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考题4-6 2015 AMC 10B

2018-08-06 重点归纳

AMC10数学竞赛是美国高中数学竞赛中的一项，是针对高中一年级及初中三年级学生的数学测试，该竞赛开始于2000年，分A赛和B赛，于每年的2月初和2月中举行，学生可任选参加一项即可。不管是对高校申请还是今后在数学领域的发展都极其有利！那么接下来跟随小编来看一下AMC10的官方真题以及官方解答吧：

Problem 4

Four siblings ordered an extra large pizza. Alex ate $\frac15$ , Beth $\frac13$ , and Cyril $\frac14$ of the pizza. Dan got the leftovers. What is the sequence of the siblings in decreasing order of the part of pizza they consumed?

$\textbf{(A) } \text{Alex, Beth, Cyril, Dan}$ $考题4-6 2015 AMC 10B$

$\textbf{(C) } \text{Beth, Cyril, Dan, Alex}$ $\textbf{(D) } \text{Beth, Dan, Cyril, Alex}$

$\textbf{(E) } \text{Dan, Beth, Cyril, Alex}$

Solution

Let the pizza have $60$ slices, since the least common multiple of $(5,3,4)=60$ . Therefore, Alex ate $\frac{1}{5}\times60=12$ slices, Beth ate $\frac{1}{3}\times60=20$ slices, and Cyril ate $\frac{1}{4}\times60=15$ slices. Dan must have eaten $60-(12+20+15)=13$ slices. In decreasing order, we see the answer is $考题4-6 2015 AMC 10B$ .

Problem 5

David, Hikmet, Jack, Marta, Rand, and Todd were in a $12$ -person race with $6$ other people. Rand finished $6$ places ahead of Hikmet. Marta finished $1$ place behind Jack. David finished $2$ places behind Hikmet. Jack finished $2$ places behind Todd. Todd finished $1$ place behind Rand. Marta finished in $6$ th place. Who finished in $8$ th place?

$\textbf{(A) } \text{David} \qquad\textbf{(B) } \text{Hikmet} \qquad\textbf{(C) } \text{Jack} \qquad\textbf{(D) } \text{Rand} \qquad\textbf{(E) } \text{Todd}$

Solution

Because Marta was $6$ th, Jack was $5$ th, so Todd was $3$ rd. Thus, Rand was $2$ nd and the 8th place finisher was $\boxed{\mathbf{(B)}\ \mathrm{Hikmet}}$

Problem 6

Marley practices exactly one sport each day of the week. She runs three days a week but never on two consecutive days. On Monday she plays basketball and two days later golf. She swims and plays tennis, but she never plays tennis the day after running or swimming. Which day of the week does Marley swim?

$考题4-6 2015 AMC 10B$

Solution

Marley does basketball on Monday and golf on Wednesday. Since there are four consecutive days between Wednesday and Monday exclusive, she must run on Tuesday. Thus, she must also run on Thursday and Saturday or Friday and Sunday. She can't run on Thursday and Saturday because she would have to do tennis after running, which is not allowed. Thus, she runs on Friday and Sunday, so she must do tennis on Thursday, so she swims on $\boxed{\mathbf{(E)}\ \mathrm{Saturday}}$