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AMC数学竞赛真题2016年10A 13

2018-10-29 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 13

Five friends sat in a movie theater in a row containing $5$ seats, numbered $1$ to $5$ from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?

$\textbf{(A) }1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$

Solution 1

Bash: we see that the following configuration works.

Bea - Ada - Ceci - Dee - Edie

After moving, it becomes

Ada - Ceci - Bea - Edie - Dee.

Thus, Ada was in seat $\boxed{\textbf{(B) }2}$ .

Solution 2

Process of elimination of possible configurations.

Let's say that Ada= $A$ , Bea= $B$ , Ceci= $C$ , Dee= $D$ , and Edie= $E$ . Since $B$ moved more to the right than $C$ did left, this implies that $B$ was in a LEFT end seat originally:

$B,-,C\rightarrow -,C,B$

This is affirmed because $DE\rightarrow ED$ , which there is no new seats uncovered. So $A,B,C$ are restricted to the same $1,2,3$ seats. Thus, it must be $B,A,C\rightarrow A,C,B$ , and more specifically:

$B,A,C,D,E\rightarrow A,C,B,E,D$

So $A$ , Ada, was originally in seat $\boxed{\textbf{(B)}\text{ 2}}$ .

Solution 3

The seats are numbered 1 through 5, so let each letter ( $A,B,C,D,E$ ) correspond to a number. Let a move to the left be subtraction and a move to the right be addition.

We know that $1+2+3+4+5=A+B+C+D+E=15$ . After everyone moves around, however, our equation looks like $(A+x)+B+2+C-1+D+E=15$ because $D$ and $E$ switched seats, $B$ moved two to the right, and $C$ moved 1 to the left.

For this equation to be true, $x$ has to be -1, meaning $A$ moves 1 left from her original seat. Since $A$ is now sitting in a corner seat, the only possible option for the original placement of $A$ is in seat number $\boxed{\textbf{(B)}\text{ 2}}$ .