1990 年 AMC 8 真题（P版）

1 答案 C

要让和最小，两个最小的数字应放在百位，接下来的两个数字放在十位，最大的两个数字放在个位。

例如 \(468 + 579 = 1047\)，这种分配方式都会得到同样的和。

所以正确答案是 C。

English

To make the sum small, the two smallest digits go in the hundreds places, the next two in the tens places, and the two largest in the units places.

One such arrangement is \(468 + 579 = 1047\), and every arrangement of this type gives the same sum.

Thus, the correct answer is C .

2 答案 A

改变十分位上的数字，比改变右边任何一位的影响都大。十分位上的数字是 \(1\)。

把它改成九后得到 \(0.92345\)，这是可能得到的最大数。

所以正确答案是 A。

English

Changing a digit in the tenths place changes the number more than changing any digit farther to the right. The digit in the tenths place is \(1\).

Changing it gives \(0.92345\), which is the largest possible result.

Thus, the correct answer is A .

3 答案 E

从一个顶点到对角顶点的对角线把正方形分成两个面积相等的部分。对角线一侧的每个涂色部分，都是另一侧某个等面积未涂色部分的镜像。

因此涂色面积和未涂色面积相等，正好有 \(\dfrac12\) 的正方形被涂色。

所以正确答案是 E。

English

The diagonal from one corner to the opposite corner splits the square into two equal halves. Every shaded piece on one side of the diagonal is the mirror image of an equal unshaded piece on the other side.

So the shaded and unshaded areas are equal, and exactly \(\dfrac12\) of the square is shaded.

Thus, the correct answer is E .

4 答案 E

平方数的个位数字由原数的个位数字决定。\(0\) 到 \(9\) 平方后的个位数字依次是 \(0\)、\(1\)、\(4\)、\(9\)、\(6\)、\(5\)、\(6\)、\(9\)、\(4\)、\(1\)。

所以平方数只能以 \(0\)、\(1\)、\(4\)、\(5\)、\(6\) 或 \(9\) 结尾；不可能以 \(2\)、\(3\)、\(7\) 或 \(8\) 结尾。选项中只有 \(8\) 不可能。

所以正确答案是 E。

English

The units digit of a square is determined by the units digit of the number squared. Squaring \(0\) through \(9\) gives units digits \(0,\) \(1,\) \(4,\) \(9,\) \(6,\) \(5,\) \(6,\) \(9,\) \(4,\) \(1\).

So a square can only end in \(0,\) \(1,\) \(4,\) \(5,\) \(6,\) or \(9\); it can never end in \(2,\) \(3,\) \(7,\) or \(8\). Among the choices, only \(8\) is impossible.

Thus, the correct answer is E .

5 答案 B

因为 \(0.48017\) 接近 \(\dfrac12\)，所以这个乘积接近

\[\left(\dfrac12\right)^3 = \dfrac18 = 0.125\text{。}\]

选项中最接近 \(0.125\) 的是 \(0.110\)。

所以正确答案是 B。

English

Since \(0.48017\) is close to \(\dfrac12\), the product is close to

\[\left(\dfrac12\right)^3 = \dfrac18 = 0.125.\]

The choice closest to \(0.125\) is \(0.110\).

Thus, the correct answer is B .

6 答案 D

选项 A、B 和 E 都非常接近 \(13579\)。选项 C 把 \(13579\) 乘以一个很小的数，会变得更小。

选项 D 是除以 \(\dfrac{1}{2468}\)，也就是乘以 \(2468\)。它比 \(13579\) 大几千倍，所以最大。

所以正确答案是 D。

English

Choices A, B, and E are all very close to \(13579\), and choice C multiplies \(13579\) by a tiny number, making it much smaller.

Choice D divides by \(\dfrac{1}{2468}\), which is the same as multiplying by \(2468\). This makes it thousands of times larger than \(13579\), so it is the largest.

Thus, the correct answer is D .

7 答案 C

三个数的乘积要为正，要么三个全是正数，要么恰好两个是负数。这里只有两个正数，所以必须用两个负数和一个正数。

为了使乘积最大，取两个最小的负数和最大的正数：\((-3)(-2)(5) = 30\)。

所以正确答案是 C。

English

For the product of three numbers to be positive, either all three are positive or exactly two are negative. There are only two positive numbers, so we must use two negatives and one positive.

To maximize, take the two most negative numbers and the largest positive: \((-3)(-2)(5) = 30\).

Thus, the correct answer is C .

8 答案 D

打折后的价格是 \(\dfrac34 \cdot \$80 = \$60\)。

税率是 \(10\%\)，所以对 \(\$60\) 征收的税是 \(\$6\)，总价为 \(\$60 + \$6 = \$66\)。

所以正确答案是 D。

English

The sale price is \(\dfrac34 \cdot \$80 = \$60.\)

The tax is \(10\%\) of \(\$60,\) which is \(\$6,\) so the total is \(\$60 + \$6 = \$66.\)

Thus, the correct answer is D .

9 答案 D

C 对应的分数范围是 \(75\) 到 \(84\)。符合的分数是 \(77\)、\(75\)、\(84\)、\(78\)、\(80\)，共有 \(5\) 名学生。

所以百分比是 \(\dfrac{5}{15} = \dfrac13 = 33\dfrac13\%\)。

所以正确答案是 D。

English

A grade of C corresponds to scores from \(75\) to \(84\). The qualifying scores are \(77,\) \(75,\) \(84,\) \(78,\) \(80\), which is \(5\) students.

So the percent is \(\dfrac{5}{15} = \dfrac13 = 33\dfrac13\%.\)

Thus, the correct answer is D .

10 答案 A

向右移动一列日期增加 \(1\)，向下移动一行日期增加 \(7\)。设 \(C\) 盖住的日期为 \(n\)，则右边一列的 \(A\) 盖住 \(n+1\)。\(P\) 在 \(C\) 正下方两行，所以 \(P = n+14\)；\(B\) 在底行且位于 \(P\) 左边一列，所以它盖住 \(n+13\)。

设所求日期为 \(d\)，则 \(d + n = (n+1) + (n+13)\)，所以 \(d = n + 14\)。这个日期被 \(P\) 盖住。

所以正确答案是 A。

English

Moving one column right adds \(1\) to the date, and moving one row down adds \(7\). Let \(C\) have date \(n\). Then \(A\), one column to the right, is \(n+1\). The letter \(P\) sits two rows directly below \(C\), so \(P = n+14\), and \(B\), one column left of \(P\) in the bottom row, is \(n+13\).

We need a letter whose date \(d\) satisfies \(d + n = (n+1) + (n+13)\), so \(d = n + 14\). That date is behind \(P\).

Thus, the correct answer is A .

11 答案 E

六个连续整数包含 \(11\)、\(14\)、\(15\)，所以其中五个数是 \(11\)、\(12\)、\(13\)、\(14\)、\(15\)，第六个数是 \(10\) 或 \(16\)。

如果第六个数是 \(10\)，为了让相对面和相等，配对应为 \((10,15)\)、\((11,14)\)、\((12,13)\)，这会使 \(11\) 和 \(14\) 相对。但图中标有 \(11\)、\(14\)、\(15\) 的三个面在同一个顶点相遇，所以它们任意两个都不相对。因此第六个数是 \(16\)，相对面可配成 \((11,16)\)、\((12,15)\)、\((13,14)\)，每对和为 \(27\)。

六个面的总和是 \(3 \cdot 27 = 81\)。

所以正确答案是 E。

English

The six consecutive numbers include \(11,\) \(14,\) \(15\), so five of the faces are \(11,\) \(12,\) \(13,\) \(14,\) \(15\) and the sixth is \(10\) or \(16\).

If the sixth were \(10\), equal opposite sums would force the pairs \((10,15),\) \((11,14),\) \((12,13)\), making \(11\) and \(14\) opposite. But the figure shows \(11,\) \(14,\) \(15\) meeting at one corner, so no two of them are opposite. Hence the sixth number is \(16\), with pairs \((11,16),\) \((12,15),\) \((13,14)\), each summing to \(27\).

The total is \(3 \cdot 27 = 81\).

Thus, the correct answer is E .

12 答案 B

每个首位数字对应 \(6\) 个四位数。全部 \(24\) 个数中，第 \(1\) 到 \(6\) 个以 \(2\) 开头，第 \(7\) 到 \(12\) 个以 \(4\) 开头，第 \(13\) 到 \(18\) 个以 \(5\) 开头。

所以第 \(17\) 个数是以 \(5\) 开头的第 \(5\) 个，依次为 \(5247\)、\(5274\)、\(5427\)、\(5472\)、\(5724\)。第五个是 \(5724\)。

所以正确答案是 B。

English

Each leading digit accounts for \(6\) of the \(24\) numbers. Positions \(1\)–\(6\) begin with \(2\), positions \(7\)–\(12\) begin with \(4\), and positions \(13\)–\(18\) begin with \(5\).

So the \(17\)th number is the \(5\)th one beginning with \(5\): \(5247,\) \(5274,\) \(5427,\) \(5472,\) \(5724\). The fifth is \(5724\).

Thus, the correct answer is B .

13 答案 C

第一盎司花 \(30\) 美分。剩下的 \(3.5\) 盎司要按 \(4\) 个额外盎司收费，因为不足一盎司也按一整盎司算。

额外邮资是 \(4 \cdot 22\) 美分 \(= 88\) 美分，总邮资是 \(30\) 美分 \(+ 88\) 美分 \(= 118\) 美分 \(= \$1.18\)。

所以正确答案是 C。

English

The first ounce costs \(30\)¢. The remaining \(3.5\) ounces are charged as \(4\) additional ounces, since any fraction rounds up to a full ounce.

That is \(4 \cdot 22\)¢ \(= 88\)¢, so the total is \(30\)¢ \(+ 88\)¢ \(= 118\)¢ \(= \$1.18.\)

Thus, the correct answer is C .

14 答案 B

蓝球占袋中球总数的 \(\dfrac14\)，而蓝球有 \(6\) 个，所以总数是 \(6 \cdot 4 = 24\) 个。

绿球数为 \(24 - 6 = 18\)。

所以正确答案是 B。

English

Since blue balls make up \(\dfrac14\) of the bag and there are \(6\) of them, the total is \(6 \cdot 4 = 24\) balls.

So the number of green balls is \(24 - 6 = 18\).

Thus, the correct answer is B .

15 答案 E

每个小正方形的面积是 \(\dfrac{100}{4} = 25 \text{ cm}^2\)，所以边长是 \(5 \text{ cm}\)。

这个阶梯形的外边界由 \(10\) 条这样的边组成，所以周长是 \(10 \cdot 5 = 50 \text{ cm}\)。

所以正确答案是 E。

English

Each of the four squares has area \(\dfrac{100}{4} = 25 \text{ cm}^2\), so each side is \(5 \text{ cm}.\)

The outline of this staircase shape is made up of \(10\) such sides, so the perimeter is \(10 \cdot 5 = 50 \text{ cm}.\)

Thus, the correct answer is E .

16 答案 D

把式子分组为 \((1990 - 1980)\) \(+ (1970 - 1960)\) \(+ \cdots\) \(+ (30 - 20)\) \(+ 10\)。每个括号里的差都是 \(10\)。

各组的第一项为 \(1990, 1970, \ldots, 30\)，共 \(99\) 项，所以共有 \(99\) 组，贡献 \(990\)；再加最后的 \(+10\)，总数为 \(1000\)。

所以正确答案是 D。

English

Group the terms as \((1990 - 1980)\) \(+ (1970 - 1960)\) \(+ \cdots\) \(+ (30 - 20)\) \(+ 10.\) Each parenthesized pair equals \(10\).

The first elements \(1990, 1970, \ldots, 30\) number \(99\), so there are \(99\) pairs, giving \(990\), plus the leftover \(+10\) at the end for a total of \(1000\).

Thus, the correct answer is D .

17 答案 A

厚度 \(3\) 英寸 \(= \dfrac14\) 英尺，所以体积是 \(3 \cdot 60 \cdot \dfrac14 = 45\) 立方英尺。

因为 \(1\) 立方码 \(= 27\) 立方英尺，所以体积是 \(\dfrac{45}{27} = 1\dfrac23\) 立方码。按整立方码订购时必须向上取整，需订购 \(2\) 立方码。

所以正确答案是 A。

English

The thickness is \(3\) inches \(= \dfrac14\) foot, so the volume is \(3 \cdot 60 \cdot \dfrac14 = 45\) cubic feet.

Since \(1\) cubic yard \(= 27\) cubic feet, this is \(\dfrac{45}{27} = 1\dfrac23\) cubic yards. Rounding up to a whole number, the contractor must order \(2\) cubic yards.

Thus, the correct answer is A .

18 答案 C

长方体原来有 \(12\) 条棱。切掉顶点只会把这些棱截短，不会把它们完全去掉。

共有 \(8\) 个顶点，每个顶点切口会产生一个小三角形面，带来 \(3\) 条新棱。所以新增 \(8 \cdot 3 = 24\) 条棱，总数是 \(12 + 24 = 36\)。

所以正确答案是 C。

English

A rectangular prism starts with \(12\) edges, and cutting corners only shortens them without removing any.

Each of the \(8\) corner cuts creates a small triangular face with \(3\) new edges, adding \(8 \cdot 3 = 24\) edges. The total is \(12 + 24 = 36\).

Thus, the correct answer is C .

19 答案 B

要迫使下一个人坐在别人旁边，每个空座位都必须与至少一个已坐座位相邻。每个已坐座位最多覆盖它自己及左右两个相邻座位，所以至少要坐满 \(\frac{120}{3} = 40\) 个座位。

这个下界可以达到：重复采用“空、已坐、空”的排列，每三个座位只坐中间一个。这样恰好有 \(40\) 个已坐座位。

所以正确答案是 B。

English

To force the next person next to someone, every empty seat must be adjacent to an occupied one. Each occupied seat can account for at most itself and its two neighboring seats, so at least \(\frac{120}{3} = 40\) seats must be occupied.

This bound is attainable: use a repeating pattern of empty-occupied-empty, filling the middle seat of every group of three. This uses exactly \(40\) occupied seats.

Thus, the correct answer is B .

20 答案 A

只有一个数据被改错。错误总和比实际总和多 \(\$980{,}000 - \$98{,}000 = \$882{,}000\)。

这多出的金额分摊到 \(1000\) 个家庭上，平均数相差 \(\dfrac{\$882{,}000}{1000} = \$882\)。

所以正确答案是 A。

English

Only one entry changed. The incorrect total exceeds the actual total by \(\$980{,}000 - \$98{,}000 = \$882{,}000.\)

Since this extra amount is spread over \(1000\) families, the means differ by \(\dfrac{\$882{,}000}{1000} = \$882.\)

Thus, the correct answer is A .

21 答案 B

因为每一项都是前两项的乘积，所以用某一项除以紧前一项，可以得到它前面第二项。从最后三个数 \(16\)、\(64\)、\(1024\) 倒推：

\[ \begin{gathered} 64 \div 16 = 4, \quad 16 \div 4 = 4, \\ 4 \div 4 = 1, \quad 4 \div 1 = 4, \\ 1 \div 4 = \dfrac14 \end{gathered}\text{。}\]

因此完整数列为 \(\dfrac14\)、\(4\)、\(1\)、\(4\)、\(4\)、\(16\)、\(64\)、\(1024\)，第一个数是 \(\dfrac14\)。

所以正确答案是 B。

English

Since each term is the product of the two before it, dividing any term by the term just before it recovers the term two positions earlier. Working backwards from the last three terms \(16,\) \(64,\) \(1024\):

\[ \begin{gathered} 64 \div 16 = 4, \quad 16 \div 4 = 4, \\ 4 \div 4 = 1, \quad 4 \div 1 = 4, \\ 1 \div 4 = \dfrac14. \end{gathered} \]

These fill in the earlier terms, giving the full list \(\dfrac14,\) \(4,\) \(1,\) \(4,\) \(4,\) \(16,\) \(64,\) \(1024\). The first number is \(\dfrac14\).

Thus, the correct answer is B .

22 答案 B

Chris 拿第一颗糖，然后袋子一圈圈传递，直到 Chris 拿到最后一颗，也就是第 \(100\) 颗。第一次和最后一次之间相隔 \(99\) 次取糖。

因此 Chris 第一次之后的取糖过程必须正好经过若干整圈后回到 Chris。这说明学生人数必须整除 \(99\)；选项中只有 \(11\) 能整除 \(99\)。

所以正确答案是 B。

English

Chris takes the first piece, and then the bag goes around until Chris takes the last (\(100\)th) piece. So the \(99\) pieces after Chris’s first must be exactly a whole number of trips around the table back to Chris.

This means the number of students divides \(99\). Among the choices, only \(11\) divides \(99\).

Thus, the correct answer is B .

23 答案 B

任意一个小时内的平均速度等于这一个小时内飞行的距离，也就是图上对应一小时区间的竖直上升量。因此最大平均速度出现在曲线最陡的一小时。

从 \(t=1\) 到 \(t=2\)，曲线从约 \(500\) 英里升到约 \(1000\) 英里，上升约 \(500\) 英里，平均速度约 \(500\) 英里每小时。其他小时的上升量都小于约 \(350\) 英里。所以最大平均速度出现在第二个小时。

正确答案是 B。

English

The average speed during any single hour equals the distance the airplane covers in that hour, which on the graph is the vertical rise of the curve over that one-hour interval. So the largest average speed happens during the hour where the curve is steepest.

From \(t=1\) to \(t=2\) the curve climbs from about \(500\) miles to about \(1000\) miles, a rise of roughly \(500\) miles, giving an average speed near \(500\) mph. During every other hour the curve rises by less than \(350\) miles, so those average speeds are all smaller. The steepest climb, and hence the largest average speed, is during the second hour.

Thus, the correct answer is B .

24 答案 C

设三角形、菱形和圆点的重量分别为 \(t\)、\(d\) 和 \(1\)。题意给出 \(3t + d = 9\) 和 \(t = d + 1\)。

代入得 \(3(d+1) + d = 9\)，所以 \(4d + 3 = 9\)，\(d = \dfrac32\)。两个菱形重 \(2d = 3\) 个圆点。

所以正确答案是 C。

English

Let a triangle, diamond, and dot weigh \(t\), \(d\), and \(1\). The conditions give \(3t + d = 9\) and \(t = d + 1\).

Substituting, \(3(d+1) + d = 9\), so \(4d + 3 = 9\) and \(d = \dfrac32\). Then two diamonds weigh \(2d = 3\) dots.

Thus, the correct answer is C .

25 答案 C

先分类含有角格的图案：两个相邻角格、两个对角角格、角格与中心格、角格与相邻边中格、角格与较远边中格，共 \(5\) 种。

不含角格的图案有：两个相邻边中格、两个相对边中格、一个边中格与中心格，共 \(3\) 种。

总共有 \(5 + 3 = 8\) 种不同图案。

所以正确答案是 C。

English

Classify the two shaded squares up to rotations and reflections. Patterns that include a corner: two adjacent corners (same edge), two diagonally opposite corners, corner with the center, corner with an adjacent edge-middle, and corner with a far edge-middle. That is \(5\) patterns.

Patterns with no corner: two adjacent edge-middles, two opposite edge-middles, and an edge-middle with the center. That is \(3\) more.

In total there are \(5 + 3 = 8\) distinct patterns.

Thus, the correct answer is C .