (some unit squares may have no diagonals drawn, but two diagonals cannot be both drawn in a unit square), so that the obtained graph has an Eulerian cycle.
graph theorygridcombinatorics
Equal angles in a donut

Source: 2022 China TST, Test 3 P1

4/30/2022
Given two circles ω1\omega_1 and ω2\omega_2 where ω2\omega_2 is inside ω1\omega_1. Show that there exists a point PP such that for any line ℓ\ell not passing through PP, if ℓ\ell intersects circle ω1\omega_1 at A,BA,B and ℓ\ell intersects circle ω2\omega_2 at C,DC,D, where A,C,D,BA,C,D,B lie on ℓ\ell in this order, then ∠APC=∠BPD\angle APC=\angle BPD.
geometrycirclesequal angles
Color switching operation terminates early

Source: 2022 China TST, Test 4 P1

4/30/2022
Initially, each unit square of an n×nn \times n grid is colored red, yellow or blue. In each round, perform the following operation for every unit square simultaneously:
[*] For a red square, if there is a yellow square that has a common edge with it, then color it yellow. [*] For a yellow square, if there is a blue square that has a common edge with it, then color it blue. [*] For a blue square, if there is a red square that has a common edge with it, then color it red.
It is known that after several rounds, all unit squares of this n×nn \times n grid have the same color. Prove that the grid has became monochromatic no later than the end of the (2n−2)(2n-2)-th round.
combinatoricsColoring