Problem of the Month

With our Problem of the Month, we would like to stimulate mathematical discussion among Hawaii's high school students. Others are not excluded from submitting solutions. We are planning to post a new problem each month, and solutions should be submitted by the end of the month to

When you submit a solution, please provide your name, address and other contact information, the name of your school, and a statement that you found the solution that you are submitting.

You are encouraged to use the literature, but you must acknowledge all sources that have helped you to obtain your solution. After solving the problem, explain your solution to a friend and polish your write-up before submitting it. Clean up the logic of your argument, and explain it in good, grammatically correct English.

At the end of the academic year we may hold a competition to which we invite students who have submitted solutions throughout the year, with a prize for the students with the best solutions.

February's problem:

Let n > m be two positive integers. Assume that we have n numbers such that the sum of any m of them is positive. Prove that the sum of any m+1 of these numbers is positive.

January's problem:

There is a path that a knight can take to go over a chessboard visiting each square exactly once. The specific path in not easy to find, but it is also not important for the problem. Now, remove the upper right and lower left hand squares from the board. Is it possible for the knight to go over the board, visiting each of the remaining square exactly once?

December's problem:

Are there two positive powers of 2, so that their difference is divisible by 2007?
(Are there positive numbers m and n, so that 2ⁿ – 2^m is an integer multiple of 2007? You may assume that n and m are integers.)

November's Problem:

There is a well known game called Gomoku. Its variations are also known as Renju, Five-In-A-Row, Caro, Gobang, etc. Different variations correspond to slightly different rules of the game. The players have stones of different color, usually black and white.

They take turns playing their stones, one at a time, into the squares of the board. The stones on the board are never moved and never taken off during the game. One cannot place a stone into a square that is already occupied. The player who gets five (or more) stones of his/her color in a row (vertical, horizontal, or horizontal and with no empty squares in between) wins.

Let us consider a variation of the basic game. Assume that to win one has to get five stones of ones color in either a row or a column. (So, five stones diagonally do not count as a winning combination.) Assume also that the size of the board is even. It may be of size 6x6, 8x8, 10x10, ... . Show that each player has a way to escape losing the game.

Historical Remark: It was known for centuries, and has been proven recently by Victor Allis, that the player who moves first has a winning strategy. This player has a way to win, no matter how creative, smart, or imaginative the opponent is. This fact was the reason for the appearance of the different additional sets of rules and variations that somehow balance the advantage of the first move.

October's problem:

You have exactly one game piece on each field of a chessboard of size 17 by 17. Each piece is supposed to go to a neighboring field, either one step left, right, up, or down. No piece is allowed to stay where it is, and after the pieces moved there should be again exactly one piece on each field. Is this possible? Justify your answer.

For those who gave up on September's problem, the October one is much simpler.

September's problem:

In acute triangle ABC with circumcenter O and altitude AP, angle C is greater or equal to angle B plus 30 degrees. Prove that angle A plus angle COP is less than 90 degrees.

Have fun solving the problem.