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2015 AMC 8考题11-12

2018-08-06 重点归纳

AMC 8数学竞赛专为8年级及以下的初中学生设计，但近年来的数据显示，越来越多小学4-6年级的考生加入到AMC 8级别的考试行列中，而当这些学生能在成绩中取得“A”类标签，则是对孩子数学天赋的优势证明，不管是对美高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AMC 8的官方真题以及官方解答吧：

Problem 11

In the small country of Mathland, all automobile license plates have four symbols. The first must be a vowel (A, E, I, O, or U), the second and third must be two different letters among the 21 non-vowels, and the fourth must be a digit (0 through 9). If the symbols are chosen at random subject to these conditions, what is the probability that the plate will read "AMC8"?

Solution 1

There is one favorable case, which is the license plate says "AMC 8". We must now find how many total cases there are. There are5choices for the first letter (since it must be a vowel),21choices for the second letter (since it must be of21consonants),20choices for the third letter (since it must differ from the second letter), and10choices for the number. This leads to 5*21*20*10=21000total possible license plates. That means the probability of a license plate saying "AMC8" is （B）1/21000.

Solution 2

The probability of choosing A as the first letter is 1/5. The probability of choosing M next is 1/21. The probability of choosing C as the third letter is 1/20(since there are 20 other consonants to choose from other then M). The probability of having 8 as the last number is 1/10. We multiply all these to obtain

Problem 12

How many pairs of parallel edges, such as AB and GH or EH and FG, does a cube have?

$\textbf{(A) }6 \quad\textbf{(B) }12 \quad\textbf{(C) } 18 \quad\textbf{(D) } 24 \quad \textbf{(E) } 36$

[asy] import three; currentprojection=orthographic(1/2,-1,1/2); /* three - currentprojection, orthographic */ draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle); draw((0,0,0)--(0,0,1)); draw((0,1,0)--(0,1,1)); draw((1,1,0)--(1,1,1)); draw((1,0,0)--(1,0,1)); draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle); label(

Solution 1

We first count the number of pairs of parallel lines that are in the same direction as AB. The pairs of parallel lines are AB, CD and GH, AB and CD, EF and GH, AB and GH, and CD and EF. These are 6 pairs total. We can do the same for the lines in the same direction as AE and AD. This means there are 6*3=（C）18 total pairs of parallel

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