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AMC12 2010 B

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AMC12 · 2010 (B)

Q1
Makarla attended two meetings during her $9$-hour work day. The first meeting took $45$ minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?
Makayla 在她 $9$ 小时的工作日中参加了两个会议。第一个会议持续 $45$ 分钟，第二个会议持续时间是第一个的两倍。她工作日中用于参加会议的时间占百分之多少？
Q2
A big $L$ is formed as shown. What is its area?
如图所示，形成了一个大 $L$ 形。其面积是多少？
stem
Q3
A ticket to a school play cost $x$ dollars, where $x$ is a whole number. A group of 9th graders buys tickets costing a total of $\$48$, and a group of 10th graders buys tickets costing a total of $\$64$. How many values for $x$ are possible?
学校戏剧的门票价格为 $x$ 美元，其中 $x$ 是整数。一群 9 年级学生买票总共花费 $\$48$，一群 10 年级学生买票总共花费 $\$64$。$x$ 有多少可能的值？
Q4
A month with $31$ days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?
一个有 $31$ 天的月份，星期一和星期三的数量相同。这个月的第一天可能是星期中的哪几天？
Q5
Lucky Larry's teacher asked him to substitute numbers for $a$, $b$, $c$, $d$, and $e$ in the expression $a-(b-(c-(d+e)))$ and evaluate the result. Larry ignored the parenthese but added and subtracted correctly and obtained the correct result by coincidence. The number Larry substituted for $a$, $b$, $c$, and $d$ were $1$, $2$, $3$, and $4$, respectively. What number did Larry substitute for $e$?
Lucky Larry 的老师让他在表达式 $a-(b-(c-(d+e)))$ 中用数字替换 $a$、$b$、$c$、$d$ 和 $e$ 并计算结果。Larry 忽略了括号但正确地进行了加减法，并碰巧得到了正确结果。Larry 分别用 $1$、$2$、$3$ 和 $4$ 替换了 $a$、$b$、$c$ 和 $d$。Larry 用什么数字替换了 $e$？
Q6
At the beginning of the school year, $50\%$ of all students in Mr. Well's class answered "Yes" to the question "Do you love math", and $50\%$ answered "No." At the end of the school year, $70\%$ answered "Yes" and $30\%$ answered "No." Altogether, $x\%$ of the students gave a different answer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of $x$?
在学年开始时，Wells先生班上所有学生中有$50\%$对问题“你爱数学吗”回答“是”，另有$50\%$回答“否”。在学年结束时，$70\%$回答“是”，$30\%$回答“否”。总共有$x\%$的学生在学年开始和结束时给出了不同的答案。$x$的最大可能值与最小可能值之差是多少？
Q7
Shelby drives her scooter at a speed of $30$ miles per hour if it is not raining, and $20$ miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of $16$ miles in $40$ minutes. How many minutes did she drive in the rain?
Shelby骑她的滑板车在不下雨时以每小时$30$英里的速度行驶，下雨时以每小时$20$英里的速度行驶。今天她早上在晴天行驶，晚上在雨中行驶，总共在$40$分钟内行驶了$16$英里。她在雨中行驶了多少分钟？
Q8
Every high school in the city of Euclid sent a team of $3$ students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed $37$th and $64$th, respectively. How many schools are in the city?
Euclid市的每所高中都派出一个由$3$名学生组成的队伍参加数学竞赛。竞赛中每位参赛者的得分都不同。Andrea的得分在所有学生中是中位数，并且她的得分是她所在队伍中的最高分。Andrea的队友Beth和Carla分别获得第$37$名和第$64$名。该市有多少所学校？
Q9
Let $n$ be the smallest positive integer such that $n$ is divisible by $20$, $n^2$ is a perfect cube, and $n^3$ is a perfect square. What is the number of digits of $n$?
设$n$为满足以下条件的最小正整数：$n$能被$20$整除，$n^2$是完全立方数，且$n^3$是完全平方数。$n$有多少位数字？
Q10
The average of the numbers $1, 2, 3,\cdots, 98, 99,$ and $x$ is $100x$. What is $x$?
数$1, 2, 3,\cdots, 98, 99,$和$x$的平均数是$100x$。$x$是多少？
Q11
A palindrome between $1000$ and $10,000$ is chosen at random. What is the probability that it is divisible by $7$?
在 $1000$ 到 $10,000$ 之间随机选择一个回文数。它能被 $7$ 整除的概率是多少？
Q12
For what value of $x$ does \[\log_{\sqrt{2}}\sqrt{x}+\log_{2}{x}+\log_{4}{x^2}+\log_{8}{x^3}+\log_{16}{x^4}=40?\]
对于什么值的 $x$，有 \[\log_{\sqrt{2}}\sqrt{x}+\log_{2}{x}+\log_{4}{x^2}+\log_{8}{x^3}+\log_{16}{x^4}=40?\]
Q13
In $\triangle ABC$, $\cos(2A-B)+\sin(A+B)=2$ and $AB=4$. What is $BC$?
在 $\triangle ABC$ 中，$\cos(2A-B)+\sin(A+B)=2$ 且 $AB=4$。求 $BC$？
Q14
Let $a$, $b$, $c$, $d$, and $e$ be positive integers with $a+b+c+d+e=2010$ and let $M$ be the largest of the sum $a+b$, $b+c$, $c+d$ and $d+e$. What is the smallest possible value of $M$?
设 $a$, $b$, $c$, $d$, 和 $e$ 是正整数，且 $a+b+c+d+e=2010$，令 $M$ 为 $a+b$, $b+c$, $c+d$ 和 $d+e$ 这四个和中的最大值。$M$ 的最小可能值是多少？
Q15
For how many ordered triples $(x,y,z)$ of nonnegative integers less than $20$ are there exactly two distinct elements in the set $\{i^x, (1+i)^y, z\}$, where $i=\sqrt{-1}$?
有多少个由小于 $20$ 的非负整数组成的有序三元组 $(x,y,z)$，使得集合 $\{i^x, (1+i)^y, z\}$ 中恰有两种不同元素，其中 $i=\sqrt{-1}$？
Q16
Positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set $\{1, 2, 3,\dots, 2010\}$. What is the probability that $abc + ab + a$ is divisible by $3$?
正整数 $a$、$b$ 和 $c$ 从集合 $\{1, 2, 3,\dots, 2010\}$ 中随机且独立地有放回选取。$abc + ab + a$ 能被 $3$ 整除的概率是多少？
Q17
The entries in a $3 \times 3$ array include all the digits from $1$ through $9$, arranged so that the entries in every row and column are in increasing order. How many such arrays are there?
一个 $3 \times 3$ 阵列的条目包含从 $1$ 到 $9$ 的所有数字，排列使得每行和每列的条目递增有序。有多少个这样的阵列？
Q18
A frog makes $3$ jumps, each exactly $1$ meter long. The directions of the jumps are chosen independently at random. What is the probability that the frog's final position is no more than $1$ meter from its starting position?
一只青蛙做 $3$ 次跳跃，每次恰好 $1$ 米长。每次跳跃的方向独立且随机选择。青蛙最终位置与起始位置的距离不超过 $1$ 米的概率是多少？
Q19
A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than $100$ points. What was the total number of points scored by the two teams in the first half?
Raiders 队与 Wildcats 队的高中篮球比赛在第一节结束时打平。Raiders 队在四节中的得分构成一个递增的等比数列，Wildcats 队在四节中的得分构成一个递增的等差数列。第四节结束时，Raiders 队以 $1$ 分优势获胜。两队的总得分都不超过 $100$ 分。两队在上半场的总得分是多少？
Q20
A geometric sequence $(a_n)$ has $a_1=\sin x$, $a_2=\cos x$, and $a_3= \tan x$ for some real number $x$. For what value of $n$ does $a_n=1+\cos x$?
一个等比数列 $(a_n)$ 满足 $a_1=\sin x$，$a_2=\cos x$，$a_3= \tan x$，其中 $x$ 为某个实数。问当 $n$ 为多少时有 $a_n=1+\cos x$？
Q21
Let $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that $P(1) = P(3) = P(5) = P(7) = a$, and $P(2) = P(4) = P(6) = P(8) = -a$. What is the smallest possible value of $a$?
设 $a > 0$，且 $P(x)$ 是一个具有整数系数的多项式，使得 $P(1) = P(3) = P(5) = P(7) = a$， 并且 $P(2) = P(4) = P(6) = P(8) = -a$。 $a$ 的最小可能值是多少？
Q22
Let $ABCD$ be a cyclic quadrilateral. The side lengths of $ABCD$ are distinct integers less than $15$ such that $BC\cdot CD=AB\cdot DA$. What is the largest possible value of $BD$?
设 $ABCD$ 是一个循环四边形。四边形 $ABCD$ 的边长是互不相同的小于 $15$ 的整数，且满足 $BC\cdot CD=AB\cdot DA$。$BD$ 的最大可能值是多少？
Q23
Monic quadratic polynomial $P(x)$ and $Q(x)$ have the property that $P(Q(x))$ has zeros at $x=-23, -21, -17,$ and $-15$, and $Q(P(x))$ has zeros at $x=-59,-57,-51$ and $-49$. What is the sum of the minimum values of $P(x)$ and $Q(x)$?
首一二次多项式 $P(x)$ 和 $Q(x)$ 具有如下性质：$P(Q(x))$ 在 $x=-23, -21, -17,$ 和 $-15$ 处有零点，且 $Q(P(x))$ 在 $x=-59,-57,-51$ 和 $-49$ 处有零点。$P(x)$ 与 $Q(x)$ 的最小值之和是多少？
Q24
The set of real numbers $x$ for which \[\dfrac{1}{x-2009}+\dfrac{1}{x-2010}+\dfrac{1}{x-2011}\ge1\] is the union of intervals of the form $a<x\le b$. What is the sum of the lengths of these intervals?
满足 \[\dfrac{1}{x-2009}+\dfrac{1}{x-2010}+\dfrac{1}{x-2011}\ge1\] 的实数 $x$ 的集合是形如 $a<x\le b$ 的区间的并集。这些区间的长度之和是多少？
Q25
For every integer $n\ge2$, let $\text{pow}(n)$ be the largest power of the largest prime that divides $n$. For example $\text{pow}(144)=\text{pow}(2^4\cdot3^2)=3^2$. What is the largest integer $m$ such that $2010^m$ divides $\prod_{n=2}^{5300}\text{pow}(n)$?
对每个整数 $n\ge2$，令 $\text{pow}(n)$ 表示 $n$ 的最大素因子的最大幂。例如 $\text{pow}(144)=\text{pow}(2^4\cdot3^2)=3^2$。求最大的整数 $m$，使得 $2010^m$ 整除 $\prod_{n=2}^{5300}\text{pow}(n)$。
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