2009 年 AMC 8 真题（P版）

1 答案 E

可以倒推，从布里奇特自己留下的 \(4\) 个苹果开始。加上她给凯茜的 \(3\) 个苹果，这一半共有 \(7\) 个苹果。

因为她把原来苹果的一半给了安，所以再乘以 \(2\)。\(7 \cdot 2 = 14\)，因此布里奇特一开始有 \(14\) 个苹果。

所以正确答案是 E。

English

We can work backwards, starting with the \(4\) apples that Bridget kept for herself. Adding the \(3\) apples that she gave Cassie, she now has \(7\) apples.

Finally, we multiply this value by \(2\) since she gave half of her initial apples to Ann. \(7 \cdot 2 = 14,\) so Bridget started off with \(14\) apples.

Thus, E is the correct answer.

2 答案 D

列出比例 \[ \dfrac{4}{7} = \dfrac{28}{x}\text{。}\]交叉相乘得 \[ 4x = 7 \cdot 28 \] 因而 \[ x = 49\text{。}\]

所以正确答案是 D。

English

Set up the proportion \[ \dfrac{4}{7} = \dfrac{28}{x}. \] Cross-multiplying gives \[ 4x = 7 \cdot 28 \] and hence \[ x = 49. \]

Thus, D is the correct answer.

3 答案 C

从图中可知，苏珊娜骑了 \(3\) 英里，用时 \(15\) 分钟。因此在 \(30\) 分钟内，她会骑 \(6\) 英里。

所以正确答案是 C。

English

From the graph, we can see that Suzanna rides \(3\) miles in \(15\) minutes. This means that in \(30\) minutes, she will have ridden \(6\) miles.

Thus, C is the correct answer.

4 答案 B

注意选项 B 中没有任何长 \(5\) 个方格的连续线段。这说明那块长 \(5\) 个方格的拼片无法放入该图形。

所以正确答案是 B。

English

Note that option B does not have any segments in it that are \(5\) blocks long. This means that it is impossible to arrange the \(5\) block long piece to fit within the figure.

Thus, B is the correct answer.

5 答案 D

数列开始为 \(1\)、\(2\)、\(3\)。

接下来的项为 \(1+2+3=6\)、\(2+3+6=11\)、\(3+6+11=20\)、\(6+11+20=37\)、\(11+20+37=68\)。

所以正确答案是 D。

English

The sequence begins \(1,\) \(2,\) \(3.\)

The next terms are \(1+2+3=6,\) \(2+3+6=11,\) \(3+6+11=20,\) \(6+11+20=37,\) and \(11+20+37=68.\)

Thus, D is the correct answer.

6 答案 A

\(4\) 根水管合起来每分钟注入 \[2.5 \cdot 4 = 10\] 加仑水。

要注满 \(24{,}000\) 加仑，水管需要 \[24{,}000 \div 10 = 2{,}400\] 分钟。

\(2{,}400\) 分钟等于 \[2{,}400 \div 60 = 40\] 小时。

所以正确答案是 A。

English

The \(4\) hoses together fill the pool with \[2.5 \cdot 4 = 10\] gallons of water per minute.

To fill \(24{,}000\) gallons, it will take the hoses \[24{,}000 \div 10 = 2{,}400\] minutes to fill the pool.

\(2{,}400\) minutes is the same as \[2{,}400 \div 60 = 40\] hours.

Thus, A is the correct answer.

7 答案 C

三角形 \(ADC\) 的底边是 \(CD\)，长度为 \(3\)。高也是 \(3\)。

因此 \(ADC\) 的面积为 \(\dfrac{1}{2} \cdot 3 \cdot 3 = 4.5\)。

所以正确答案是 C。

English

The base of \(ADC\) is \(CD,\) which is \(3.\) The altitude is \(3\) as well.

Therefore, the area of \(ADC\) is \(\dfrac{1}{2} \cdot 3 \cdot 3 = 4.5.\)

Thus, C is the correct answer.

8 答案 B

设原来的长和宽为 \(l\) 和 \(w\)，则原面积为 \(lw\)。

新的长为 \(1.1l\)，新的宽为 \(0.9w\)。因此新面积为 \(1.1\cdot0.9lw=0.99lw\)。

这说明新面积是原面积的 \(99\%\)。

所以正确答案是 B。

English

Let the old length and width be \(l\) and \(w,\) so the old area is \(lw.\)

The new length is \(1.1l,\) and the new width is \(0.9w.\) Thus the new area is \(1.1\cdot0.9lw=0.99lw.\)

This shows that the new area is \(99\%\) of the old area.

Thus, B is the correct answer.

9 答案 B

三角形和八边形在链的两端，所以各有一条边成为共享边而不在外边界上。正方形、五边形、六边形和七边形在中间，所以各有两条边成为共享边。

所得图形有 \[ \begin{aligned} &(3-1)+(4-2) \\ &\quad {}+(5-2)+(6-2) \\ &\quad {}+(7-2)+(8-1)=23 \end{aligned} \] 条边。

所以正确答案是 B。

English

The triangle and octagon are at the ends of the chain, so each loses one side to a shared edge. The square, pentagon, hexagon, and heptagon are in the middle, so each loses two sides to shared edges.

The resulting polygon has \[ \begin{aligned} &(3-1)+(4-2) \\ &\quad {}+(5-2)+(6-2) \\ &\quad {}+(7-2)+(8-1)=23 \end{aligned} \] sides.

Thus, B is the correct answer.

10 答案 D

内部正方形共有 \[(8 - 2)^2 = 6^2 = 36\] 个。

因此选中其中一个的概率是 \[ \dfrac{36}{64} = \dfrac{9}{16}\text{。}\]

所以正确答案是 D。

English

There are \[(8 - 2)^2 = 6^2 = 36\] squares on the interior.

This means that the probability of choosing one of these squares is \[ \dfrac{36}{64} = \dfrac{9}{16}. \]

Thus, D is the correct answer.

11 答案 D

买铅笔的七年级学生人数等于 \(143\) 除以一支铅笔的价格。同样，买铅笔的六年级学生人数等于 \(195\) 除以一支铅笔的价格。

因此铅笔价格同时整除 \(143\) 和 \(195\)。质因数分解得 \[ 143 = 11 \cdot 13 \] 和 \[ 195 = 3 \cdot 5 \cdot 13\text{。}\]同时整除 \(143\) 和 \(195\) 的正整数只有 \(1\) 和 \(13\)。

如果一支铅笔价格是 \(1\) 美分，那么会有 \(195\) 名六年级学生买铅笔，这是不可能的。因此一支铅笔价格是 \(13\) 美分。

于是 \(143 \div 13 = 11\) 名七年级学生买铅笔，\(195 \div 13 = 15\) 名六年级学生买铅笔。所以六年级学生比七年级学生多 \(4\) 人。

所以正确答案是 D。

English

The number of seventh graders that bought a pencil is \(143\) divided by the price of a pencil. Similarly, the number of sixth graders that bought a pencil is \(195\) divided by the price of a pencil.

This means that the price of a pencil divides both \(143\) and \(195.\) Prime factorizing, we get \[ 143 = 11 \cdot 13 \] and \[ 195 = 3 \cdot 5 \cdot 13. \] The only numbers that divide both \(143\) and \(195\) are \(1\) and \(13.\)

If \(1\) cent was the price of the pencil, that means \(195\) sixth graders bought pencils, which is not possible. Therefore, the price of a pencil is \(13\) cents.

This means that \(143 \div 13 = 11\) seventh graders bought a pencil, and \(195 \div 13 = 15\) sixth graders bought a pencil. Therefore, \(4\) more sixth graders than seventh graders bought pencils.

Thus, D is the correct answer.

12 答案 D

可以求出每种可能结果中两个数字的和。

\[ \begin{gather*} 1 + 2 = 3 \\ 1 + 4 = 5 \\ 1 + 6 = 7 \\ 3 + 2 = 5 \\ 3 + 4 = 7 \\ 3 + 6 = 9 \\ 5 + 2 = 7 \\ 5 + 4 = 9 \\ 5 + 6 = 11 \end{gather*}\text{。}\]

只有 \(2\) 个结果的和不是质数，也就是和为 \(9\) 的两种情况。因此和为质数的概率是 \(\dfrac{7}{9}\)。

所以正确答案是 D。

English

We can find the sum of the two numbers in every possible outcome.

\[ \begin{gather*} 1 + 2 = 3 \\ 1 + 4 = 5 \\ 1 + 6 = 7 \\ 3 + 2 = 5 \\ 3 + 4 = 7 \\ 3 + 6 = 9 \\ 5 + 2 = 7 \\ 5 + 4 = 9 \\ 5 + 6 = 11. \end{gather*} \]

There are only \(2\) outcomes where the sum is not prime (the two instances when the sum is \(9\)). Therefore, the probability that the sum is prime is \(\dfrac{7}{9}.\)

Thus, D is the correct answer.

13 答案 B

这个数以 \(1\)、\(3\) 或 \(5\) 结尾的可能性相同。

只有当末位是 \(5\) 时，它才能被 \(5\) 整除，概率为 \(\dfrac{1}{3}\)。

所以正确答案是 B。

English

The number is equally likely to end in \(1,\) \(3,\) or \(5.\)

It is divisible by \(5\) only if the last digit is \(5,\) which happens with probability \(\dfrac{1}{3}.\)

Thus, B is the correct answer.

14 答案 B

从奥斯汀到坦普尔用时 \(50 \div 60 = \dfrac{5}{6}\) 小时。从坦普尔到奥斯汀用时 \(50 \div 40 = \dfrac{5}{4}\) 小时。因此往返总时间为 \(\dfrac{5}{6} + \dfrac{5}{4} = \dfrac{25}{12}\) 小时。

往返总路程为 \(2 \cdot 50 = 100\) 英里。因此往返平均速度为 \(100 \div \dfrac{25}{12} = 48\) 英里每小时。

所以正确答案是 B。

English

The trip from Austin to Temple took \(50 \div 60 = \dfrac{5}{6}\) hours. The trip from Temple to Austin took \(50 \div 40 = \dfrac{5}{4}\) hours. This means that the total time for the round trip was \(\dfrac{5}{6} + \dfrac{5}{4} = \dfrac{25}{12}\) hours.

The total distance of the round trip was \(2 \cdot 50 = 100\) miles. Therefore, the average speed for the round trip was \(100 \div \dfrac{25}{12} = 48\) miles per hour.

Thus, B is the correct answer.

15 答案 D

需要找出哪种配料是限制因素。

乔丹的巧克力足够做 \(5\div2=\dfrac{5}{2}\) 批，糖足够做 \(2\div\dfrac{1}{4}=8\) 批，牛奶足够做 \(7\div4=\dfrac{7}{4}\) 批。

牛奶是限制因素，所以乔丹可以做 \(5\cdot\dfrac{7}{4}=\dfrac{35}{4}=8\dfrac{3}{4}\) 份。

所以正确答案是 D。

English

We need to find which ingredient is the limiting factor.

Jordan has enough chocolate for \(5\div2=\dfrac{5}{2}\) batches, enough sugar for \(2\div\dfrac{1}{4}=8\) batches, and enough milk for \(7\div4=\dfrac{7}{4}\) batches.

The milk is limiting, so Jordan can make \(5\cdot\dfrac{7}{4}=\dfrac{35}{4}=8\dfrac{3}{4}\) servings.

Thus, D is the correct answer.

16 答案 D

小于 \(10\) 且乘积为 \(24\) 的整数三元组只有 \[ (1, 3, 8), (1, 4, 6), (2, 2, 6)\text{，}\]\[ (2, 3, 4)\text{。}\]

有 \(3\) 个数字互不相同的三元组，每个可以重新排列成 \(6\) 个不同的 \(3\) 位正整数。另一个三元组可以排列成 \(3\) 个不同的 \(3\) 位正整数。

总数为 \(3 \cdot 6 + 3 = 21\)。

所以正确答案是 D。

English

The only triples of integers less than \(10\) that multiply to \(24\) are \[ (1, 3, 8), (1, 4, 6), (2, 2, 6), \]\[ (2, 3, 4). \]

The triples with \(3\) distinct numbers can be rearranged to form \(6\) distinct \(3\)-digit positive integers. The other triple can be arranged to form \(3\) distinct \(3\)-digit positive integers.

This leaves a total of \(3 \cdot 6 + 3 = 21\) integers.

Thus, D is the correct answer.

17 答案 B

一个数是完全平方数时，质因数分解中每个指数都必须是偶数。一个数是立方数时，指数都必须能被 \(3\) 整除。

分解 \(360\)，得 \[ 360 = 2^3 \cdot 3^2 \cdot 5\text{。}\]要让 \(360x\) 成为完全平方数且 \(x\) 最小，\(x\) 必须提供一个因子 \(2\) 和一个因子 \(5\)。因此可取 \(x = 10\)。

要让 \(360y\) 成为立方数，\(y\) 必须提供一个因子 \(3\) 和两个因子 \(5\)。因此可取 \(y = 75\)，所以 \(x + y = 85\)。

所以正确答案是 B。

English

For a number to be a perfect square, every exponent in the prime factorization must be even. For it to be a cube, the exponents must be divisible by \(3.\)

We can factor \(360\) to get \[ 360 = 2^3 \cdot 3^2 \cdot 5. \] For \(360x\) to be a perfect square and \(x\) to be minimized, \(x\) must have one factor of \(2\) and one factor of \(5.\) Therefore, we can let \(x = 10.\)

For \(360y\) to be a cube, \(y\) must have one factor of \(3\) and two factors of \(5.\) Therefore, we can let \(y = 75,\) suggesting \(x + y = 85.\)

Thus, B is the correct answer.

18 答案 C

在 \(7\times7\) 的例子中，有 \(4\) 行各含 \(4\) 块非阴影瓷砖，所以共有 \(4^2=16\) 块非阴影瓷砖。

对 \(15\times15\) 且模式相同的地板，会有 \(8\) 行各含 \(8\) 块非阴影瓷砖。

因此非阴影瓷砖数为 \(8^2=64\)。

所以正确答案是 C。

English

In the \(7\times7\) example, there are \(4\) rows that contain \(4\) unshaded tiles, for \(4^2=16\) unshaded tiles.

For a \(15\times15\) floor with the same pattern, there will be \(8\) such rows with \(8\) unshaded tiles each.

Thus the number of unshaded tiles is \(8^2=64.\)

Thus, C is the correct answer.

19 答案 D

所有可能情形如下图所示。

在第一种情形中，由等腰三角形性质可得 \(x = 70\)。

在第二种情形中，有 \[ 70 \cdot 2 + x = 180\text{，}\]因此 \(x = 40\)。

在第三种情形中，有 \[ 2x + 70 = 180\text{，}\]因此 \(x = 55\)。

这些值之和为 \[ 70 + 40 + 55 = 165\text{。}\]

所以正确答案是 D。

English

All the following possibilities are shown below.

In the first scenario, we get \(x = 70\) by the properties of the isosceles triangle.

In the second scenario, we get that \[ 70 \cdot 2 + x = 180, \] from which we get that \(x = 40.\)

From the third scenario, we get that \[ 2x + 70 = 180, \] from which we get that \(x = 55.\)

The sum of these values yields \[ 70 + 40 + 55 = 165. \]

Thus, D is the correct answer.

20 答案 D

由对称性，只需列出每种可能三角形形状的一个代表。

一份完整的不全等三角形代表列表是 \(ABE\)、\(ABG\)、\(ACE\)、\(ABH\)、\(AEH\)、\(ACF\)、\(ACH\) 和 \(ADF\)。

八个点形成的其他三角形都与这 \(8\) 个三角形中的某一个全等。

所以正确答案是 D。

English

By symmetry, it is enough to list one representative of each possible triangle shape.

One complete list of non-congruent possibilities is \(ABE,\) \(ABG,\) \(ACE,\) \(ABH,\) \(AEH,\) \(ACF,\) \(ACH,\) and \(ADF.\)

Every other triangle formed from the eight points is congruent to one of these \(8\) triangles.

Thus, D is the correct answer.

21 答案 D

设 \(S\) 是阵列中所有数的和。

安迪的 \(40\) 个行和加起来也是 \(S\)，所以平均数为 \(A=\dfrac{S}{40}\)。贝瑟妮的 \(75\) 个列和加起来也是 \(S\)，所以平均数为 \(B=\dfrac{S}{75}\)。

因此 \(\dfrac{A}{B}=\dfrac{\frac{S}{40}}{\frac{S}{75}}=\dfrac{75}{40}=\dfrac{15}{8}\)。

所以正确答案是 D。

English

Let \(S\) be the sum of all numbers in the array.

Andy’s \(40\) row sums also add to \(S,\) so their average is \(A=\dfrac{S}{40}.\) Bethany’s \(75\) column sums also add to \(S,\) so their average is \(B=\dfrac{S}{75}.\)

Therefore \(\dfrac{A}{B}=\dfrac{\frac{S}{40}}{\frac{S}{75}}=\dfrac{75}{40}=\dfrac{15}{8}.\)

Thus, D is the correct answer.

22 答案 D

可以按位数分类。

一位数共有 \(8\) 个不含数字 \(1\)。

两位数共有 \(8 \cdot 9 = 72\) 个不含数字 \(1\)。

三位数共有 \(8 \cdot 9 \cdot 9 = 648\) 个不含数字 \(1\)。

因此不含数字 \(1\) 的数共有 \[ 8 + 72 + 648 = 728 \] 个。

所以正确答案是 D。

English

We can case on the number of digits.

There are \(8\) one digit numbers excluding \(1.\)

There are \(8 \cdot 9 = 72\) two digit numbers that lack the digit \(1.\)

There are \(8 \cdot 9 \cdot 9 = 648\) three digit numbers that do not include \(1.\)

This yields a total of \[ 8 + 72 + 648 = 728 \] numbers that do not contain the digit \(1.\)

Thus, D is the correct answer.

23 答案 B

设班上男生人数为 \(b\)，女生人数为 \(g\)。由题意，\(b = g + 2\)。

如果每个男生得到 \(b\) 颗果冻豆，那么给所有男生共发出 \(b^2\) 颗。同样，给所有女生共发出 \(g^2\) 颗。

因此 \[ \begin{gather*} b^2 + g^2 = 400 - 6 \\ (g + 2)^2 + g^2 = 394 \\ 2g^2 + 4g + 4 = 394 \\ g^2 + 2g - 195 = 0 \\ (g + 15)(g - 13) = 0 \end{gather*}\text{。}\]因为 \(g\) 不能为负，所以 \(g = 13\)。这意味着 \(b = 15\)，所以 \(b + g = 28\)。

所以正确答案是 B。

English

Let \(b\) be the number of boys in the class and \(g\) be the number of girls. From the problem, we get that \(b = g + 2.\)

If each boy gets \(b\) jelly beans, then Mrs. Wonderful will give out a total of \(b^2\) jelly beans to all the boys. Similarly, she will give out \(g^2\) jelly beans to all the girls.

Therefore, \[ \begin{gather*} b^2 + g^2 = 400 - 6 \\ (g + 2)^2 + g^2 = 394 \\ 2g^2 + 4g + 4 = 394 \\ g^2 + 2g - 195 = 0 \\ (g + 15)(g - 13) = 0. \end{gather*} \] Since \(g\) cannot be negative, we get that \(g = 13.\) This means that \(b = 15,\) so \(b + g = 28.\)

Thus, B is the correct answer.

24 答案 E

从加法的个位可知，\(A+B\) 的个位仍是 \(A\)，所以 \(B=0\)。

在减法 \(AB-CA=A\) 中，现在有 \(A0-CA=A\)。个位必须借位，所以 \(10-A=A\)，得 \(A=5\)。

于是 \(50-C5=5\)，所以 \(C=4\)。在加法中，\(50+45=95\)，所以 \(D=9\)。

所以正确答案是 E。

English

From the ones column of the addition, \(A+B\) ends in \(A,\) so \(B=0.\)

In the subtraction \(AB-CA=A,\) we now have \(A0-CA=A.\) The ones column requires a borrow, so \(10-A=A,\) giving \(A=5.\)

Then \(50-C5=5,\) so \(C=4.\) In the addition, \(50+45=95,\) so \(D=9.\)

Thus, E is the correct answer.

25 答案 E

从六个坐标方向观察这个立体。

从任一端观察时，外露竖直面的总高度等于 \(A\) 块的厚度，即 \(\dfrac{1}{2}\) 英尺。因此每个端向视图的面积为 \(\dfrac{1}{2}\) 平方英尺。

从每个侧面看，四块拼成原单位立方体的侧视图，所以每个侧视图面积为 \(1\) 平方英尺。

从上方和下方看，每个视图都显示四个 \(1\) 乘 \(1\) 的面，所以每个面积为 \(4\) 平方英尺。

总表面积为 \(\dfrac{1}{2}+\dfrac{1}{2}+1+1+4+4=11\) 平方英尺。

所以正确答案是 E。

English

Look at the solid from the six coordinate directions.

Viewed from either end, the exposed vertical faces have total height equal to the thickness of piece \(A,\) namely \(\dfrac{1}{2}\) foot. Thus each end view has area \(\dfrac{1}{2}\) square foot.

From each side, the four pieces stack to the side view of the original unit cube, so each side view has area \(1\) square foot.

From the top and bottom, each view shows four \(1\)-by-\(1\) faces, so each has area \(4\) square feet.

The total surface area is \(\dfrac{1}{2}+\dfrac{1}{2}+1+1+4+4=11\) square feet.

Thus, E is the correct answer.