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AMC数学竞赛真题2016年10A 9-10

2018-10-26 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 9

A triangular array of $2016$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$ th row. What is the sum of the digits of $N$ ?

$AMC真题$

Solution

We are trying to find the value of $N$ such that

Noticing that $\frac{63\cdot 64}{2}=2016,$ we have $N=63,$ so our answer is $\boxed{\textbf{(D) } 9}.$

Notice that we were attempting to solve $AMC数学竞赛,AMC真题$ . Approximating $N(N+1) \approx N^2$ , we were looking for a square that is close to, but less than, $4032$ . Since $64^2 = 4096$ , we see that $N = 63$ is a likely candidate. Multiplying $63\cdot64$ confirms that our assumption is correct.

Problem 10

A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is $1$ foot wide on all four sides. What is the length in feet of the inner rectangle?

$\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } 6 \qquad \textbf{(E) }8$

Solution

Let the length of the inner rectangle be $x$ .

Then the area of that rectangle is $x\cdot1 = x$ .

The second largest rectangle has dimensions of $x+2$ and $3$ , making its area $3x+6$ . The area of the second shaded area, therefore, is $3x+6-x = 2x+6$ .

The largest rectangle has dimensions of $x+4$ and $5$ , making its area $5x + 20$ . The area of the largest shaded region is the largest rectangle minus the second largest rectangle, which is $(5x+20) - (3x+6) = 2x + 14$ .