MathDB

2

Part of 2017 Korea Winter Program Practice Test

Problems(4)

Game on a perfect bipartite graph

Source: 2017 Korean Winter Program Practice Test 1 Day 1 #2

1/18/2017
There are m≥2m \ge 2 blue points and n≥2n \ge 2 red points in three-dimensional space, and no four points are coplanar. Geoff and Nazar take turns, picking one blue point and one red point and connecting the two with a straight-line segment. Assume that Geoff starts first and the one who first makes a cycle wins. Who has the winning strategy?
combinatoricsCombinatorial gamesgraph theory
Almost multiplicative function with an iterative condition

Source: 2017 Korea Winter Program Practice Test 1 Day 2 #2

1/21/2017
Find all functions f:N→Nf : \mathbb{N} \to \mathbb{N} satisfying the following conditions:
[*]For every n∈Nn \in \mathbb{N}, f(n)(n)=nf^{(n)}(n) = n. (Here f(1)=ff^{(1)} = f and f(k)=f(k−1)∘ff^{(k)} = f^{(k-1)} \circ f.) [*]For every m,n∈Nm, n \in \mathbb{N}, ∣f(mn)−f(m)f(n)∣<2017\lvert f(mn) - f(m) f(n) \rvert < 2017.
functionnumber theory
Game with three kinds of coins

Source: 2017 Korea Winter Program Practice Test 2 #2

8/14/2019
Alice and Bob play a game. There are 100100 gold coins, 100100 silver coins, and 100100 bronze coins. Players take turns to take at least one coin, but they cannot take two or more coins of same kind at once. Alice goes first. The player who cannot take any coin loses. Who has a winning strategy?
combinatoricsCombinatorial games
Circle passing C with center B

Source: 2017 Korea Winter Program Practice Test 2 #6

8/14/2019
ABCABC is an obtuse triangle satisfying ∠A>90∘\angle A>90^\circ, and its circumcenter OO and circumcircle ω1\omega_1. Let ω2\omega_2 be a circle passing CC with center BB. ω2\omega_2 meets BCBC at DD. ω1\omega_1 meets ADAD and ω2\omega_2 at EE and F(≠C)F(\neq C), respectively. AFAF meets ω2\omega_2 at G(≠F)G(\neq F). Prove that the intersection of CECE and BGBG lies on the circumcircle of AOBAOB.
geometry