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AMC12 2003 B

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AMC12 · 2003 (B)

Q1
Which of the following is the same as \[\frac{2-4+6-8+10-12+14}{3-6+9-12+15-18+21}?\]
以下哪个与 \[\frac{2-4+6-8+10-12+14}{3-6+9-12+15-18+21}?\]
Q2
Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs $\$1$ more than a pink pill, and Al's pills cost a total of $\$546$ for the two weeks. How much does one green pill cost?
Al 得了 algebritis 病，必须每天服用一颗绿色药丸和一颗粉色药丸，持续两周。一颗绿色药丸的价格比一颗粉色药丸贵 $\$1$，两周的药费总共是 $\$546$。一颗绿色药丸多少钱？
Q3
Rose fills each of the rectangular regions of her rectangular flower bed with a different type of flower. The lengths, in feet, of the rectangular regions in her flower bed are as shown in the figure. She plants one flower per square foot in each region. Asters cost $\$1$ each, begonias $\$1.50$ each, cannas $\$2$ each, dahlias $\$2.50$ each, and Easter lilies $\$3$ each. What is the least possible cost, in dollars, for her garden?
Rose 在她的矩形花坛中，把每个矩形区域都种上不同种类的花。图中给出了花坛中各矩形区域的长度（单位：英尺）。她在每个区域中每平方英尺种一朵花。紫菀每朵 $\$1$，秋海棠每朵 $\$1.50$，美人蕉每朵 $\$2$，大丽花每朵 $\$2.50$，复活节百合每朵 $\$3$。她的花园最少可能花费多少美元？
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Q4
Moe uses a mower to cut his rectangular $90$-foot by $150$-foot lawn. The swath he cuts is $28$ inches wide, but he overlaps each cut by $4$ inches to make sure that no grass is missed. He walks at the rate of $5000$ feet per hour while pushing the mower. Which of the following is closest to the number of hours it will take Moe to mow the lawn?
Moe 用割草机修剪他 $90$ 英尺乘 $150$ 英尺的矩形草坪。他每次割出的条带宽 $28$ 英寸，但为了确保不漏草，每次割草都会重叠 $4$ 英寸。他推着割草机行走的速度是每小时 $5000$ 英尺。以下哪个最接近 Moe 修剪完草坪所需的小时数？
Q5
Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is $4:3$. The horizontal length of a "$27$-inch" television screen is closest, in inches, to which of the following?
许多电视屏幕是矩形，通常用其对角线长度来衡量。标准电视屏幕的水平长度与高度之比为 $4:3$。“$27$ 英寸”电视屏幕的水平长度最接近以下哪个数（单位：英寸）？
Q6
The second and fourth terms of a geometric sequence are $2$ and $6$. Which of the following is a possible first term?
一个等比数列的第二项和第四项分别是 $2$ 和 $6$。以下哪一项可能是首项？
Q7
Penniless Pete's piggy bank has no pennies in it, but it has 100 coins, all nickels,dimes, and quarters, whose total value is \$8.35. It does not necessarily contain coins of all three types. What is the difference between the largest and smallest number of dimes that could be in the bank?
身无分文的皮特的小猪存钱罐里没有一分钱，但有 100 个硬币，全是镍币、角币和25美分币，总价值为 $8.35。它不一定包含所有三种类型的硬币。存钱罐里可能存在的角币数量的最大值和最小值之差是多少？
Q8
Let $\clubsuit(x)$ denote the sum of the digits of the positive integer $x$. For example, $\clubsuit(8)=8$ and $\clubsuit(123)=1+2+3=6$. For how many two-digit values of $x$ is $\clubsuit(\clubsuit(x))=3$?
设 $\clubsuit(x)$ 表示正整数 $x$ 的数字之和。例如，$\clubsuit(8)=8$ 且 $\clubsuit(123)=1+2+3=6$。有多少个两位数的 $x$ 满足 $\clubsuit(\clubsuit(x))=3$？
Q9
Let $f$ be a linear function for which $f(6) - f(2) = 12.$ What is $f(12) - f(2)?$
设 $f$ 是一个一次函数，且 $f(6) - f(2) = 12.$ 求 $f(12) - f(2)$。
Q10
Several figures can be made by attaching two equilateral triangles to the regular pentagon ABCDE in two of the five positions shown. How many non-congruent figures can be constructed in this way?
在图示的五个位置中选取两个位置，把两个等边三角形分别贴在正五边形 $ABCDE$ 上，可以构成若干图形。用这种方式能构造出多少个互不全等的图形？
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Q11
Cassandra sets her watch to the correct time at noon. At the actual time of 1:00 PM, she notices that her watch reads 12:57 and 36 seconds. Assuming that her watch loses time at a constant rate, what will be the actual time when her watch first reads 10:00 PM?
Cassandra 在中午将她的手表调到正确时间。在实际时间下午 1:00 时，她注意到她的手表显示 12:57 和 36 秒。假设她的手表以恒定速率走慢，当她的手表第一次显示晚上 10:00 时，实际时间是几点？
Q12
What is the largest integer that is a divisor of \[(n+1)(n+3)(n+5)(n+7)(n+9)\] for all positive even integers $n$?
对于所有正偶整数 $n$，使得 \[(n+1)(n+3)(n+5)(n+7)(n+9)\] 都能被整除的最大整数是多少？
Q13
An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly fill the cone. Assume that the melted ice cream occupies $75\%$ of the volume of the frozen ice cream. What is the ratio of the cone’s height to its radius?
一个冰淇淋甜筒由一个香草冰淇淋球和一个与球直径相同的直圆锥组成。如果冰淇淋融化，它将正好填满圆锥。假设融化后的冰淇淋体积是冻结时体积的 $75\%$。圆锥的高与其半径之比是多少？
Q14
In rectangle $ABCD, AB=5$ and $BC=3$. Points $F$ and $G$ are on $\overline{CD}$ so that $DF=1$ and $GC=2$. Lines $AF$ and $BG$ intersect at $E$. Find the area of $\triangle AEB$.
在矩形 $ABCD$ 中，$AB=5$ 且 $BC=3$。点 $F$ 和 $G$ 在 $\overline{CD}$ 上，使得 $DF=1$ 且 $GC=2$。直线 $AF$ 与 $BG$ 相交于点 $E$。求 $\triangle AEB$ 的面积。
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Q15
A regular octagon $ABCDEFGH$ has an area of one square unit. What is the area of the rectangle $ABEF$?
正八边形 $ABCDEFGH$ 的面积为 1 平方单位。矩形 $ABEF$ 的面积是多少？
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Q16
Three semicircles of radius $1$ are constructed on diameter $\overline{AB}$ of a semicircle of radius $2$. The centers of the small semicircles divide $\overline{AB}$ into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles?
在半径为 $2$ 的半圆的直径 $\overline{AB}$ 上作三个半径为 $1$ 的半圆。小半圆的圆心将 $\overline{AB}$ 分成四段等长线段，如图所示。求阴影区域的面积：该区域位于大半圆内但在小半圆外。
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Q17
If $\log (xy^3) = 1$ and $\log (x^2y) = 1$, what is $\log (xy)$?
若 $\log (xy^3) = 1$ 且 $\log (x^2y) = 1$，求 $\log (xy)$？
Q18
Let $x$ and $y$ be positive integers such that $7x^5 = 11y^{13}.$ The minimum possible value of $x$ has a prime factorization $a^cb^d.$ What is $a + b + c + d?$
设 $x$ 和 $y$ 为正整数，满足 $7x^5 = 11y^{13}$。$x$ 的最小可能值的质因数分解为 $a^cb^d$。求 $a + b + c + d$？
Q19
Let $S$ be the set of permutations of the sequence $1,2,3,4,5$ for which the first term is not $1$. A permutation is chosen randomly from $S$. The probability that the second term is $2$, in lowest terms, is $a/b$. What is $a+b$?
设 $S$ 为序列 $1,2,3,4,5$ 的所有排列中第一项不是 $1$ 的那些排列的集合。从 $S$ 中随机选取一个排列。第二项为 $2$ 的概率化为最简分数为 $a/b$。求 $a+b$。
Q20
Part of the graph of $f(x) = ax^3 + bx^2 + cx + d$ is shown. What is $b$?
函数 $f(x) = ax^3 + bx^2 + cx + d$ 的图像如图所示的一部分。求 $b$。
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Q21
An object moves $8$ cm in a straight line from $A$ to $B$, turns at an angle $\alpha$, measured in radians and chosen at random from the interval $(0,\pi)$, and moves $5$ cm in a straight line to $C$. What is the probability that $AC < 7$?
一个物体沿直线从 $A$ 移动 $8$ cm 到 $B$，然后转一个角度 $\alpha$（以弧度为单位，从区间 $(0,\pi)$ 中随机选择），再沿直线移动 $5$ cm 到 $C$。$AC < 7$ 的概率是多少？
Q22
Let $ABCD$ be a rhombus with $AC = 16$ and $BD = 30$. Let $N$ be a point on $\overline{AB}$, and let $P$ and $Q$ be the feet of the perpendiculars from $N$ to $\overline{AC}$ and $\overline{BD}$, respectively. Which of the following is closest to the minimum possible value of $PQ$?
设 $ABCD$ 是一个菱形，$AC = 16$，$BD = 30$。令 $N$ 为 $\overline{AB}$ 上的点，$P$ 和 $Q$ 分别为从 $N$ 到 $\overline{AC}$ 和 $\overline{BD}$ 的垂足。以下哪项最接近 $PQ$ 的最小可能值？
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Q23
The number of $x$-intercepts on the graph of $y=\sin(1/x)$ in the interval $(0.0001,0.001)$ is closest to
函数 $y=\sin(1/x)$ 的图像在区间 $(0.0001,0.001)$ 中的 $x$ 截距个数最接近于
Q24
Positive integers $a,b,$ and $c$ are chosen so that $a<b<c$, and the system of equations $2x + y = 2003 \quad$ and $\quad y = |x-a| + |x-b| + |x-c|$ has exactly one solution. What is the minimum value of $c$?
选择正整数 $a,b,$ 和 $c$，使得 $a<b<c$，且方程组 $2x + y = 2003 \quad$ 和 $\quad y = |x-a| + |x-b| + |x-c|$ 恰有一个解。$c$ 的最小值为多少？
Q25
Three points are chosen randomly and independently on a circle. What is the probability that all three pairwise distances between the points are less than the radius of the circle?
在圆上随机且相互独立地选取三个点。三点之间所有两两距离都小于圆的半径的概率是多少？
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