Problem Set One

Apparently, according to a reader of my blog, I have written enough about how to solve problems and have not given actual problems for you to try.

Here are some interesting ones, spanning several "topics" from Mathematical Circles (Russian Experience).

Problem 1: Two children take turns breaking up a rectangular chocolate bar 6 squares wide by 8 squares long. They may break the bar only along the divisions between the squares. If the bar breaks into several pieces, they keep breaking the pieces up until only the individual squares remain. The player who cannot make a break loses the game. Who will win?

Problem 2 (16): Prove that the number n^3 + 2n is divisble by 3 for any natural number n.

Problem 3 (26): How many six-digit numbers have atleast one even digit?

Problem 4 (16): Can one make change of a 25-ruble bill, using in all ten bills each having a value of 1, 3, or 5 rubles?

Problem 5 (5): Find the smallest natural number n such that n! is divisible by 990.

I have intentionally spread these problems across a variety of topics to minimize the mindlessness that many of you experience in a test all on the same specific topic. Try using the four phases. If these are easy, awesome! If not, keep on trying. Either way, I hope the problems above at least exercised your mind a bit.

-Andrew