MathDB

7

Part of 2019 South East Mathematical Olympiad

Problems(2)

Strange Geometry

Source: CSMO 2019 Grade 10 Problem 7

7/31/2019
Let ABCDABCD be a given convex quadrilateral in a plane. Prove that there exist a line with four different points P,Q,R,SP,Q,R,S on it and a square A’B’C’D’A’B’C’D’ such that PP lies on both line ABAB and A’B’,A’B’, QQ lies on both line BCBC and B’C’,B’C’, RR lies on both line CDCD and C’D’,C’D’, SS lies on both line DADA and D’A’.D’A’.
geometry
choose numbers from 0,1,2,...,81

Source: CSMO 2019 Grade 11 Problem 7

7/31/2019
Amy and Bob choose numbers from 0,1,2,⋯ ,810,1,2,\cdots,81 in turn and Amy choose the number first. Every time the one who choose number chooses one number from the remaining numbers. When all 8282 numbers are chosen, let AA be the sum of all the numbers Amy chooses, and let BB be the sum of all the numbers Bob chooses. During the process, Amy tries to make gcd⁡(A,B)\gcd(A,B) as great as possible, and Bob tries to make gcd⁡(A,B)\gcd(A,B) as little as possible. Suppose Amy and Bob take the best strategy of each one, respectively, determine gcd⁡(A,B)\gcd(A,B) when all 8282 numbers are chosen.
combinatoricsCombinatorial games