MathDB

2

Part of 2018 Korea Winter Program Practice Test

Problems(2)

Playing Game on Grid

Source: 2018 Korea Winter Program Practice Test #2

2/1/2018
For odd integers n,n, two people play the game on 2×n2\times n grid. Each people color one cell that is not colored before with white and black. When coloring is done, they count the number of ordered pairs of neighboring cells that have the same color and different color, respectively. If same color neighboring ordered pair of cells are more than different color neighboring ordered pair of cells, the person who first starts win and lose otherwise. (If the number is same, they are tied.) If both of them use the best strategy, find the result of the game.
combinatorics
Geometric Inequality

Source: 2018 Korea Winter Program Practice Test 1 #6

1/7/2018
Let ΔABC\Delta ABC be a triangle and PP be a point in its interior. Prove that [BPC]PA2+[CPA]PB2+[APB]PC2≥[ABC]R2 \frac{[BPC]}{PA^2}+\frac{[CPA]}{PB^2}+\frac{[APB]}{PC^2} \ge \frac{[ABC]}{R^2} where RR is the radius of the circumcircle of ΔABC\Delta ABC, and [XYZ][XYZ] is the area of ΔXYZ\Delta XYZ.
geometric inequalityareasgeometrycircumcircleinequalitiesKorea