首页> 重点归纳 > AMC数学竞赛真题2016年10A 14-15

AMC数学竞赛真题2016年10A 14-15

2018-10-30 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数学竞赛真题以及官方解答吧：

Problem 14

How many ways are there to write $2016$ as the sum of twos and threes, ignoring order? (For example, $1008\cdot 2 + 0\cdot 3$ and $402\cdot 2 + 404\cdot 3$ are two such ways.)

$\textbf{(A)}\ 236\qquad\textbf{(B)}\ 336\qquad\textbf{(C)}\ 337\qquad\textbf{(D)}\ 403\qquad\textbf{(E)}\ 672$

Solution 1

The amount of twos in our sum ranges from $0$ to $1008$ , with differences of $3$ because $2 \cdot 3 = \operatorname{lcm}(2, 3)$ .

The possible amount of twos is $AMC数学竞赛$ .

Solution 2

You can also see that you can rewrite the word problem into an equation $2x$ + $3y$ = $2016$ . Therefore the question is just how many multiples of $3$ subtracted from 2016 will be an even number. We can see that $x = 1008$ , $y = 0$ all the way to $x = 0$ , and $y = 672$ works, with $y$ being incremented by $2$ 's.Therefore, between $0$ and $672$ , the number of multiples of $2$ is $\boxed{\textbf{(C)}337}$ .

Solution 3

We can utilize the stars-and-bars distribution technique to solve this problem. We have 2 "buckets" in which we will distribute parts of our total sum, 2016. By doing this, we know we will have $(2016-1) C (2-1)$ "total" answers. We want every third x and second y, so we divide our previous total by 6, which will result in $2015/6$ . We have to round down to the nearest integer, and we have to add 2 because we did not consider the 2 solutions involving x or y being 0. So, $2015/6\implies335\implies335+2=\boxed{\textbf{(C)}337}$ .

Problem 15

Seven cookies of radius $1$ inch are cut from a circle of cookie dough, as shown. Neighboring cookies are tangent, and all except the center cookie are tangent to the edge of the dough. The leftover scrap is reshaped to form another cookie of the same thickness. What is the radius in inches of the scrap cookie?

AMC竞赛

$\textbf{(A) } \sqrt{2} \qquad \textbf{(B) } 1.5 \qquad \textbf{(C) } \sqrt{\pi} \qquad \textbf{(D) } \sqrt{2\pi} \qquad \textbf{(E) } \pi$

Solution

The big cookie has radius $3$ , since the center of the center cookie is the same as that of the large cookie. The difference in areas of the big cookie and the seven small ones is $2 \pi$ . The scrap cookie has this area, so its radius must be $\boxed{\textbf{(A) }\sqrt 2}$ .
以上就是小编对AMC10数学竞赛真题以及解析的介绍，希望对你有所帮助，更多学习资料请持续关注AMC数学竞赛网