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AMC数学竞赛真题2016年B 11-12

2018-11-27 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 11

Carl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl’s garden?

$\textbf{(A)}\ 256\qquad\textbf{(B)}\ 336\qquad\textbf{(C)}\ 384\qquad\textbf{(D)}\ 448\qquad\textbf{(E)}\ 512$

Solution

If the dimensions are $a\times b$ , then one side will have $a+1$ posts (including corners) and the other $b+1$ .

The total number of posts is $2(a+b)=20$ .

美国数学竞赛AMC8

Solve the system $AMC美国数学竞赛$ to get $a=3,\ b=7$ . Then the area is $4a\cdot 4b=336$ which is $\mathbf{(B)}$ .

Solution 2

To do this problem, we have to draw a rectangle. We also have to keep track of the fence posts. Putting a post on each corner leaves us with only $16$ posts. Now there are twice as many posts on the longer side then the shorter side. From this we can see that we can put $8$ posts on the longer side and $4$ posts on the shorter side. On the shorter side, we have $3$ spaces between the $4$ posts. On the longer side, we have 7 spaces between the 8 fence posts. There are $4$ yards between each post. Therefore, the answer is $12*28=336$

$\textbf{(B)}\ 336$ .

Problem 12

Two different numbers are selected at random from $\{1, 2, 3, 4, 5\}$ and multiplied together. What is the probability that the product is even?

$\textbf{(A)}\ 0.2\qquad\textbf{(B)}\ 0.4\qquad\textbf{(C)}\ 0.5\qquad\textbf{(D)}\ 0.7\qquad\textbf{(E)}\ 0.8$

Solution

The product will be even if at least one selected number is even, and odd if none are. Using complementary counting, the chance that both numbers are odd is $美国数学竞赛AMC有用吗$ , so the answer is $1-0.3$ which is $\textbf{(D)}\ 0.7$ .