2010 AMC 12A Problem 5

Problem 5 of 25EasierAlgebra

Halfway through a

100

-shot archery tournament, Chelsea leads by

50

points. For each shot a bullseye scores

10

points, with other possible scores being

8,

4,

2,

and

0

points. Chelsea always scores at least

4

points on each shot. If Chelsea’s next

n

shots are bullseyes she will be guaranteed victory. What is the minimum value for

n?

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Solution

The opponent can score at most

50\cdot10=500

on the last

50

shots. Since Chelsea leads by

50,

she must score more than

500-50=450

points on her remaining shots to guarantee victory. Her

n

bullseyes give

10n

points, and her other

50-n

shots give at least

4(50-n)

points, so

10n+4(50-n)\gt450.

This simplifies to

6n\gt250,

i.e.

n\gt41\tfrac23.

Therefore Chelsea needs at least

42

bullseyes. Thus, C is the correct answer.