Question

在一场 $100$ 箭的射箭比赛进行到一半时，Chelsea 领先 $50$ 分。每箭射中靶心得 $10$ 分，其他可能得分为 $8$ 、 $4$ 、 $2$ 和 $0$ 分。Chelsea 每箭至少得 $4$ 分。如果 Chelsea 接下来 $n$ 箭都射中靶心，她就能保证获胜。 $n$ 的最小值是多少？

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 $n?$

$38$

$40$

$42$

$44$

$46$

答案：C

知识点：不等式极限情形界定

难度评级：1350

解答：

对手在最后 $50$ 箭中最多可得 $50\cdot10=500$ 分。Chelsea 已领先 $50$ 分，所以她还需要超过 $500-50=450$ 分。

设 Chelsea 有 $n$ 箭靶心，得 $10n$ 分，其余 $50-n$ 箭至少得 $4(50-n)$ 分。要保证获胜，需要化简得 $6n\gt250$ ，即 $n\gt41\tfrac23$ 。 $10n+4(50-n)\gt450.$

因此至少需要 $42$ 箭靶心。

所以正确答案是 C。

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 第 5 题

答案：C

难度评级：1350

解答：

其他年份的第 5 题