Math Field Day 2009
Team Medley
Varsity Level
1. A game is played as follows. There is a pile of counters. The first player may take any number of
counters, but not the entire pile. Thereafter, either player may take up to twice as many counters as his
opponent just took. The player taking the last counter wins. Thus, for example, if we start with 4
counters, the first player can win only by taking 1 counter; if we start with 5 counters the first player
cannot win if the opponent plays optimally.
How many counters should the first player take if the number of counters is:
(A) 9
(B) 15
(C) 25
(D) 52
2. Prove that if I multiply four consecutive integers and then add 1, I get a perfect square.
3. For what values of a (if any) does the system of equations below have no solutions? For what
values (if any) are there an infinite number of solutions?
ax  y  2
2 x  ay  a 2
4. Prove that if n is an integer then n 3  5n is evenly divisible by 6.
5. Draw or describe the graph of the equation: x  y  | x |  | y |  0
6. For what positive values of x is
1
1

1
x x 1
7. Three of the roots of x 4  ax3  bx 2  cx  15 are 1, -1, and 5. What is the value of a?
8. Find all pairs of positive integers x and y, with x  y such that
1 1 1
 
x y 6