MathDB

3

Part of 2014 BAMO

Problems(2)

Symmetric square root equality

Source: 2014 BAMO-12 #3

2/22/2016
Suppose that for two real numbers xx and yy the following equality is true: (x+1+x2)(y+1+y2)=1(x+ \sqrt{1+ x^2})(y+\sqrt{1+y^2})=1 Find (with proof) the value of x+yx+y.
algebrapolynomialsquare roots
Cookie game

Source: 2014 BAMO-8 #3, 2014 BAMO-12 #1

2/22/2016
Amy and Bob play a game. They alternate turns, with Amy going first. At the start of the game, there are 2020 cookies on a red plate and 1414 on a blue plate. A legal move consists of eating two cookies taken from one plate, or moving one cookie from the red plate to the blue plate (but never from the blue plate to the red plate). The last player to make a legal move wins; in other words, if it is your turn and you cannot make a legal move, you lose, and the other player has won. Which player can guarantee that they win no matter what strategy their opponent chooses? Prove that your answer is correct.
gamesGame TheoryCombinatorial gamesgameOlympiadProofcombinatorics