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AMC10 2010 A

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AMC10 · 2010 (A)

Q1
Mary’s top book shelf holds five books with the following widths, in centimeters: 6, $\frac{1}{2}$, 1, 2.5, and 10. What is the average book width, in centimeters?
Mary 的顶层书架上有五本书，它们的宽度（厘米）分别是：6、$\frac{1}{2}$、1、2.5 和 10。这些书的平均宽度是多少厘米？
Q2
Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width?
四个相同的正方形和一个矩形拼在一起形成一个大正方形，如图所示。矩形的长是其宽的几倍？
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Q3
Tyrone had 97 marbles and Eric had 11 marbles. Tyrone then gave some of his marbles to Eric so that Tyrone ended with twice as many marbles as Eric. How many marbles did Tyrone give to Eric?
Tyrone 原来有 97 颗弹珠，Eric 有 11 颗。Tyrone 给了 Eric 一些弹珠，使得 Tyrone 最后剩下的弹珠是 Eric 的两倍。Tyrone 给了 Eric 多少颗弹珠？
Q4
A book that is to be recorded onto compact discs takes 412 minutes to read aloud. Each disc can hold up to 56 minutes of reading. Assume that the smallest possible number of discs is used and that each disc contains the same length of reading. How many minutes of reading will each disc contain?
一本将要录制到光盘上的书朗读需要 412 分钟。每张光盘最多可容纳 56 分钟的朗读。假设使用最少的光盘数，且每张光盘包含相同长度的朗读。每张光盘将包含多少分钟的朗读？
Q5
The area of a circle whose circumference is $24\pi$ is $k\pi$. What is the value of $k$?
周长为 $24\pi$ 的圆的面积是 $k\pi$。$k$ 的值为多少？
Q6
For positive numbers $x$ and $y$ the operation $\spadesuit(x, y)$ is defined as $\spadesuit(x, y) = x - 1/y$. What is $\spadesuit(2, \spadesuit(2, 2))$?
对于正数$x$和$y$，操作$\spadesuit(x, y)$定义为$\spadesuit(x, y) = x - 1/y$。求$\spadesuit(2, \spadesuit(2, 2))$的值。
Q7
Crystal has a running course marked out for her daily run. She starts this run by heading due north for one mile. She then runs northeast for one mile, then southeast for one mile. The last portion of her run takes her on a straight line back to where she started. How far, in miles, is this last portion of her run?
Crystal有一个标记好的日常跑步路线。她从正北方向跑1英里开始。然后向东北跑1英里，再向东南跑1英里。跑步的最后一段是直线回到起点。这最后一段跑步有多远，单位英里？
Q8
Tony works 2 hours a day and is paid \$0.50 per hour for each full year of his age. During a six month period Tony worked 50 days and earned \$630. How old was Tony at the end of the six month period?
Tony每天工作2小时，每满岁按年龄的每小时0.50美元付工资。在六个月期间，Tony工作了50天，赚了630美元。六个月期末Tony多大了？
Q9
A palindrome, such as 83438, is a number that remains the same when its digits are reversed. The numbers $x$ and $x + 32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of $x$?
回文数，如83438，是一个数字反转其各位数字后仍相同的数。数$x$和$x + 32$分别是三位数和四位数的回文数。$x$的各位数字之和是多少？
Q10
Marvin had a birthday on Tuesday, May 27 in the leap year 2008. In what year will his birthday next fall on a Saturday?
Marvin的生日是闰年2008年的星期二，5月27日。他的生日下次落在星期六是哪一年？
Q11
The length of the interval of solutions of the inequality $a \leq 2x + 3 \leq b$ is 10. What is $b - a$?
不等式 $a \leq 2x + 3 \leq b$ 的解集区间长度为 10。$b - a$ 是多少？
Q12
Logan is constructing a scaled model of his town. The city’s water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan’s miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower?
Logan 正在制作他城镇的缩比模型。城市的水塔高 40 米，上部是一个容纳 100,000 升水的球体。Logan 的迷你水塔容纳 0.1 升水。Logan 应该把他的水塔做多高（米）？
Q13
Angelina drove at an average rate of 80 kph and then stopped 20 minutes for gas. After the stop, she drove at an average rate of 100 kph. Altogether she drove 250 km in a total trip time of 3 hours including the stop. Which equation could be used to solve for the time $t$ in hours that she drove before her stop?
Angelina 以平均速度 80 千米/时行驶，然后停下 20 分钟加油。停下后，她以平均速度 100 千米/时行驶。总共她行驶了 250 千米，总行程时间为 3 小时（包括停顿）。哪一个方程可以用来解她在停下前行驶的时间 $t$（小时）？
Q14
Triangle ABC has AB = 2 · AC. Let D and E be on AB and BC, respectively, such that ∠BAE = ∠ACD. Let F be the intersection of segments AE and CD, and suppose that △CFE is equilateral. What is ∠ACB?
三角形 ABC 有 AB = 2 · AC。D 和 E 分别在 AB 和 BC 上，使得 ∠BAE = ∠ACD。F 是线段 AE 和 CD 的交点，且假设 △CFE 是等边三角形。∠ACB 是多少度？
Q15
In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in this swamp, and they make the following statements. Brian: “Mike and I are different species.” Chris: “LeRoy is a frog.” LeRoy: “Chris is a frog.” Mike: “Of the four of us, at least two are toads.” How many of these four amphibians are frogs?
在一个神奇的沼泽中有两种会说话的两栖动物：蟾蜍，它们的陈述总是真实的；青蛙，它们的陈述总是假的。有四只两栖动物，Brian、Chris、LeRoy 和 Mike 一起生活在这个沼泽，它们做了以下陈述。Brian：“Mike 和我是不同种。” Chris：“LeRoy 是青蛙。” LeRoy：“Chris 是青蛙。” Mike：“我们四个中至少有两个是蟾蜍。” 这四只两栖动物中有多少是青蛙？
Q16
Nondegenerate △ABC has integer side lengths, BD is an angle bisector, AD = 3, and DC = 8. What is the smallest possible value of the perimeter?
非退化三角形△ABC具有整数边长，BD是角平分线，AD = 3，DC = 8。周长的最小可能值是多少？
Q17
A solid cube has side length 3 inches. A 2-inch by 2-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid?
一个边长3英寸的实心立方体。在每个面上中心切出一个2英寸×2英寸的正方形孔。每个切口的边平行于立方体的边，每个孔贯穿整个立方体。剩余固体的体积是多少立方英寸？
Q18
Bernardo randomly picks 3 distinct numbers from the set ${1, 2, 3, 4, 5, 6, 7, 8, 9}$ and arranges them in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbers from the set ${1, 2, 3, 4, 5, 6, 7, 8}$ and also arranges them in descending order to form a 3-digit number. What is the probability that Bernardo’s number is larger than Silvia’s number?
Bernardo从集合${1, 2, 3, 4, 5, 6, 7, 8, 9}$中随机挑选3个不同的数，并按降序排列形成一个3位数。Silvia从集合${1, 2, 3, 4, 5, 6, 7, 8}$中随机挑选3个不同的数，也按降序排列形成一个3位数。Bernardo的数大于Silvia的数的概率是多少？
Q19
Equiangular hexagon ABCDEF has side lengths AB = CD = EF = 1 and BC = DE = FA = r. The area of △ACE is 70% of the area of the hexagon. What is the sum of all possible values of r?
等角六边形ABCDEF有边长AB = CD = EF = 1且BC = DE = FA = r。△ACE的面积是六边形面积的70%。所有可能r值的和是多少？
Q20
A fly trapped inside a cubical box with side length 1 meter decides to relieve its boredom by visiting each corner of the box. It will begin and end in the same corner and visit each of the other corners exactly once. To get from a corner to any other corner, it will either fly or crawl in a straight line. What is the maximum possible length, in meters, of its path?
一只苍蝇被困在一个边长1米的立方体盒子里，它决定通过访问盒子的每个顶点来消磨无聊。它从同一个顶点开始并结束，并恰好访问每个其他顶点一次。从一个顶点到另一个顶点，它要么飞要么爬直线行进。其路径的最大可能长度是多少米？
Q21
The polynomial $x^3 - a x^2 + b x - 2010$ has three positive integer zeros. What is the smallest possible value of $a$?
多项式 $x^3 - a x^2 + b x - 2010$ 有三个正整数零点。$a$ 的最小可能值为多少？
Q22
Eight points are chosen on a circle, and chords are drawn connecting every pair of points. No three chords intersect in a single point inside the circle. How many triangles with all three vertices in the interior of the circle are created?
在圆周上选 8 个点，并连接每对点的弦。没有三条弦在圆内单一点相交。圆内顶点全在圆内的三角形有多少个？
Q23
Each of 2010 boxes in a line contains a single red marble, and for 1 ≤ k ≤ 2010, the box in the kth position also contains k white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let P(n) be the probability that Isabella stops after drawing exactly n marbles. What is the smallest value of n for which P(n) < 1/2010?
有 2010 个盒子排成一行，每个盒子含一个红弹珠，对于 $1 \leq k \leq 2010$，第 $k$ 个盒子还含 $k$ 个白弹珠。Isabella 从第一个盒子开始，依次从每个盒子随机抽取一颗弹珠。她在第一次抽到红弹珠时停止。设 $P(n)$ 为 Isabella 恰好抽取 $n$ 颗弹珠后停止的概率。求 $P(n) < 1/2010$ 的最小 $n$ 值。
Q24
The number obtained from the last two nonzero digits of 90! is equal to n. What is n?
90! 的最后两个非零数字组成的数等于 $n$。$n$ 是多少？
Q25
Jim starts with a positive integer n and creates a sequence of numbers. Each successive number is obtained by subtracting the largest possible integer square less than or equal to the current number until zero is reached. For example, if Jim starts with n = 55, then his sequence contains 5 numbers: 55, 55−7²=6, 6−2²=2, 2−1²=1, 1−1²=0. Let N be the smallest number for which Jim’s sequence has 8 numbers. What is the units digit of N?
Jim 从正整数 $n$ 开始，生成数列。每个后续数由减去当前数小于或等于的最大完全平方整数得到，直到达到零。例如，若 Jim 从 $n = 55$ 开始，则数列含 5 个数：55, 55−7²=6, 6−2²=2, 2−1²=1, 1−1²=0。设 $N$ 为 Jim 的数列有 8 个数的数的最小值。$N$ 的个位数是多少？
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