/

AMC10 2010 B

You are not logged in. After submit, your report may not be available on other devices. Login

AMC10 · 2010 (B)

Q1
What is $100(100-3)-(100\cdot 100-3)$?
计算 $100(100-3)-(100\cdot 100-3)$ 的值？
Q2
Makayla 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 在她的 9 小时工作日中参加了两个会议。第一个会议用了 45 分钟，第二个会议用了两倍的时间。她工作日中参加会议所占的百分比是多少？
Q3
A drawer contains red, green, blue and white socks with at least 2 of each color. What is the minimum number of socks that must be pulled from the drawer to guarantee a matching pair?
抽屉中有红、绿、蓝、白四种颜色的袜子，每种颜色至少有 2 双。要保证抽出一双同色袜子，至少需要抽出多少双袜子？
Q4
For a real number $x$, define $\heartsuit(x)$ to be the average of $x$ and $x^2$. What is $\heartsuit(1)+\heartsuit(2)+\heartsuit(3)$?
对于实数 $x$，定义 $\heartsuit(x)$ 为 $x$ 和 $x^2$ 的平均值。求 $\heartsuit(1)+\heartsuit(2)+\heartsuit(3)$？
Q5
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 天的月份中，周一和周三的数量相同。这个月的第一天可能是星期中的哪几天？
Q6
A circle is centered at $O$, $AB$ is a diameter and $C$ is a point on the circle with $\angle COB=50^\circ$. What is the degree measure of $\angle CAB$?
一个圆以 $O$ 为圆心，$AB$ 是直径，$C$ 是圆上一点，且 $\angle COB=50^\circ$。$\angle CAB$ 的度数是多少？
Q7
A triangle has side lengths 10, 10, and 12. A rectangle has width 4 and area equal to the area of the triangle. What is the perimeter of this rectangle?
一个三角形边长为 10、10 和 12。一个矩形宽为 4，面积等于该三角形的面积。这个矩形的周长是多少？
Q8
A ticket to a school play costs $x$ 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 美元，一群 10 年级学生买票总共花费 64 美元。$x$ 可能取的值有多少个？
Q9
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 parentheses but added and subtracted correctly and obtained the correct result by coincidence. The numbers Larry substituted for $a,b,c,$ and $d$ were 1, 2, 3, and 4, respectively. What number did Larry substitute for $e$?
幸运拉里的老师让他在表达式 $a-(b-(c-(d+e)))$ 中代入数字代替 $a,b,c,d,$ 和 $e$ 并计算结果。拉里忽略了括号但加减正确，碰巧得到了正确结果。拉里代入 $a,b,c,$ 和 $d$ 的数字分别是 1、2、3 和 4。那么拉里代入 $e$ 的数字是多少？
Q10
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?
谢尔比骑滑板车在不下雨时速度为 30 英里/小时，下雨时为 20 英里/小时。今天她早上在阳光下开车，晚上在雨中开车，总共 16 英里，用时 40 分钟。她在雨中开了多少分钟？
Q11
A shopper plans to purchase an item that has a listed price greater than \$100 and can use any one of three coupons. Coupon A gives 15\% off the listed price, Coupon B gives \$30 off the listed price, and Coupon C gives 25\% off the amount by which the listed price exceeds \$100. Let $x$ and $y$ be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or C. What is $y-x$?
一位购物者计划购买一件标价大于 $100$ 的商品，可以使用以下三种优惠券中的任意一种。优惠券 A 打八五折（15% off），优惠券 B 减 $30$，优惠券 C 对标价超过 $100$ 的部分打七五折（25% off）。设 $x$ 和 $y$ 分别为优惠券 A 节省的金额至少与 B 或 C 一样多的最小和最大价格。求 $y-x$？
Q12
At the beginning of the school year, 50\% of all students in Mr. Wells’ math 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 $x$?
在学年初，Wells 先生数学班上 50% 的学生回答“爱数学”，50% 回答“不爱”。学年末，70% 回答“是”，30% 回答“否”。总共有 $x$% 的学生前后答案不同。$x$ 的最大和最小可能值的差是多少？
Q13
What is the sum of all the solutions of $x=|2x-|60-2x||$?
求方程 $x=|2x-|60-2x||$ 所有解之和。
Q14
The average of the numbers $1,2,3,\dots,98,99,$ and $x$ is $100x$. What is $x$?
数 $1,2,3,\dots,98,99,$ 和 $x$ 的平均数是 $100x$。求 $x$。
Q15
On a 50-question multiple choice math contest, students receive 4 points for a correct answer, 0 points for an answer left blank, and −1 point for an incorrect answer. Jesse’s total score on the contest was 99. What is the maximum number of questions that Jesse could have answered correctly?
在一场 50 题选择题数学竞赛中，正确得 4 分，空题 0 分，错误 −1 分。Jesse 总分为 99。Jesse 最多能正确答多少题？
Q16
A square of side length 1 and a circle of radius $\sqrt{3}/3$ share the same center. What is the area inside the circle, but outside the square?
边长为 1 的正方形和半径为 $\sqrt{3}/3$ 的圆共享同一个中心。圆内、正方形外的面积是多少？
Q17
Every high school in the city of Euclid sent a team of 3 students to a math contest. Each participant received a different score. Andrea’s score was the median among all students, and hers was the highest on her team. Andrea’s teammates Beth and Carla placed 37th and 64th, respectively. How many schools are in the city?
Euclid 市每所高中派一个 3 人学生队参加数学竞赛。每位参赛者得分均不同。Andrea 的得分是所有学生中的中位数，且是她队内最高。Andrea 的队友 Beth 和 Carla 分别排第 37 和第 64 名。该市有多少所学校？
Q18
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 $a,b,c$ 从集合 $\{1,2,3,\dots,2010\}$ 中独立随机有放回选取。$abc+ab+a$ 能被 3 整除的概率是多少？
Q19
A circle with center $O$ has area $156\pi$. Triangle $ABC$ is equilateral, $BC$ is a chord on the circle, $OA=4\sqrt{3}$, and point $O$ is outside $\triangle ABC$. What is the side length of $\triangle ABC$?
圆心为 $O$ 的圆面积为 $156\pi$。等边三角形 $ABC$，$BC$ 为圆上弦，$OA=4\sqrt{3}$，且点 $O$ 在 $\triangle ABC$ 外。求 $\triangle ABC$ 的边长。
Q20
Two circles lie outside regular hexagon $ABCDEF$. The first is tangent to $AB$, and the second is tangent to $DE$. Both are tangent to lines $BC$ and $FA$. What is the ratio of the area of the second circle to that of the first circle?
两个圆位于正六边形 $ABCDEF$ 外。第一圆切 $AB$，第二圆切 $DE$。两者均切直线 $BC$ 和 $FA$。第二圆与第一圆面积之比是多少？
Q21
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？
Q22
Seven distinct pieces of candy are to be distributed among three bags. The red bag and the blue bag must each receive at least one piece of candy; the white bag may remain empty. How many arrangements are possible?
有7块不同的糖果要分给三个袋子。红袋和蓝袋各至少得一块糖果；白袋可以为空。可能的分法有多少种？
Q23
The entries in a $3\times3$ 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 $3\times3$ 阵列的条目包含1到9的所有数字，排列使得每行每列的条目递增有序。有多少种这样的阵列？
Q24
A high school basketball game between the Raiders and the 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队的高中篮球比赛第一节结束时打平。Raiders队在四个节得分构成递增等比数列，Wildcats队构成递增等差数列。第四节结束时，Raiders队以1分优势获胜。两队均未得分超过100分。两队上半场总得分是多少？
Q25
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 $a>0$，$P(x)$ 是具有整数系数的多项式，使得 $P(1)=P(3)=P(5)=P(7)=a$，且 $P(2)=P(4)=P(6)=P(8)=-a$。$a$ 的最小可能值是多少？
Time Left 75:00