Problem Set #1

1.  Prove that every 6 digit integer of the form abcabc (ex. 241241, 638638) is divisible by 13. 

2.  Find the maximum value of the positive integer n for which n²+2 divides f(n)=(n+1)(n⁴+2n)+3(n³+57).

3.  How can a cube be cut up into 20 cubes (not necessarily the same size)?

4.  How can you bring up from a river exactly 6 quarts of water when you have only two containers, a four quart pail and a  nine quart pail, to measure with.

5.  Describe how to construct a square inscribed in a triangle such that two vertices of the square are on the base of the triangle and the other two vertices are on each of the other two sides of the triangle.

6.  In numbering the pages of a book, a printer used 3289 digits.  How many pages were in the book, assuming that the first page of the book was number 1?

7.  Three people play a game in which one person loses and two people win each game.  The one who loses must double the amount of money that each of the other two players has at that time.  The three players agree to play three games.  At the end of 3 games, each player has lost one game and each player has $8.  What was the original stake of each player?

8.  You are given a sack of 24 coins. Twenty-three of the coins have the same weight and one is heavier than the others.  You are given a balance which will compare the weights of any two sets of coins out of the total set of 24 coins.  What is the minimum number of weighings needed to determine which coin is heavier?