Austin Math Circle

(As usual, put solutions/suggestions in comments. A solution will be posted June 5.) The arithmetic, geometric, and harmonic mean of two positive integers are themselves three distinct positive integers. What is the smallest possible value of the arithmetic mean. (Source: Iberoamerican Olympiad 2010, Q4. But don't look! Solve it on your own.)

First, a reminder of the problem: Problem : The number 10 2007 is written on the blackboard. Anne and Berit play a two-player game in which players alternate performing one of the following operations: Replace any number x on the blackboard with two integers a ,  b  > 1 such that a b  =  x . Erase one or both of any two equal numbers on the blackboard. The person who first cannot perform any operations loses the game. Who has the winning strategy if Anne moves first, and why? Solution : We’ll show that Anne has the winning strategy by using the idea of mirroring moves . Mirroring moves is a nice technique for producing winning strategies in games. Let’s start with a simple classic example. Consider a game in which players alternate placing knights on a standard chessboard, with the requirement that each new knight be placed on a square that is not under attack from a knight already on the board. There’s a simple strategy that gives the win to the se...

(Note: the math symbols in this post will come out more cleanly if you open up the single post by itself, rather than viewing it within the whole blog.) Welcome to the Austin Math Circle problem of the week. Each week we'll put up a new problem here. (Mostly drawn from old contests, but maybe some original ones from time to time.) Put answers/solutions in the comments, and the next week we'll put up a solution, along with discussion of any particularly interesting approaches that people took. The number 10 2007 is written on the blackboard. Anne and Berit play a two-player game in which players alternate performing one of the following operations: Replace any number x on the blackboard with two integers a ,  b  > 1 such that a b  =  x . Erase one or both of any two equal numbers on the blackboard. The person who first cannot perform any operations loses the game. Who has the winning strategy if Anne moves first, and why? [Source: Nordic Mathematical Competi...