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AMC12 2000 A

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AMC12 · 2000 (A)

Q1
In the year $2001$, the United States will host the International Mathematical Olympiad. Let $I,M,$ and $O$ be distinct positive integers such that the product $I \cdot M \cdot O = 2001$. What is the largest possible value of the sum $I + M + O$?
在 $2001$ 年，美国将举办国际数学奥林匹克竞赛。设 $I,M,$ 和 $O$ 是互不相同的正整数，使得乘积 $I \cdot M \cdot O = 2001$。求和 $I + M + O$ 的最大可能值。
Q2
$2000(2000^{2000}) = x$ Find x.
$2000(2000^{2000}) = x$ 求 $x$。
Q3
Each day, Jenny ate $20\%$ of the jellybeans that were in her jar at the beginning of that day. At the end of the second day, $32$ remained. How many jellybeans were in the jar originally?
每天，Jenny 吃掉当天开始时罐子里果冻豆的 $20\%$。第二天结束时，还剩 $32$ 颗。最初罐子里有多少颗果冻豆？
Q4
The Fibonacci sequence $1,1,2,3,5,8,13,21,\ldots$ starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence?
斐波那契数列 $1,1,2,3,5,8,13,21,\ldots$ 以两个 1 开头，其后每一项是前两项之和。十个数字中，哪个数字最后出现在斐波那契数列中某一项的个位数上？
Q5
If $|x - 2| = p$, where $x < 2$, then $x - p =$
如果 $|x - 2| = p$，其中 $x < 2$，则 $x - p =$
Q6
Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?
在 $4$ 到 $18$ 之间选取两个不同的质数。当它们的和从它们的积中减去时，下列哪个数可能得到？
Q7
How many positive integers $b$ have the property that $\log_{b} 729$ is a positive integer?
有多少个正整数 $b$ 满足 $\log_{b} 729$ 是一个正整数？
Q8
Figures $0$, $1$, $2$, and $3$ consist of $1$, $5$, $13$, and $25$ nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure 100?
图形 $0$、$1$、$2$ 和 $3$ 分别由 $1$、$5$、$13$ 和 $25$ 个互不重叠的单位正方形组成。如果继续这个模式，图形 100 中会有多少个互不重叠的单位正方形？
stem
Q9
Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were $71$, $76$, $80$, $82$, and $91$. What was the last score Mrs. Walter entered?
Walter 夫人在一个有五名学生的数学班里进行了一次考试。她以随机顺序将分数输入电子表格，每输入一个分数后，表格都会重新计算班级平均分。Walter 夫人注意到，每次输入分数后，平均分总是整数。分数（按从小到大排列）为 $71$、$76$、$80$、$82$ 和 $91$。Walter 夫人最后输入的分数是多少？
Q10
The point $P = (1,2,3)$ is reflected in the $xy$-plane, then its image $Q$ is rotated by $180^\circ$ about the $x$-axis to produce $R$, and finally, $R$ is translated by 5 units in the positive-$y$ direction to produce $S$. What are the coordinates of $S$?
点 $P = (1,2,3)$ 先关于 $xy$ 平面反射得到其像 $Q$，再将 $Q$ 绕 $x$ 轴旋转 $180^\circ$ 得到 $R$，最后将 $R$ 沿正 $y$ 方向平移 5 个单位得到 $S$。求 $S$ 的坐标。
Q11
Two non-zero real numbers, $a$ and $b,$ satisfy $ab = a - b$. Which of the following is a possible value of $\frac {a}{b} + \frac {b}{a} - ab$?
两个非零实数 $a$ 和 $b,$ 满足 $ab = a - b$。以下哪一项可能是 $\frac {a}{b} + \frac {b}{a} - ab$ 的值？
Q12
Let $A, M,$ and $C$ be nonnegative integers such that $A + M + C = 12$. What is the maximum value of $A \cdot M \cdot C + A \cdot M + M \cdot C + A \cdot C$?
设 $A, M,$ 和 $C$ 为非负整数，且 $A + M + C = 12$。$A \cdot M \cdot C + A \cdot M + M \cdot C + A \cdot C$ 的最大值是多少？
Q13
One morning each member of Angela's family drank an 8-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family?
一天早上，Angela 家的每个成员都喝了一杯 8 盎司的咖啡加牛奶混合饮料。每杯中咖啡和牛奶的量各不相同，但都不为零。Angela 喝了总牛奶量的四分之一和总咖啡量的六分之一。这个家庭有多少人？
Q14
When the mean, median, and mode of the list \[10,2,5,2,4,2,x\] are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of $x$?
当列表 \[10,2,5,2,4,2,x\] 的均值、中位数和众数按从小到大排列时，它们构成一个非恒定的等差数列。所有可能的实数 $x$ 的和是多少？
Q15
Let $f$ be a function for which $f\left(\dfrac{x}{3}\right) = x^2 + x + 1$. Find the sum of all values of $z$ for which $f(3z) = 7$.
设函数 $f$ 满足 $f\left(\dfrac{x}{3}\right) = x^2 + x + 1$。求所有满足 $f(3z) = 7$ 的 $z$ 的和。
Q16
A checkerboard of $13$ rows and $17$ columns has a number written in each square, beginning in the upper left corner, so that the first row is numbered $1,2,\ldots,17$, the second row $18,19,\ldots,34$, and so on down the board. If the board is renumbered so that the left column, top to bottom, is $1,2,\ldots,13,$, the second column $14,15,\ldots,26$ and so on across the board, some squares have the same numbers in both numbering systems. Find the sum of the numbers in these squares (under either system).
一个有$13$行$17$列的棋盘，每个格子中写有一个数字，从左上角开始编号，使第一行编号为$1,2,\ldots,17$，第二行编号为$18,19,\ldots,34$，依此类推向下编号。如果将棋盘重新编号，使得最左列从上到下为$1,2,\ldots,13,$，第二列为$14,15,\ldots,26$，依此类推向右编号，则有些格子在两种编号系统中数字相同。求这些格子中的数字之和（在任一系统下相同）。
Q17
A circle centered at $O$ has radius $1$ and contains the point $A$. The segment $AB$ is tangent to the circle at $A$ and $\angle AOB = \theta$. If point $C$ lies on $\overline{OA}$ and $\overline{BC}$ bisects $\angle ABO$, then $OC =$
以$O$为圆心、半径为$1$的圆包含点$A$。线段$AB$在$A$点与圆相切，且$\angle AOB = \theta$。若点$C$在$\overline{OA}$上，且$\overline{BC}$平分$\angle ABO$，则$OC =$
stem
Q18
In year $N$, the $300^{\text{th}}$ day of the year is a Tuesday. In year $N+1$, the $200^{\text{th}}$ day is also a Tuesday. On what day of the week did the $100$th day of year $N-1$ occur?
在年份$N$中，第$300^{\text{th}}$天是星期二。在年份$N+1$中，第$200^{\text{th}}$天也是星期二。年份$N-1$的第$100$天是星期几？
Q19
In triangle $ABC$, $AB = 13$, $BC = 14$, $AC = 15$. Let $D$ denote the midpoint of $\overline{BC}$ and let $E$ denote the intersection of $\overline{BC}$ with the bisector of angle $BAC$. Which of the following is closest to the area of the triangle $ADE$?
在三角形$ABC$中，$AB = 13$，$BC = 14$，$AC = 15$。设$D$为$\overline{BC}$的中点，$E$为$\overline{BC}$与$\angle BAC$的角平分线的交点。以下哪一项最接近三角形$ADE$的面积？
Q20
If $x,y,$ and $z$ are positive numbers satisfying \[x + \frac{1}{y} = 4,\qquad y + \frac{1}{z} = 1, \qquad \text{and} \qquad z + \frac{1}{x} = \frac{7}{3}\] Then what is the value of $xyz$ ?
若$x,y,$和$z$为正数，满足 \[x + \frac{1}{y} = 4,\qquad y + \frac{1}{z} = 1, \qquad \text{and} \qquad z + \frac{1}{x} = \frac{7}{3}\] 则$xyz$的值是多少？
Q21
Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is $m$ times the area of the square. The ratio of the area of the other small right triangle to the area of the square is
通过直角三角形的斜边上的一点，作两条分别平行于该三角形两条直角边的直线，使三角形被分成一个正方形和两个更小的直角三角形。其中一个小直角三角形的面积是正方形面积的 $m$ 倍。另一个小直角三角形面积与正方形面积的比值为
Q22
The graph below shows a portion of the curve defined by the quartic polynomial $P(x) = x^4 + ax^3 + bx^2 + cx + d$. Which of the following is the smallest?
下图显示了由四次多项式 $P(x) = x^4 + ax^3 + bx^2 + cx + d$ 定义的曲线的一部分。以下哪一个最小？
stem
Q23
Professor Gamble buys a lottery ticket, which requires that he pick six different integers from $1$ through $46$, inclusive. He chooses his numbers so that the sum of the base-ten logarithms of his six numbers is an integer. It so happens that the integers on the winning ticket have the same property— the sum of the base-ten logarithms is an integer. What is the probability that Professor Gamble holds the winning ticket?
甘布尔教授买了一张彩票，要求他从 $1$ 到 $46$（含）中选出六个不同的整数。他选择这些数使得这六个数的以 10 为底的对数之和为整数。碰巧，中奖彩票上的整数也具有同样的性质——以 10 为底的对数之和为整数。甘布尔教授持有中奖彩票的概率是多少？
Q24
If circular arcs $\widehat{AC}$ and $\widehat{BC}$ have centers at $B$ and $A$, respectively, then there exists a circle tangent to both $\widehat{AC}$ and $\widehat{BC}$, and to $\overline{AB}$. If the length of $\overline{BC}$ is $12$, then the circumference of the circle is
如果圆弧 $\widehat{AC}$ 和 $\widehat{BC}$ 的圆心分别在 $B$ 和 $A$，那么存在一个圆同时与圆弧 $\widehat{AC}$、圆弧 $\widehat{BC}$ 以及线段 $\overline{AB}$ 相切。若线段 $\overline{BC}$ 的长度为 $12$，则该圆的周长为（ ）
stem
Q25
Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.)
用八个全等的正三角形（每个颜色都不同）构造一个正八面体。有多少种可区分的方式来构造该八面体？（若两个彩色八面体无法通过旋转使其看起来完全相同，则它们是可区分的。）
stem
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