1997 AMC 12 Problem 11

Question

In the sixth, seventh, eighth, and ninth basketball games of the season, a player scored

23,

14,

11,

and

20

points, respectively. Her points-per-game average was higher after nine games than it was after the first five games. If her average after ten games was greater than

18,

what is the least number of points she could have scored in the tenth game?

26

27

28

29

30

Answer: D

Concepts:averages inequalities

Difficulty rating: 1440

Small hint:

Let

S

be her point total in the first five games

Big hint:

Use the nine-game comparison to maximize

S

, then apply the strict ten-game average bound

Solution:

Games

6

–

9

total

68.

The condition

\frac{S+68}{9}>\frac{S}{5}

gives

S<85,

so the greatest integral

S

is

84.

A ten-game average above

18

requires a total at least

181,

hence the tenth score is at least

181-(84+68)=29.

This is attainable, so D is correct.