Question

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 $n?$

$38$

$40$

$42$

$44$

$46$

Answer: C

Concepts:inequality bounding to limit cases

Difficulty rating: 1350

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.

2010 AMC 12A Problem 5

Answer: C

Difficulty rating: 1350

Solution:

Problem 5 in Other Years