Math Field Day 2009 Team Medley Junior 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 only win by taking 1 counter; if we start with 5 counters, the first player must lose 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. Solve the system:
1 x

1
 y
2
1

1
2 x y
2

6
3. An isosceles triangle has sides 10, 13, and 13. Another triangle has sides x, 13 and 13 AND has the same area. If x

10 then what is x?
4. Draw or describe the graph of the equation: x
 
| x |

| y |

0
5. Find all pairs of positive integers x and y, with x
 y such that 1/x + 1/y = 1/6
6. What is the largest number of regions a plane can be divided into
A. using 5 lines? B. using 6 lines? C. using n lines?
7. A point on one side of a triangle is equidistant from the three vertices. Prove that the triangle is a right triangle.
8. Solve: 2

2
 x
2  x