2013 AMC 10A Problem 4

Question

A softball team played ten games, scoring

1,

2,

3,

4,

5,

6,

7,

8,

9,

and

10

runs. They lost by one run in exactly five games. In each of their other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?

35

40

45

50

55

Answer: C

Concepts:parity casework

Difficulty rating: 1070

Small hint:

The doubled-score games must be the even scoring games

Big hint:

Pair odd scores with one-run losses and even scores with halved opponent scores

Solution:

Note that if they scored twice as many runs as their opponents, then they scored an even number of runs.

This means in the games where they scored

2, 4, 6, 8,

and

10

runs, their opponents scored

1, 2, 3, 4,

and

5

runs respectively.

This sums to

1 + 2 + 3 + 4 + 5 = 15.

In the other games, their opponents scored

2, 4, 6, 8,

and

10

runs.