2017 年 AMC 8 真题（P版）

1 答案 A

选项 \((A)\) 的值为 \(10\)。

选项 \((B)\) 的值为 \(0 + 1 + 7 = 8\)。

选项 \((C)\) 的值为 \(2 + 0 + 7 = 9\)。

选项 \((D)\) 的值为 \(2 + 0 + 7 = 9\)。

选项 \((E)\) 的值为 \(0\)。

所以正确答案是 A。

English

Option \((A)\) evaluates to \(10.\)

Option \((B)\) evaluates to \(0 + 1 + 7 = 8.\)

Option \((C)\) evaluates to \(2 + 0 + 7 = 9.\)

Option \((D)\) evaluates to \(2 + 0 + 7 = 9.\)

Option \((E)\) evaluates to \(0.\)

Thus, A is the correct answer.

2 答案 E

如果 \(36\) 票是总票数的 \(30\%\)，那么总票数的 \(10\%\) 是 \(12\) 票。总票数就是 \(10 \cdot 12 = 120\)。

所以正确答案是 E。

English

If \(36\) votes is \(30\%\) of the total votes, then \(10\%\) of the total votes is \(12\) votes. The number of total votes would then be \(10 \cdot 12 = 120.\)

Thus, E is the correct answer.

3 答案 C

这个表达式可化简为 \[ \begin{align*} \sqrt{16\sqrt{8\sqrt{4}}} &= \sqrt{16\sqrt{16}} \\ &= \sqrt{64}\\ &= 8 \end{align*} \]

所以正确答案是 C。

English

This expression can be reduced as follows: \[ \begin{align*} \sqrt{16\sqrt{8\sqrt{4}}} &= \sqrt{16\sqrt{16}} \\ &= \sqrt{64}\\ &= 8 \end{align*} \]

Thus, C is the correct answer.

4 答案 D

可以将乘积近似为 \[ \begin{align*} (3 \cdot 10^{-4})(8 \cdot 10^6) &= 24 \cdot 10^2 \\ &= 2{,}400 \end{align*}\text{。}\]

所以正确答案是 D。

English

We can approximate the product as \[ \begin{align*} (3 \cdot 10^{-4})(8 \cdot 10^6) &= 24 \cdot 10^2 \\ &= 2{,}400. \end{align*} \]

Thus, D is the correct answer.

5 答案 B

分母为 \(1+2+\cdots+8=36=2\cdot3\cdot6\)。从分子中约去这三个因数，剩下 \(4\cdot5\cdot7\cdot8=1120\)。

所以正确答案是 B。

English

The denominator is \(1+2+\cdots+8=36=2\cdot3\cdot6.\) Canceling those three factors from the numerator leaves \(4\cdot5\cdot7\cdot8=1120.\)

Thus, B is the correct answer.

6 答案 D

设三个角为 \(3x\)、\(3x\) 和 \(4x\)。它们的和为 \(180\)，所以 \(3x + 3x + 4x = 180\)，即 \(10x = 180\)，从而 \(x = 18\)。

最大角为 \(4x = 72\)。

所以正确答案是 D。

English

We can let the three angles be equal to \(3x,\) \(3x,\) and \(4x.\) Then we know that their sum equals \(180.\) From this we can set \(3x + 3x + 4x = 180,\) and solving this, we get \(10x = 180\) and \(x = 18.\)

The largest angle is \(4x = 72.\)

Thus, D is the correct answer.

7 答案 A

设 \(n\) 为任意一个重复数块所组成的三位数。那么 \[\begin{aligned} Z&=1000n+n\\ &=1001n\\ &=7\cdot11\cdot13\cdot n \end{aligned}\text{。}\]因此，\(11\) 一定是 \(Z\) 的因数。

所以正确答案是 A。

English

Let \(n\) be the three-digit number formed by either repeated block. Then \[\begin{aligned} Z&=1000n+n\\ &=1001n\\ &=7\cdot11\cdot13\cdot n. \end{aligned}\] Therefore, \(11\) must be a factor of \(Z.\)

Thus, A is the correct answer.

8 答案 D

陈述 \((1)\) 和 \((2)\) 不能同时为真，因为唯一的偶质数不是两位数。陈述 \((1)\) 和 \((3)\) 也不能同时为真，因为能被 \(7\) 整除的两位数不是质数。由于只有一个陈述为假，陈述 \((1)\) 必须为假，而陈述 \((2)\)、\((3)\) 和 \((4)\) 为真。

门牌号能被 \(2\) 和 \(7\) 整除，所以能被 \(14\) 整除。在两位的 \(14\) 的倍数中，唯一含有数字 \(9\) 的是 \(98\)。因此个位数字是 \(8\)。

所以正确答案是 D。

English

Statements \((1)\) and \((2)\) cannot both be true, because the only even prime is not two-digit. Statements \((1)\) and \((3)\) also cannot both be true, because a two-digit number divisible by \(7\) would not be prime. Since only one statement is false, statement \((1)\) must be false, while statements \((2),\) \((3),\) and \((4)\) are true.

The house number is divisible by \(2\) and \(7,\) so it is divisible by \(14.\) Among the two-digit multiples of \(14,\) the only one with a digit of \(9\) is \(98.\) Therefore, the units digit is \(8.\)

Thus, D is the correct answer.

9 答案 D

如果弹珠总数同时能被 \(3\) 和 \(4\) 整除，那么它必须能被 \(12\) 整除。测试 \(12\) 时，蓝色有 \(4\) 颗，红色有 \(3\) 颗，最多只剩 \(5\) 颗绿色，不可能。

若总数为 \(24\)，则蓝色有 \(8\) 颗，红色有 \(6\) 颗。黄色弹珠数为 \(24 - 8 - 6 - 6 = 4\)。

所以正确答案是 D。

English

If the number of marbles is divisible by both \(3\) and \(4,\) then the number must be divisible by \(12.\) If we test \(12,\) we get that there are \(4\) blue marbles and \(3\) red marbles. This leaves a maximum of \(5\) green marbles, which is not possible.

If there are \(24\) marbles, then there are \(8\) blue marbles and \(6\) red marbles. To find the number of yellow marbles, we get \(24 - 8 - 6 - 6 = 4.\)

Thus, D is the correct answer.

10 答案 C

从 \(5\) 张卡中选 \(3\) 张共有 \({5 \choose 3} = 10\) 种方式。如果 \(4\) 是所选卡片中的最大值，那么另外两张必须从 \(\{1, 2, 3\}\) 中选，有 \({3 \choose 2} = 3\) 种方式。因此概率为 \(\dfrac{3}{10}\)。

所以正确答案是 C。

English

The number of ways to choose \(3\) cards from \(5\) is \({5 \choose 3} = 10.\) If \(4\) is the largest value selected, then the other two cards have to be chosen from \(\{1, 2, 3\}.\) There are \({3 \choose 2} = 3\) ways to do this. The probability is then \(\dfrac{3}{10}.\)

Thus, C is the correct answer.

11 答案 C

两条对角线共有 \(37\) 块瓷砖，表示每条对角线上有 \(19\) 块瓷砖，因为中间有一块重叠。每行瓷砖数等于一条对角线上的瓷砖数，所以总瓷砖数为 \(19^2 = 361\)。

所以正确答案是 C。

English

\(37\) tiles on both diagonals imply that there are \(19\) tiles on each diagonal, since one tile overlaps in the middle. The total number of tiles would then be \(19^2 = 361\) since the number of tiles in each row is equal to the number of tiles in one diagonal.

Thus, C is the correct answer.

12 答案 D

如果一个数除以 \(4\)、\(5\) 和 \(6\) 时余数都是 \(1\)，那么它比这些数的最小公倍数多一。最小公倍数是 \(60\)，所以最小的这类正整数是 \(60 + 1 = 61\)。

所以正确答案是 D。

English

If a number leaves a remainder of \(1\) when divided by \(4,\) \(5,\) and \(6,\) then it is one more than the least common multiple of these numbers. The least common multiple is \(60,\) so the smallest such positive integer is \(60 + 1 = 61.\)

Thus, D is the correct answer.

13 答案 B

把三个人的记录合在一起，每一场胜利都对应一场失败；和棋若有，则对胜负总数都没有贡献。总负场数为 \(2+3+3=8\)，所以凯勒赢了 \(8-4-3=1\) 局。

所以正确答案是 B。

English

Across all players, every win is matched by one loss; drawn games, if any, contribute neither. The number of losses is \(2+3+3=8,\) so the number of games Kyler won is \(8-4-3=1.\)

Thus, B is the correct answer.

14 答案 C

因为题目数量不影响答案，可以假设作业有 \(100\) 道题。\(80\%\) 的 \(50\) 是 \(40\)，所以克洛伊独立答对 \(40\) 道。\(88\%\) 的 \(100\) 是 \(88\)，所以克洛伊总共答对 \(88\) 道。因此共同完成部分她答对 \(88 - 40 = 48\) 道。

\(90\%\) 的 \(50\) 是 \(45\)，所以佐伊独立答对 \(45\) 道。共同完成部分她也答对 \(48\) 道，所以总共答对 \(45 + 48 = 93\) 道，整体正确率为 \(93\%\)。

所以正确答案是 C。

English

Since the answer is the same regardless of the number of problems, we can assume that there were \(100\) problems on the assignment. \(80\%\) of \(50\) is \(40,\) so Chloe answered \(40\) questions correctly alone. \(88\%\) of \(100\) is \(88,\) so Chloe answered \(88\) questions correctly in total. This means that Chloe answered \(88 - 40 = 48\) together with Zoe.

\(90\%\) of \(50\) is \(45,\) so Zoe answered \(45\) questions correctly by herself. We know that she answered \(48\) questions correctly with Chloe, so she answered \(45 + 48 = 93\) correctly in total. This means that her overall percentage is \(93\%.\)

Thus, C is the correct answer.

15 答案 D

从 \(A\) 出发，有 \(4\) 种方式到达一个 \(M\)。从每个 \(M\) 出发，有 \(3\) 种方式到达一个 \(C\)。从每个 \(C\) 出发，有 \(2\) 种方式到达一个 \(8\)。因此路径总数为 \(4 \cdot 3 \cdot 2 = 24\)。

所以正确答案是 D。

English

Starting from \(A,\) there are \(4\) ways to reach an \(M.\) From each \(M,\) there are \(3\) ways to reach a \(C.\) From each \(C,\) there are \(2\) ways to reach an \(8.\) Multiplying all these possibilities, we get \(4 \cdot 3 \cdot 2 = 24.\)

Thus, D is the correct answer.

16 答案 D

要使两个三角形周长相等，\(\overline{BC}\) 必须分成 \(\overline{CD} = 3\) 和 \(\overline{BD} = 2\)。

\(\triangle ACD\) 和 \(\triangle ABD\) 有相同的高，所以面积与底边长度成比例。这说明 \(\triangle ABD\) 的面积是 \(\dfrac{2}{5}\) 倍的 \(\triangle ABC\) 的面积，即 \[\dfrac{2}{5} \cdot 3 \cdot \dfrac{4}{2} = \dfrac{12}{5}\text{。}\]

所以正确答案是 D。

English

The only way to split \(\overline{BC}\) into two parts such that the two triangles have the same perimeter is if \(\overline{CD} = 3\) and \(\overline{BD} = 2.\)

\(\triangle ACD\) and \(\triangle ABD\) have the same altitudes, so their areas are proportional to their bases. This means that the area of \(\triangle ABD\) is \(\dfrac{2}{5}\) the area of \(\triangle ABC,\) which is \[\dfrac{2}{5} \cdot 3 \cdot \dfrac{4}{2} = \dfrac{12}{5}.\]

Thus, D is the correct answer.

17 答案 C

设宝箱数为 \(n\)，金币数为 \(g\)。那么 \[ 9(n - 2) = g \] 且 \[ 6n + 3 = g\text{。}\]解这个方程组得 \(n = 7\)，所以金币数为 \(6 \cdot 7 + 3 = 45\)。

所以正确答案是 C。

English

Let \(n\) be the number of treasure chests and \(g\) be the number of gold coins. Then \[9(n - 2) = g\] and \[6n + 3 = g.\] Solving this system yields \(n = 7,\) so the number of gold coins is \(6 \cdot 7 + 3 = 45.\)

Thus, C is the correct answer.

18 答案 B

因为 \(\angle BCD\) 是直角，对 \(\triangle BCD\) 用勾股定理得 \(\overline{BD} = 5\)。又可得 \(\angle DBA\) 是直角，因为 \(\triangle BDA\) 的三边构成勾股三元组。

四边形 \(ABCD\) 的面积等于 \[\begin{align*} \text{面积}(\triangle &BDA) - \text{面积}(\triangle BCD) \\ &= \dfrac{1}{2} \cdot 12 \cdot 5 - \dfrac{1}{2} \cdot 4 \cdot 3 \\ &= 30 - 6 \\ &= 24 \end{align*}\text{。}\]

所以正确答案是 B。

English

Since \(\angle BCD\) is a right angle, we can apply the Pythagorean theorem to \(\triangle BCD\) to get that \(\overline{BD} = 5.\) We also get that \(\angle DBA\) is right since the sides of \(\triangle BDA\) form a Pythagorean triple.

Then the area of \(ABCD\) is equal to \[\begin{align*} \text{area}(\triangle &BDA) - \text{area}(\triangle BCD) \\ &= \dfrac{1}{2} \cdot 12 \cdot 5 - \dfrac{1}{2} \cdot 4 \cdot 3 \\ &= 30 - 6 \\ &= 24. \end{align*}\]

Thus, B is the correct answer.

19 答案 D

提出公因式 \(98!\)：\[ \begin{aligned} 98!+99!+100! &=98!\bigl(1+99\\ &\qquad+100\cdot99\bigr)\\ &=98!\cdot100^2 \end{aligned}\text{。}\]

\(98!\) 中因子 \(5\) 的指数是 \(\left\lfloor\frac{98}{5}\right\rfloor+\left\lfloor\frac{98}{25}\right\rfloor\)，等于 \(19+3=22\)。因子 \(100^2\) 还贡献四个因子 \(5\)，所以总指数是 \(22+4=26\)。

所以正确答案是 D。

English

Factor out \(98!\): \[\begin{aligned} 98!+99!+100! &=98!\bigl(1+99\\ &\qquad+100\cdot99\bigr)\\ &=98!\cdot100^2. \end{aligned}\]

The exponent of \(5\) in \(98!\) is \(\left\lfloor\frac{98}{5}\right\rfloor+\left\lfloor\frac{98}{25}\right\rfloor\), which equals \(19+3=22.\) The factor \(100^2\) contributes four more factors of \(5\), so the total exponent is \(22+4=26.\)

Thus, D is the correct answer.

20 答案 B

这个数是奇数，所以个位数字有 \(5\) 种选择。千位数字不能为零，也不能等于个位数字，因此有 \(8\) 种选择。百位有 \(8\) 种选择，十位有 \(7\) 种选择。符合条件的数共有 \(5 \cdot 8 \cdot 8 \cdot 7 = 2{,}240\)，所以概率为 \(\dfrac{2{,}240}{9{,}000} = \dfrac{56}{225}\)。

所以正确答案是 B。

English

Since the number is odd, the last digit is odd, giving \(5\) possibilities. The thousands digit cannot be zero or the number we already got, so that gives \(8\) possibilities. Similarly, the hundreds digit has \(8\) possibilities, and the tens digit has \(7\) possibilities. This gives a total of \(5 \cdot 8 \cdot 8 \cdot 7 = 2{,}240,\) making the probability \(\dfrac{2{,}240}{9{,}000} = \dfrac{56}{225}.\)

Thus, B is the correct answer.

21 答案 A

因为 \(a+b+c=0\) 且三个数都不为零，所以它们的符号只能是两个正数、一个负数，或两个负数、一个正数。第一种情况下，前三个符号分式的和为 \(1\)，而 \(\dfrac{abc}{|abc|}=-1\)；第二种情况下，前三个分式的和为 \(-1\)，而 \(\dfrac{abc}{|abc|}=1\)。无论哪种情况，整个表达式都等于 \(0\)。

所以正确答案是 A。

English

Because \(a+b+c=0\) and none of the numbers is zero, their signs are either two positive and one negative or two negative and one positive. In the first case, the first three sign fractions sum to \(1\), while \(\dfrac{abc}{|abc|}=-1\). In the second case, the first three sum to \(-1\), while \(\dfrac{abc}{|abc|}=1\). Either way, the entire expression equals \(0\).

Thus, A is the correct answer.

22 答案 D

设 \(O\) 为内切半圆圆心，\(D\) 为半圆与 \(\overline{AB}\) 的切点。于是 \(BD = 5\)，因为 \(\overline{BD}\) 和 \(\overline{BC}\) 都与半圆相切。由此得 \(AD = 8\)，并记 \(OD = r\)。因为 \(\overline{OD}\) 垂直于 \(\overline{AB}\)，所以 \(\triangle ADO \sim \triangle BCA\)，有 \(\dfrac{r}{8} = \dfrac{5}{12}\)，解得 \(r = \dfrac{10}{3}\)。

所以正确答案是 D。

English

Let \(O\) be the center of the inscribed semicircle and \(D\) be the tangent point of the semicircle on \(\overline{AB}.\) Then \(BD = 5\) since \(\overline{BD}\) and \(\overline{BC}\) are tangents to the semicircle. Then \(AD = 8\) and \(OD = r.\) \(\overline{OD}\) is perpendicular to \(\overline{AB}\) so \(\triangle ADO \sim \triangle BCA,\) so \(\dfrac{r}{8} = \dfrac{5}{12}.\) Solving this, we get \(r = \dfrac{10}{3}.\)

Thus, D is the correct answer.

23 答案 C

琳达每天旅行 \(60\) 分钟。因为每天每英里所需分钟数是整数，且每天行进英里数也是整数，所以每天的每英里分钟数必须是 \(60\) 的因数。\(60\) 的因数为 \(1\)、\(2\)、\(3\)、\(4\)、\(5\)、\(6\)、\(10\)、\(12\)、\(15\)、\(20\)、\(30\) 和 \(60\)。其中唯一一组四个相邻相差 \(5\) 的数是 \(5\)、\(10\)、\(15\) 和 \(20\)。四天总路程为 \[ \dfrac{60}{5} + \dfrac{60}{10} + \dfrac{60}{15} + \dfrac{60}{20} = 25 \] 英里。

所以正确答案是 C。

English

Linda traveled for \(60\) minutes every day. Since one mile was traveled in an integer amount of minutes each day, her minutes per mile every day must be a factor of \(60.\) The factors of \(60\) are \(1,\) \(2,\) \(3,\) \(4,\) \(5,\) \(6,\) \(10,\) \(12,\) \(15,\) \(20,\) \(30,\) and \(60.\) The only sequence of four of these numbers that differ by \(5\) are \(5,\) \(10,\) \(15,\) and \(20.\) For the four days, she traveled \[ \dfrac{60}{5} + \dfrac{60}{10} + \dfrac{60}{15} + \dfrac{60}{20} = 25 \] miles in total.

Thus, C is the correct answer.

24 答案 D

在一个 \(60\) 天周期中，第一个孙辈打 \(20\) 次，第二个打 \(15\) 次，第三个打 \(12\) 次。但 \(20 + 15 + 12 = 47\) 有重复计数：第一个和第二个孙辈在同一天打电话 \(\frac{60}{12} = 5\) 次，第一个和第三个在同一天打电话 \(\frac{60}{15} = 4\) 次，第二个和第三个在同一天打电话 \(\frac{60}{20} = 3\) 次。从 \(47\) 中减去这些，得 \(47 - 5 - 4 - 3 = 35\)。

第 \(60\) 天三人都打电话，被加了三次又减了三次，所以要加回一次，得到每 \(60\) 天有 \(36\) 天收到电话，也就是 \(24\) 天没有电话。这一年有 \(6\) 个完整的 \(60\) 天周期，剩下第 \(361\) 天和第 \(362\) 天没有电话，因此没有电话的天数为 \(24 \cdot 6 + 2 = 146\)。

所以正确答案是 D。

English

In a \(60\)-day period, the first child calls \(20\) times, the second child calls \(15\) times, and the third child calls \(12\) times. \(20 + 15 + 12 = 47\) overcounts, however. The first and second children call on the same day \(\frac{60}{12} = 5\) times. The first and third children call on the same day \(\frac{60}{15} = 4\) times. The second and third children call on the same day \(\frac{60}{20} = 3\) times. Subtracting these from \(47\) yields \(47 - 5 - 4 - 3 = 35.\)

The \(60\)th day is added in thrice and subtracted out thrice, so we need to add it back in. This means that for every \(60\) days, Mrs. Sanders receives a call \(36\) days, which means that she does not receive a call on \(24\) days. There are \(6\) \(60\)-day periods, and there are no calls on the \(361\)st or \(362\)nd day, which results in \(24 \cdot 6 + 2 = 146\) total days with no phone calls.

Thus, D is the correct answer.

25 答案 B

可以延长 \(\overline{SU}\) 和 \(\overline{TU}\)，形成如下图形。

所求面积等于边长为 \(4\) 的等边三角形面积，减去两个半径为 \(2\) 的六分之一圆面积。边长为 \(s\) 的等边三角形面积为 \(\dfrac{s^2\sqrt{3}}{4}\)。因此总面积为 \(\dfrac{4^2 \sqrt{3}}{4} - \dfrac{1}{3} \pi \cdot 2^2 = 4 \sqrt{3} - \dfrac{4}{3} \pi\)。

所以正确答案是 B。

English

We can extend \(\overline{SU}\) and \(\overline{TU}\) to form the following picture.

The area of this region is the area of an equilateral triangle with side length of \(4\) minus the area of two-sixths of a circle with radius \(2.\) The area for an equilateral triangle with side length \(s\) is \(\dfrac{s^2\sqrt{3}}{4}.\) This means that the total area is \(\dfrac{4^2 \sqrt{3}}{4} - \dfrac{1}{3} \pi \cdot 2^2 = 4 \sqrt{3} - \dfrac{4}{3} \pi.\)

Thus, B is the correct answer.