/

AMC10 2012 A

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

AMC10 · 2012 (A)

Q1
Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many cupcakes can they frost in 5 minutes?
Cagney 每20秒可以给一个杯子涂霜，Lacey 每30秒可以给一个杯子涂霜。他们一起工作，5分钟内可以涂霜多少个杯子？
Q2
A square with side length 8 is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles?
一个边长为8的正方形被切成两半，得到两个全等的矩形。其中一个矩形的尺寸是多少？
Q3
A bug crawls along a number line, starting at −2. It crawls to −6, then turns around and crawls to 5. How many units does the bug crawl altogether?
一只虫子沿着数轴爬行，从−2开始。它爬到−6，然后掉头爬到5。虫子总共爬了多少单位距离？
Q4
Let $\angle ABC = 24^\circ$ and $\angle ABD = 20^\circ$. What is the smallest possible degree measure for $\angle CBD$?
设$\angle ABC = 24^\circ$且$\angle ABD = 20^\circ$。$\angle CBD$的最小可能度数是多少？
Q5
Last year 100 adult cats, half of whom were female, were brought into the Smallville Animal Shelter. Half of the adult female cats were accompanied by a litter of kittens. The average number of kittens per litter was 4. What was the total number of cats and kittens received by the shelter last year?
去年，有100只成年猫被带到Smallville动物收容所，其中一半是母猫。成年母猫中有一半带着一窝小猫。每窝小猫的平均数量是4。去年收容所收到的猫和小猫总数是多少？
Q6
The product of two positive numbers is 9. The reciprocal of one of these numbers is 4 times the reciprocal of the other number. What is the sum of the two numbers?
两个正数的乘积是9。其中一个数的倒数是另一个数的倒数的4倍。这两个数的和是多少？
Q7
In a bag of marbles, $\frac{3}{5}$ of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red?
一袋弹珠中，$\frac{3}{5}$是蓝色的，其余是红色的。如果红色的弹珠数量加倍，蓝色的数量不变，那么红色的弹珠占总数的几分之几？
Q8
The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number?
三个整数两两相加的和分别是12、17和19。这三个整数中的中间大小的数是多少？
Q9
A pair of six-sided fair dice are labeled so that one die has only even numbers (two each of 2, 4, and 6), and the other die has only odd numbers (two each of 1, 3, and 5). The pair of dice is rolled. What is the probability that the sum of the numbers on the tops of the two dice is 7?
一对六面的公平骰子被标记，其中一个骰子只有偶数（各有两个2、4和6），另一个骰子只有奇数（各有两个1、3和5）。掷这对骰子，两个骰子上面数字之和为7的概率是多少？
Q10
Mary divides a circle into 12 sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?
Mary将一个圆分成12个扇形。这些扇形的圆心角（以度为单位）都是整数，并且形成一个等差数列。可能的最小扇形角的度量是多少度？
Q11
Externally tangent circles with centers at points A and B have radii of lengths 5 and 3, respectively. A line externally tangent to both circles intersects ray AB at point C. What is BC?
中心在点 A 和 B 的外部相切圆分别有半径 5 和 3。一条与两个圆都外部相切的直线与射线 AB 相交于点 C。求 BC 的长度。
Q12
A year is a leap year if and only if the year number is divisible by 400 (such as 2000) or is divisible by 4 but not by 100 (such as 2012). The 200th anniversary of the birth of novelist Charles Dickens was celebrated on February 7, 2012, a Tuesday. On what day of the week was Dickens born?
一年是闰年当且仅当年份能被 400 整除（如 2000 年）或能被 4 整除但不能被 100 整除（如 2012 年）。小说家查尔斯·狄更斯的诞生 200 周年纪念于 2012 年 2 月 7 日星期二庆祝。狄更斯出生于星期几？
Q13
An iterative average of the numbers 1, 2, 3, 4, and 5 is computed in the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?
对数字 1、2、3、4 和 5 进行迭代平均的计算方式如下。将五个数字按某种顺序排列。先求前两个数的平均数，然后将该平均数与第三个数的平均数，再将结果与第四个数的平均数，最后与第五个数的平均数。使用此过程可能得到的最大值与最小值之差是多少？
Q14
Chubby makes nonstandard checkerboards that have 31 squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. How many black squares are there on such a checkerboard?
Chubby 制作非标准棋盘，每边有 31 个方格。棋盘四个角都是黑方格，每行每列红黑方格交替。这样的棋盘上有多少黑方格？
Q15
Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of $\triangle ABC$?
给出三个单位正方形和连接两对顶点的两条线段。求 $\triangle ABC$ 的面积。
stem
Q16
Three runners start running simultaneously from the same point on a 500-meter circular track. They each run clockwise around the course maintaining constant speeds of 4.4, 4.8, and 5.0 meters per second. The runners stop once they are all together again somewhere on the circular course. How many seconds do the runners run?
三名跑步者同时从500米圆形跑道上的同一点开始顺时针跑步，他们的速度分别是4.4、4.8和5.0米/秒。他们一直跑到再次在圆形跑道上的某处同时到达为止。他们跑了多少秒？
Q17
Let a and b be relatively prime integers with a > b > 0 and $\frac{a^3 - b^3}{(a - b)^3} = \frac{73}{3}$. What is a − b?
设$a$和$b$是互质的正整数，且$a > b > 0$，$\frac{a^3 - b^3}{(a - b)^3} = \frac{73}{3}$。求$a - b$的值。
Q18
The closed curve in the figure is made up of 9 congruent circular arcs each of length $\frac{2\pi}{3}$, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side 2. What is the area enclosed by the curve?
图中的闭合曲线由9个全等的圆弧组成，每个圆弧长度为$\frac{2\pi}{3}$，对应圆心是边长为2的正六边形的顶点之一。求该曲线围成的面积。
stem
Q19
Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:00 am, and all three always take the same amount of time to eat lunch. On Monday the three of them painted 50% of a house, quitting at 4:00 pm. On Tuesday, when Paula wasn’t there, the two helpers painted only 24% of the house and quit at 2:12 pm. On Wednesday Paula worked by herself and finished the house by working until 7:12 pm. How long, in minutes, was each day’s lunch break?
画家Paula和她的两名助手以恒定但不同的速率作画。他们总是从上午8:00开始，三人午饭时间相同。周一三人画了房子50%，下午4:00停止。周二Paula不在，两名助手只画了24%，下午2:12停止。周三Paula独自工作，到下午7:12完成房子。每日的午饭休息时间有多少分钟？
Q20
A 3 × 3 square is partitioned into 9 unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated 90° clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability that the grid is now entirely black?
一个3×3方格被分成9个单位方格。每个单位方格独立随机涂成白色或黑色，两种颜色等概率。方格绕中心顺时针旋转90°，每个原先被黑色方格占据位置的白色方格涂成黑色，其他方格颜色不变。求现在整个格子全黑的概率。
Q21
Let points A = (0, 0, 0), B = (1, 0, 0), C = (0, 2, 0), and D = (0, 0, 3). Points E, F, G, and H are midpoints of line segments BD, AB, AC, and DC respectively. What is the area of EFGH?
设点 A = (0, 0, 0), B = (1, 0, 0), C = (0, 2, 0), D = (0, 0, 3)。点 E, F, G, H 分别是线段 BD, AB, AC, DC 的中点。EFGH 的面积是多少？
Q22
The sum of the first m positive odd integers is 212 more than the sum of the first n positive even integers. What is the sum of all possible values of n?
前 m 个正奇数的和比前 n 个正偶数的和多 212。所有可能 n 值之和是多少？
Q23
Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen?
Adam, Benin, Chiang, Deshawn, Esther, 和 Fiona 有网络账户。他们有些但不是全部是彼此的网络朋友，且他们都没有该组外的网络朋友。他们每个人有相同数目的网络朋友。这种情况可以有多少种不同的方式发生？
Q24
Let a, b, and c be positive integers with a ≥ b ≥ c such that $a^2 - b^2 - c^2 + ab = 2011$ and $a^2 + 3b^2 + 3c^2 - 3ab - 2ac - 2bc = -1997$. What is a?
设 a, b, c 为正整数且 a ≥ b ≥ c，使得 $a^2 - b^2 - c^2 + ab = 2011$ 且 $a^2 + 3b^2 + 3c^2 - 3ab - 2ac - 2bc = -1997$。a 是多少？
Q25
Real numbers x, y, and z are chosen independently and at random from the interval [0, n] for some positive integer n. The probability that no two of x, y, and z are within 1 unit of each other is greater than $\frac{1}{2}$. What is the smallest possible value of n?
实数 x, y, z 独立均匀随机从区间 [0, n] 中选取，其中 n 为正整数。x, y, z 中任意两个不相距 1 个单位以内的概率大于 $\frac{1}{2}$。n 的最小可能值是多少？
Time Left 75:00