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AMC12 2002 B

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AMC12 · 2002 (B)

Q1
The arithmetic mean of the nine numbers in the set $\{9, 99, 999, 9999, \ldots, 999999999\}$ is a $9$-digit number $M$, all of whose digits are distinct. The number $M$ doesn't contain the digit
集合 $\{9, 99, 999, 9999, \ldots, 999999999\}$ 中九个数的算术平均数是一个九位数 $M$，其所有数字均不相同。数 $M$ 不包含数字
Q2
What is the value of $(3x - 2)(4x + 1) - (3x - 2)4x + 1$ when $x=4$?
当 $x=4$ 时，$(3x - 2)(4x + 1) - (3x - 2)4x + 1$ 的值是多少？
Q3
For how many positive integers $n$ is $n^2 - 3n + 2$ a prime number?
有多少个正整数 $n$ 使得 $n^2 - 3n + 2$ 是质数？
Q4
Let $n$ be a positive integer such that $\frac 12 + \frac 13 + \frac 17 + \frac 1n$ is an integer. Which of the following statements is not true:
设 $n$ 是正整数，使得 $\frac 12 + \frac 13 + \frac 17 + \frac 1n$ 是整数。以下哪个陈述不正确：
Q5
Let $v, w, x, y,$ and $z$ be the degree measures of the five angles of a pentagon. Suppose that $v < w < x < y < z$ and $v, w, x, y,$ and $z$ form an arithmetic sequence. Find the value of $x$.
设 $v, w, x, y,$ 和 $z$ 是一个五边形的五个角的度数。假设 $v < w < x < y < z$ 且 $v, w, x, y,$ 和 $z$ 构成一个等差数列。求 $x$ 的值。
Q6
Suppose that $a$ and $b$ are nonzero real numbers, and that the equation $x^2 + ax + b = 0$ has solutions $a$ and $b$. Then the pair $(a,b)$ is
假设 $a$ 和 $b$ 是非零实数，且方程 $x^2 + ax + b = 0$ 的解为 $a$ 和 $b$。则有序对 $(a,b)$ 是
Q7
The product of three consecutive positive integers is $8$ times their sum. What is the sum of their squares?
三个连续正整数的乘积是它们和的 $8$ 倍。它们的平方和是多少？
Q8
Suppose July of year $N$ has five Mondays. Which of the following must occur five times in the August of year $N$? (Note: Both months have $31$ days.)
假设 $N$ 年的 7 月有五个星期一。以下哪一项一定在 $N$ 年的 8 月出现五次？（注：两个月都有 $31$ 天。）
Q9
If $a,b,c,d$ are positive real numbers such that $a,b,c,d$ form an increasing arithmetic sequence and $a,b,d$ form a geometric sequence, then $\frac ad$ is
若 $a,b,c,d$ 是正实数，且 $a,b,c,d$ 形成递增等差数列，$a,b,d$ 形成等比数列，则 $\frac ad$ 是
Q10
How many different integers can be expressed as the sum of three distinct members of the set $\{1,4,7,10,13,16,19\}$?
有多少个不同的整数可以表示为集合 $\{1,4,7,10,13,16,19\}$ 中三个不同成员之和？
Q11
The positive integers $A, B, A-B,$ and $A+B$ are all prime numbers. The sum of these four primes is
正整数 $A, B, A-B,$ 和 $A+B$ 都是素数。这四个素数的和是
Q12
For how many integers $n$ is $\dfrac n{20-n}$ the square of an integer?
有几个整数 $n$ 使得 $\dfrac n{20-n}$ 是一个整数的平方？
Q13
The sum of $18$ consecutive positive integers is a perfect square. The smallest possible value of this sum is
18 个连续正整数的和是一个完全平方数。这个和的最小可能值是
Q14
Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?
平面上画了 4 个不同的圆。至少有两个圆相交的点的最大数目是多少？
Q15
How many four-digit numbers $N$ have the property that the three-digit number obtained by removing the leftmost digit is one ninth of $N$?
有多少个四位数 $N$ 具有这样的性质：去掉最左边数字得到的三位数是 $N$ 的九分之一？
Q16
Juan rolls a fair regular octahedral die marked with the numbers $1$ through $8$. Then Amal rolls a fair six-sided die. What is the probability that the product of the two rolls is a multiple of 3?
胡安掷一个公平的标有数字$1$到$8$的正八面体骰子。然后阿迈尔掷一个公平的六面骰子。两次掷出的乘积是$3$的倍数的概率是多少？
Q17
Andy's lawn has twice as much area as Beth's lawn and three times as much area as Carlos' lawn. Carlos' lawn mower cuts half as fast as Beth's mower and one third as fast as Andy's mower. If they all start to mow their lawns at the same time, who will finish first?
安迪的草坪面积是贝丝草坪的两倍，也是卡洛斯草坪的三倍。卡洛斯的割草机割草速度是贝丝割草机的一半，也是安迪割草机的三分之一。如果他们同时开始割草，谁会最先完成？
Q18
A point $P$ is randomly selected from the rectangular region with vertices $(0,0),(2,0),(2,1),(0,1)$. What is the probability that $P$ is closer to the origin than it is to the point $(3,1)$?
从顶点为$(0,0),(2,0),(2,1),(0,1)$的矩形区域中随机选取一点$P$。$P$到原点的距离比到点$(3,1)$的距离更近的概率是多少？
Q19
If $a,b,$ and $c$ are positive real numbers such that $a(b+c) = 152, b(c+a) = 162,$ and $c(a+b) = 170$, then $abc$ is
若$a,b,$和$c$为正实数，且满足$a(b+c)=152,\ b(c+a)=162,$以及$c(a+b)=170$，则$abc$等于
Q20
Let $\triangle XOY$ be a right-angled triangle with $m\angle XOY = 90^{\circ}$. Let $M$ and $N$ be the midpoints of legs $OX$ and $OY$, respectively. Given that $XN = 19$ and $YM = 22$, find $XY$.
设$\triangle XOY$为直角三角形，且$m\angle XOY=90^{\circ}$。设$M$与$N$分别为直角边$OX$与$OY$的中点。已知$XN=19$且$YM=22$，求$XY$。
Q21
For all positive integers $n$ less than $2002$, let \begin{eqnarray*} a_n =\left\{ \begin{array}{lr} 11, & \text{if\ }n\ \text{is\ divisible\ by\ }13\ \text{and\ }14;\\ 13, & \text{if\ }n\ \text{is\ divisible\ by\ }14\ \text{and\ }11;\\ 14, & \text{if\ }n\ \text{is\ divisible\ by\ }11\ \text{and\ }13;\\ 0, & \text{otherwise}. \end{array} \right. \end{eqnarray*} Calculate $\sum_{n=1}^{2001} a_n$.
对于所有小于 $2002$ 的正整数 $n$，令 \begin{eqnarray*} a_n =\left\{ \begin{array}{lr} 11, & \text{如果 }n\ \text{能被 }13\ \text{和 }14\ \text{整除};\\ 13, & \text{如果 }n\ \text{能被 }14\ \text{和 }11\ \text{整除};\\ 14, & \text{如果 }n\ \text{能被 }11\ \text{和 }13\ \text{整除};\\ 0, & \text{否则}. \end{array} \right. \end{eqnarray*} 计算 $\sum_{n=1}^{2001} a_n$。
Q22
For all integers $n$ greater than $1$, define $a_n = \frac{1}{\log_n 2002}$. Let $b = a_2 + a_3 + a_4 + a_5$ and $c = a_{10} + a_{11} + a_{12} + a_{13} + a_{14}$. Then $b- c$ equals
对所有大于 $1$ 的整数 $n$，定义 $a_n = \frac{1}{\log_n 2002}$。令 $b = a_2 + a_3 + a_4 + a_5$，$c = a_{10} + a_{11} + a_{12} + a_{13} + a_{14}$。则 $b- c$ 等于
Q23
In $\triangle ABC$, we have $AB = 1$ and $AC = 2$. Side $\overline{BC}$ and the median from $A$ to $\overline{BC}$ have the same length. What is $BC$?
在 $\triangle ABC$ 中，$AB = 1$ 且 $AC = 2$。边 $\overline{BC}$ 与从 $A$ 到 $\overline{BC}$ 的中线长度相同。求 $BC$。
Q24
A convex quadrilateral $ABCD$ with area $2002$ contains a point $P$ in its interior such that $PA = 24, PB = 32, PC = 28, PD = 45$. Find the perimeter of $ABCD$.
一个面积为 $2002$ 的凸四边形 $ABCD$ 内部有一点 $P$，使得 $PA = 24, PB = 32, PC = 28, PD = 45$。求 $ABCD$ 的周长。
Q25
Let $f(x) = x^2 + 6x + 1$, and let $R$ denote the set of points $(x,y)$ in the coordinate plane such that \[f(x) + f(y) \le 0 \qquad \text{and} \qquad f(x)-f(y) \le 0\] The area of $R$ is closest to
设 $f(x) = x^2 + 6x + 1$，并令 $R$ 表示坐标平面中满足 \[f(x) + f(y) \le 0 \qquad \text{且} \qquad f(x)-f(y) \le 0\] 的点 $(x,y)$ 的集合。$R$ 的面积最接近
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