2006 年 AMC 8 真题（P版）

1 答案 D

这些数都接近整数美元，可以把价格四舍五入后相加。

这些价格分别约为 \(\$ 2\)、\(\$ 5\) 和 \(\$ 10\)，总和为 \(\$ 17\)。

所以正确答案是 D。

English

Since all the values are already close to a whole number, we can just add the sums of the rounded numbers.

These prices round to \(\$ 2,\) \(\$ 5,\) and \(\$ 10,\) which add to \(\$ 17.\)

Thus, D is the correct answer.

2 答案 C

AMC \(8\) 每答对一题得 \(1\) 分，所以 Billy 得 \(13\) 分。

所以正确答案是 C。

English

Since the AMC \(8\) only awards \(1\) point for each correct question, Billy will get \(13\) points.

Thus, C is the correct answer.

3 答案 A

起初 Elisa 每圈用 \(\frac{25}{10}=2.5\) 分钟。现在她每圈用 \(\frac{24}{12}=2\) 分钟。

因此每圈用时减少 \(2.5-2=0.5=\dfrac{1}{2}\) 分钟。

所以正确答案是 A。

English

Initially, Elisa swam one lap in \(\frac{25}{10}=2.5\) minutes. Now she swims one lap in \(\frac{24}{12}=2\) minutes.

Therefore, she improved her lap time by \(2.5-2=0.5=\dfrac{1}{2}\) minute.

Thus, A is the correct answer.

4 答案 B

转盘顺时针转 \(2 \frac{1}{4}\) 圈，再逆时针转 \(3 \frac{3}{4}\) 圈，等价于总共逆时针转 \(1 \frac{1}{2}\) 圈。

因此转盘最终指向东。

所以正确答案是 B。

English

If the spinner goes \(2 \frac{1}{4}\) revolutions clockwise and \(3 \frac{3}{4}\) revolutions counterclockwise, it goes \(1 \frac{1}{2}\) revolutions counterclockwise.

This means that the spinner will be pointing east.

Thus, B is the correct answer.

5 答案 D

可以把所有三角形向内折叠，正好覆盖小正方形。这说明小正方形的面积是大正方形面积的一半。

所以小正方形面积为 \(60 \div 2 = 30\)。

所以正确答案是 D。

English

Note that we can fold all the triangles in to perfectly cover the smaller square. This means that area of the smaller square is half the area of the larger square.

This makes the area of the smaller square \(60 \div 2 = 30.\)

Thus, D is the correct answer.

6 答案 C

如果分别计算两个长方形的总周长，会得到 \[ 2(2(2 + 4)) = 2 \cdot 2 \cdot 6 = 24\text{。}\]

在字母 T 中，它们的相交部分从每个长方形中去掉了一段长度 \(2\) 的边。因此 T 的周长为 \[ 24 - 2 \cdot 2 = 20\text{。}\]

所以正确答案是 C。

English

If we found the total perimeter of the two rectangles separately, we would have gotten \[ 2(2(2 + 4)) = 2 \cdot 2 \cdot 6 = 24. \]

In the letter T, we can see that their intersection removes a piece of length \(2\) from each of the rectangles. Therefore, the perimeter of the T is \[ 24 - 2 \cdot 2 = 20. \]

Thus, C is the correct answer.

7 答案 B

公式为 \(C = 2 \pi r\) 和 \(A = \pi r^2\)。由此圆 \(Y\) 的半径是 \[ 8 \pi \div (2 \pi) = 4\text{。}\]圆 \(Z\) 的半径是 \[ \sqrt{9 \pi \div \pi} = 3\text{。}\]因为 \(\pi\) 大于 \(3\) 且小于 \(4\)，所以正确顺序为 \(Z\)、\(X\)、\(Y\)。

所以正确答案是 B。

English

Recall that \(C = 2 \pi r\) and \(A = \pi r^2.\) Using these formulas we get that the radius of \(Y\) is \[ 8 \pi \div (2 \pi) = 4. \] We also get that the radius of \(Z\) is \[ \sqrt{9 \pi \div \pi} = 3. \] As \(\pi\) is greater than \(3\) and less than \(4,\) the correct order is \(Z,\) \(X,\) \(Y.\)

Thus, the answer is B .

8 答案 E

接受调查的男性总数是调查总人数减去女性人数：\(200-96=104\)。

这些男性中 \(26\) 人不收听，所以 \(104-26=78\) 名男性收听。所求百分比为 \(\dfrac{78}{104}\cdot 100\%=75\%\)。

所以正确答案是 E。

English

The total number of males surveyed is the total number surveyed minus the number of females surveyed: \(200-96=104\).

Of those males, \(26\) do not listen, so \(104-26=78\) males listen. The desired percentage is \(\dfrac{78}{104}\cdot 100\%=75\%\).

Thus, E is the correct answer.

9 答案 C

每个分数的分子都会与下一个分数的分母约掉。最后只剩 \[ \dfrac{2006}{2} = 1003\text{。}\]

所以正确答案是 C。

English

Note that the numerator of every fraction cancels with the denominator of the following fraction. This leaves two numbers: \[ \dfrac{2006}{2} = 1003. \]

Thus, C is the correct answer.

10 答案 A

满足 \(wl=12\) 的正整数因数对 \((w,l)\) 为 \((1,12)\)、\((2,6)\)、\((3,4)\)、\((4,3)\)、\((6,2)\) 和 \((12,1)\)。

这些点形成六个递减排列的点，只有图 A 与它们匹配。

所以正确答案是 A。

English

The positive integer factor pairs \((w,l)\) with \(wl=12\) are \((1,12),\) \((2,6),\) \((3,4),\) \((4,3),\) \((6,2),\) and \((12,1)\).

These points form a decreasing set of six points, and only graph A matches them.

Thus, A is the correct answer.

11 答案 C

数位和为 \(1\) 的数有 \(1\) 个：\(10\)。

数位和为 \(4\) 的数有 \(4\) 个：\(13\)、\(22\)、\(31\) 和 \(40\)。

数位和为 \(9\) 的数有 \(9\) 个：\(18\)、\(27\)、\(36\)、\(45\)、\(54\)、\(63\)、\(72\)、\(81\) 和 \(90\)。

数位和为 \(16\) 的数有 \(3\) 个：\(79\)、\(88\) 和 \(97\)。

因此共有 \(17\) 个数满足题意。

所以正确答案是 C。

English

There is \(1\) number whose digit sum is \(1\): \(10.\)

There are \(4\) numbers whose digit sum is \(4\): \(13,\) \(22,\) \(31,\) and \(40.\)

There are \(9\) numbers whose digit sum is \(9\): \(18,\) \(27,\) \(36,\) \(45,\) \(54,\) \(63,\) \(72,\) \(81,\) and \(90.\)

There are \(3\) numbers whose digit sum is \(16\): \(79,\) \(88,\) and \(97.\)

Therefore, there are \(17\) numbers that satisfy the problem statement.

Thus, C is the correct answer.

12 答案 D

Antonette 第一次测试答对 \(0.7\cdot 10=7\) 题。类似地，她第二、三次测试分别答对 \(0.8\cdot 20=16\) 题和 \(0.9\cdot 30=27\) 题。

总共答对 \(50\) 题。合并测试成绩为 \(100\cdot \dfrac{50}{60}=83\frac{1}{3}\%\)，最接近 \(83\%\)。

所以正确答案是 D。

English

Antonette got \(0.7\cdot 10=7\) questions right on the first test. Similarly, she got \(0.8\cdot 20=16\) and \(0.9\cdot 30=27\) problems right on her second and third tests.

Adding these up yields \(50\) correct questions. Her score on the combined test would have been \(100\cdot \dfrac{50}{60}=83\frac{1}{3}\%\), closest to \(83\%\).

Thus, D is the correct answer.

13 答案 D

Brian 开始骑车时，Cassie 已经骑了 \(\dfrac{1}{2}\cdot 12=6\) 英里。因此两人相距 \(62-6=56\) 英里。

他们合起来以每小时 \(12+16=28\) 英里的速度缩短距离。因此他们在上午 \(9{:}00\) 后 \(56\div 28=2\) 小时相遇。

所以他们在上午 \(11:00\) 相遇。

所以正确答案是 D。

English

By the time Brian starts biking, Cassie has already traveled \(\dfrac{1}{2}\cdot 12=6\) miles. This means that Cassie and Brian are then \(62-6=56\) miles apart.

Together, they close the distance at \(12+16=28\) miles per hour. This means that they meet \(56\div 28=2\) hours after \(9{:}00\) AM.

This means that they meet at \(11:00\) AM.

Thus, D is the correct answer.

14 答案 B

Bob 读完整本书需要 \(760 \cdot 45\) 秒，Chandra 需要 \(760 \cdot 30\) 秒。

两人阅读时间相差 \[ 760 \cdot 45 - 760 \cdot 30 \] \[= 760(45 - 30) \] \[= 760 \cdot 15 \] \[= 11,400 \] 秒。

所以正确答案是 B。

English

Bob will take \(760 \cdot 45\) seconds to read the book, and Chandra will take \(760 \cdot 30\) seconds.

The difference between the time they spent reading is \[ 760 \cdot 45 - 760 \cdot 30 \]\[= 760(45 - 30) \]\[= 760 \cdot 15 \]\[= 11,400 \] seconds.

Thus, B is the correct answer.

15 答案 C

设 Chandra 阅读 \(x\) 页，那么 Bob 阅读 \(760 - x\) 页。

为了让阅读时间相同，\[ \begin{align*} 30x &= 45(760 - x) \\ 30x &= 45 \cdot 760 - 45x \\ 75x &= 45 \cdot 760 \\ 5x &= 3 \cdot 760 \\ x &= 3 \cdot 152 \\ x &= 456 \end{align*}\text{。}\]

所以正确答案是 C。

English

Let \(x\) be the number of pages that Chandra will read. Then Bob will read \(760 - x\) pages.

For them to read for the same amount of time, \[ \begin{align*} 30x &= 45(760 - x) \\ 30x &= 45 \cdot 760 - 45x \\ 75x &= 45 \cdot 760 \\ 5x &= 3 \cdot 760 \\ x &= 3 \cdot 152 \\ x &= 456. \end{align*} \]

Thus, C is the correct answer.

16 答案 E

如果 \(3\) 人阅读相同时间，那么 Bob、Chandra 和 Alice 阅读页数之比为 \(4:6:9\)。

因此他们分别读 \(160\)、\(240\) 和 \(360\) 页。因为三人阅读时间相同，只需计算 Bob 读他那部分所需时间。

Bob 读完这部分需要 \(45 \cdot 160 = 7200\) 秒。

所以正确答案是 E。

English

If all \(3\) individuals read the same amount of time, then the number of pages Bob, Chandra, and Alice will read will be in the ratio \(4:6:9\) respectively.

This means that they will read \(160,\) \(240,\) and \(360\) pages respectively. Since they all read for the same amount of time, we can just calculate how long it takes for Bob to read his portion.

Bob will take \(45 \cdot 160 = 7200\) seconds to read his portion.

Thus, E is the correct answer.

17 答案 B

转盘 \(Q\) 总是落在偶数上，所以不改变总和的奇偶性。转盘 \(R\) 总是落在奇数上。

为了让总和为奇数，转盘 \(P\) 的数必须是偶数。转盘 \(P\) 的三个等可能区域中只有一个是偶数。

因此概率为 \(\dfrac{1}{3}\)。

所以正确答案是 B。

English

Spinner \(Q\) always lands on an even number, so it does not change the parity of the total. Spinner \(R\) always lands on an odd number.

For the full sum to be odd, the number from spinner \(P\) must be even. Only one of the three equal sectors of spinner \(P\) contains an even number.

Therefore the probability is \(\dfrac{1}{3}\).

Thus, B is the correct answer.

18 答案 D

因为每个面的黑白表面积比例相同，只需分析一个面的白色比例。

一个面上有 \(9\) 个单位正方形，其中 \(4\) 个是黑色。因此每个面有 \(\dfrac{5}{9}\) 是白色。

所以正确答案是 D。

English

Since each face has the same black and white surface area, we can analyze what fraction of one side is white.

On one side, there are \(9\) unit squares. \(4\) of them are black. This means that \(\dfrac{5}{9}\) of each face is white.

Thus, D is the correct answer.

19 答案 D

由全等条件可知 \[AB = EC = 11\text{。}\]

由等腰条件，\[BC = AB = 11\text{。}\]

因为 \(D\) 是 \(\overline{BC}\) 的中点，\[BD = BC \div 2 = 5.5\text{。}\]

所以正确答案是 D。

English

By the congruency condition, we know that \[AB = EC = 11.\]

Also from the isosceles condition, we know that \[BC = AB = 11.\]

Since \(D\) is the midpoint of \(\overline{BC},\) we know that \[BD = BC \div 2 = 5.5.\]

Thus, D is the correct answer.

20 答案 C

每场比赛恰好有一个胜者。六名选手共有 \(6 \cdot 5 \div 2 = 15\) 场比赛，因此共有 \(15\) 个胜场。

已经计入的胜场数为 \[ 4 + 3 + 2 + 2 + 2 = 13 \] 场，所以 Monica 赢了 \(15 - 13 = 2\) 场。

所以正确答案是 C。

English

In every match, there was exactly one winner. There are \(6 \cdot 5 \div 2 = 15\) games and therefore \(15\) wins.

There are already \[ 4 + 3 + 2 + 2 + 2 = 13 \] wins accounted for, so Monica won \(15 - 13 = 2\) games.

Thus, C is the correct answer.

21 答案 A

水族箱底面积为 \(100\cdot 40=4000\text{ cm}^2\)。

完全浸没的石头排开 \(1000\text{ cm}^3\) 的水，所以水位上升 \(1000\div 4000=0.25\) cm。

所以正确答案是 A。

English

The base area of the aquarium is \(100\cdot 40=4000\text{ cm}^2\).

The submerged rock displaces \(1000\text{ cm}^3\) of water, so the water level rises by \(1000\div 4000=0.25\) cm.

Thus, A is the correct answer.

22 答案 D

如果底行格子中的数依次为 \(a\)、\(b\) 和 \(c\)，那么中间一行就是 \(a+b\) 和 \(b+c\)。

因此顶格为 \(a+2b+c\)。要使它最小，把 \(b=1\) 放在中间，外侧放 \(2\) 和 \(3\)，得到顶格数 \(7\)。

要使它最大，把 \(b=9\) 放在中间，外侧放 \(7\) 和 \(8\)，得到顶格数 \(33\)。所求差为 \(33-7=26\)。

所以正确答案是 D。

English

If the lower cells contain \(a,\) \(b,\) and \(c,\) the middle row will have \(a+b\) and \(b+c\).

This means that the top row will have \(a+2b+c\). To minimize this, put \(b=1\) in the middle and \(2\) and \(3\) in the outer cells. This yields a top number of \(7\).

To maximize it, put \(b=9\) in the middle and \(7\) and \(8\) in the outer cells. This yields a top number of \(33\). The desired difference is \(33-7=26\).

Thus, D is the correct answer.

23 答案 A

除以 \(6\) 余 \(4\) 的正整数为 \[ 4, 10, 16, 22, 28, 34, 40, \cdots\text{。}\]除以 \(5\) 余 \(3\) 的正整数为 \[ 3, 8, 13, 18, 23, 28, 33, \cdots\text{。}\]

可见满足条件的最少金币数是 \(28\)。它除以 \(7\) 余 \(0\)。

所以正确答案是 A。

English

The positive integers that leave a remainder of \(4\) when divided by \(6\) are \[ 4, 10, 16, 22, 28, 34, 40, \cdots. \] The positive integers that leave a remainder of \(3\) when divided by \(5\) are \[ 3, 8, 13, 18, 23, 28, 33, \cdots. \]

From this, we can see that the smallest number of coins that work is \(28.\) This leaves a remainder of \(0\) when divided by \(7.\)

Thus, A is the correct answer.

24 答案 A

注意 \[ CDCD = 101 \cdot CD\text{。}\]这迫使 \(ABA = 101\)。所以 \(A = 1\)，\(B = 0\)，且 \(A + B = 1\)。

所以正确答案是 A。

English

Note that \[ CDCD = 101 \cdot CD. \] This forces \(ABA = 101.\) Then \(A = 1,\) \(B = 0,\) and \(A + B = 1.\)

Thus, A is the correct answer.

25 答案 B

公共和必须是奇数。如果公共和是偶数，那么 \(44\) 和 \(38\) 背后的隐藏数都必须是偶质数，但偶质数只有一个。

所以奇数 \(59\) 背后的隐藏质数必须是 \(2\)，公共和为 \(59+2=61\)。

另外两个隐藏质数是 \(61-44=17\) 和 \(61-38=23\)。

隐藏质数的平均数为 \(\dfrac{2+17+23}{3}=14\)。

所以正确答案是 B。

English

The common sum must be odd. If the common sum were even, then the hidden numbers behind \(44\) and \(38\) would both have to be even primes, but there is only one even prime.

So the prime hidden behind the odd visible number \(59\) must be \(2\), making the common sum \(59+2=61\).

The other two hidden primes are \(61-44=17\) and \(61-38=23\).

The average of the hidden primes is \(\dfrac{2+17+23}{3}=14\).

Thus, B is the correct answer.