2010年2012 美国AMC10

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amc10评分标准

amc10评分标准AMC10是美国数学竞赛的一部分，旨在评估初中学生在数学领域的能力。

在这项竞赛中，学生将面临一系列挑战性的数学问题。

为了公正地评估学生的表现，AMC10有一套评分标准，本文将介绍这些标准。

一、选择题评分标准AMC10中的多项选择题占据了竞赛的大部分内容。

这些题目提供了若干个选项供学生选择，需要学生选出正确答案。

选择题的评分标准如下：1.正确答案：如果选手选择了正确的答案，将得到4分。

2.错误答案：如果选手选择了错误的答案，不得分也不扣分。

3.未解答：如果选手未选择任何答案，将不得分也不扣分。

需要注意的是，选择题没有部分得分。

学生的分数完全取决于他们的选择是否正确。

二、填空题评分标准AMC10中的填空题考察学生解答数学问题的能力，学生需要计算出准确的答案并填入相应的空白中。

填空题的评分标准如下：1.正确答案：如果选手填入了正确的答案，将得到6分。

2.错误答案：如果选手填入了错误的答案，不得分也不扣分。

3.未解答：如果选手未填入任何答案，将不得分也不扣分。

填空题同样没有部分得分，只有完全正确才能得到分数。

三、证明题评分标准AMC10中的证明题考察学生证明数学命题的能力，学生需要利用所学的数学知识和推理能力，给出合理的证明过程。

证明题的评分标准如下：1.正确证明：如果学生给出了完全正确的证明过程，将得到7分。

2.部分正确：如果学生给出了部分正确的证明过程，将根据证明的准确性和完整性给予部分分数。

3.未解答：如果选手未给出任何证明过程，将不得分也不扣分。

需要注意的是，证明题可以有部分得分，学生的分数取决于他们的证明过程的准确性和完整性。

四、解答题评分标准AMC10中的解答题考察学生解决复杂数学问题的能力，学生需要给出详细的解题步骤和答案。

解答题的评分标准如下：1.完整解答：如果学生给出了完整的解题过程和正确的答案，将得到10分。

2.部分解答：如果学生给出了部分解题过程或者答案，将根据解题的准确性和完整性给予部分分数。

amc10公式定理

amc10公式定理AMC10公式定理AMC10是美国数学竞赛中的一项考试，涵盖了初级数学的各个领域。

在解题过程中，我们可以运用一些定理和公式来简化计算和推理，提高解题效率。

本文将介绍一些常用的AMC10公式定理，帮助读者更好地应对竞赛中的数学问题。

1. 一元二次方程的解法一元二次方程是AMC10考试中常见的题型，解题时可以运用二次方程的求根公式：对于方程 $ax^2 + bx + c = 0$，其根可以通过以下公式求得：$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$2. 直角三角形的勾股定理勾股定理是直角三角形中最基本的定理之一，可以用于计算三角形的边长关系。

对于一个直角三角形，边长分别为a、b、c，其中c 为斜边长度，则有：$a^2 + b^2 = c^2$3. 三角函数的基本关系在三角函数中，正弦、余弦和正切是常见的函数。

它们之间存在一些基本关系，可以帮助我们简化计算。

如下所示：$\sin^2(\theta) + \cos^2(\theta) = 1$$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$4. 二项式定理二项式定理是展开一个二项式的公式，可以用于求解多项式的各项系数。

对于任意实数a和b以及正整数n，有：$(a+b)^n = \sum_{k=0}^{n} C_n^k a^{n-k} b^k$其中，$C_n^k$ 表示从n个元素中选取k个元素的组合数。

5. 平均值不等式平均值不等式可以用于证明两个数之间的大小关系。

对于任意一组非负实数 $a_1, a_2, \ldots, a_n$，有：$\frac{a_1 + a_2 + \ldots + a_n}{n} \geq \sqrt[n]{a_1 a_2 \ldots a_n}$其中，等号成立当且仅当 $a_1 = a_2 = \ldots = a_n$。

2010AMC10B答案及部分问题参考解答

2010 AMC 10 B 答案
1. C 2. C 3. C 4. C 5. B 6. B 7. D 8. E 9. D 10.C 11.A 12.D 13.C 14.B 15.C 16.B 17.B 18.E 19.B 20.D 21.E 22.C 23.D 24.E 25.B
18. 译题: 正整数 a, b 和 c 是从集合{1,2,3,…,2010}中独立且可重复的随机选取，则
8=a32 或 a23. 由对称性，先考 2=a12，当 2=a21 时，排法数与之相等.
当 a12=2, 取 8=a32 时[如图一(1)], 3= a21 或 a13. 7= a23 或 a31. 当 a21=3 时，取 a23=7 [如图一(2)], 则 4,5,6 可排在余下三格中任意一格.共 A33=6 种;
25. 译题: 令 a>0, 且 P(x)为整系数多项式,且满足 P(1)=P(3)=P(5)=P(7)=a, P(2)=P(4)=P(6)=P(8)= -a. 求 a 的最小值.
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PDF 文件使用 "pdfFactory Pro" 试用版本创建
(A) 105
(B) 315
(C) 945
(D) 7!
(E) 8!
[分析]: 由题意知：1,3,5,7 为多项式 P(x)-a 的根，∴P(x)-a=(x-1)(x-3)(x-5)(x-7)Q(x)
其中 Q(x)为整系数多项式, 上式中分别令 x=2,4,6,8 , 可得：
-2a= -15Q(2)=9Q(4)= -15Q(6)=105Q(8) ∴ a 为 15,9,105 的最小公倍数 315 的倍数
∴ 其能被 7 整除的概率为:
2
=

2000-2012美国AMC10中文版试题及答案

2000到2012年AMC10美国数学竞赛0 0P 0 A 0 B 0 C 0D 0 全美中学数学分级能力测验(AMC 10)2000年第01届美国AMC10 (2000年2月日时间75分钟)1. 国际数学奥林匹亚将于2001年在美国举办，假设I 、M 、O 分别表示不同的正整数，且满足I ⨯M ⨯O =2001，则试问I +M +O 之最大值为。

(A) 23 (B) 55 (C) 99 (D) 111 (E) 6712. 2000(20002000)为。

(A) 40 (B) 50 (C) 55 (D) 60 (E) 754. Candra 每月要付给网络公司固定的月租费及上网的拨接费，已知她12月的账单为12.48元，而她1月的账单为17.54元，若她1月的上网时间是12月的两倍，试问月租费是元。

(A) 2.53 (B) 5.06 (C) 6.24 (D) 7.42 (E) 8.775. 如图M ，N 分别为PA 与PB 之中点，试问当P 在一条平行AB 的直在线移动时，下列各数值有项会变动。

(a) MN 长 (b) △P AB 之周长 (c) △P AB 之面积 (d) ABNM 之面积(A) 0项 (B) 1项 (C) 2项 (D) 3项 (E) 4项 6. 费氏数列是以两个1开始，接下来各项均为前两项之和，试问在费氏数列各项的个位数字中，最后出现的阿拉伯数字为。

(A) 0 (B) 4 (C) 6 (D) 7 (E) 97. 如图，矩形ABCD 中，AD =1，P 在AB 上，且DP 与DB 三等分∠ADC ，试问△BDP 之周长为。

2010 AMC 10A 试题及答案解析

2010 AMC 10A problems and solutions.The test was held on February 8, 2010. The first link contains the full set of test problems. The rest contain each individual problem and its solution.Problem 1Mary’s top book shelf holds five books with the follow ing widths, incentimeters: , , , , and . What is the average book width, in centimeters?SolutionTo find the average, we add up the widths , , , , and , to get a total sum of . Since there are books, the average book width isThe answer is .Problem 2Four 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?SolutionLet the length of the small square be , intuitively, the length of the big square is . It can be seen that the width of the rectangle is .Thus, the length of the rectangle is times large as the width. The answer is .Problem 3Tyrone had marbles and Eric had 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?SolutionLet be the number of marbles Tyrone gave to Eric. Then,. Solving for yields and . The answer is .Problem 4A book that is to be recorded onto compact discs takes minutes to read aloud. Each disc can hold up to 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?SolutionAssuming that there were fractions of compact discs, it would take CDs to have equal reading time. However, since the number of discs can only be a whole number, there are at least 8 CDs, in which case it would have minutes on each of the 8 discs. The answer is .Problem 5The area of a circle whose circumference is is . What is the value of ?SolutionIf the circumference of a circle is , the radius would be . Since the area of a circle is , the area is . The answer is . Problem 6For positive numbers and the operation is defined asWhat is ?Solution. Then, is The answer isProblem 7Crystal 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?SolutionCrystal first runs North for one mile. Changing directions, she runs Northeast for another mile. The angle difference between North and Northeast is 45 degrees. She then switches directions to Southeast, meaning a 90 degree angle change. The distance now from travelling North for one mile, and her current destination is miles, because it is the hypotenuse of a 45-45-90 triangle with side length one (mile). Therefore, Crystal's distance from her starting position, x, is equal to , which is equal to . The answer isTony works hours a day and is paid $per hour for each full year of his age. During a six month period Tony worked days and earned $. How old was Tony at the end of the six month period?SolutionTony worked hours a day and is paid dollars per hour for each full year of his age. This basically says that he gets a dollar for each year of his age. So if he is years old, he gets dollars a day. We also know that he worked days and earned dollars. If he was years old at the beginning of his working period, he would have earned dollars. If he was years old at the beginning of his working period, he would have earned dollars. Because he earned dollars, we know that he was for some period of time, but not the whole time, because then the money earned would be greater than or equal to . This is why he was when he began, but turned sometime in the middle and earned dollars in total. So the answer is .The answer is . We could find out for how long he was and . . Then isand we know that he was for days, and for days. Thus, the answer is .Problem 9A palindrome, such as , is a number that remains the same when its digits are reversed. The numbers and are three-digit and four-digit palindromes, respectively. What is the sum of the digits of ?Solutionis at most , so is at most . The minimum value ofis . However, the only palindrome between and is , which means that must be .It follows that is , so the sum of the digits is .Marvin had a birthday on Tuesday, May 27 in the leap year . In what year will his birthday next fall on a Saturday?Solution(E) 2017There are 365 days in a non-leap year. There are 7 days in a week. Since 365 = 52 * 7 + 1 (or 365 is congruent to 1 mod 7), the same date (after February) moves "forward" one day in the subsequent year, if that year is not a leap year.For example: 5/27/08 Tue 5/27/09 WedHowever, a leap year has 366 days, and 366 = 52 * 7 + 2. So the same date (after February) moves "forward" two days in the subsequent year, if that year is a leap year.For example: 5/27/11 Fri 5/27/12 SunYou can keep count forward to find that the first time this date falls on a Saturday is in 2017:5/27/13 Mon 5/27/14 Tue 5/27/15 Wed 5/27/16 Fri 5/27/17 Sat Problem 11The length of the interval of solutions of the inequality is . What is ?SolutionSince we are given the range of the solutions, we must re-write the inequalities so that we have in terms of and .Subtract from all of the quantities:Divide all of the quantities by .Since we have the range of the solutions, we can make them equal to .Multiply both sides by 2.Re-write without using parentheses.Simplify.We need to find for the problem, so the answer isProblem 12Logan 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?SolutionThe water tower holds times more water than Logan's miniature. Therefore, Logan should make his towertimes shorter than the actual tower. This ismeters high, or choice .Problem 13Angelina drove at an average rate of kph and then stopped minutes for gas. After the stop, she drove at an average rate of kph. Altogether she drove km in a total trip time of hours including the stop. Which equation could be used to solve for the time in hours that she drove before her stop?SolutionThe answer is （）because she drove at kmh for hours (the amount of time before the stop), and 100 kmh for because she wasn't driving for minutes, or hours. Multiplying by gives the total distance, which is kms. Therefore, the answer isProblem 14Triangle has . Let and be on and , respectively, such that . Let be the intersection of segments and , and suppose that is equilateral. What is ?SolutionLet .Since ,Problem 15In 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 amphibians are frogs?SolutionSolution 1We can begin by first looking at Chris and LeRoy.Suppose Chris and LeRoy are the same species. If Chris is a toad, then what he says is true, so LeRoy is a frog. However, if LeRoy is a frog, then he is lying, but clearly Chris is not a frog, and we have a contradiction. The same applies if Chris is a frog.Clearly, Chris and LeRoy are different species, and so we have at least frog out of the two of them.Now suppose Mike is a toad. Then what he says is true because we already have toads. However, if Brian is a frog, then he is lying, yet his statement is true, a contradiction. If Brian is a toad, then what he says is true, but once again it conflicts with his statement, resulting in contradiction.Therefore, Mike must be a frog. His statement must be false, which means that there is at most toad. Since either Chris or LeRoy is already a toad, Brain must be a frog. We can also verify that his statement is indeed false.Both Mike and Brian are frogs, and one of either Chris or LeRoy is a frog, so we have frogs total.Solution 2Start with Brian. If he is a toad, he tells the truth, hence Mike is a frog. If Brian is a frog, he lies, hence Mike is a frog, too. Thus Mike must be a frog.As Mike is a frog, his statement is false, hence there is at most one toad.As there is at most one toad, at least one of Chris and LeRoy is a frog. But then the other one tells the truth, and therefore is a toad. Hence we must have one toad and three frogs.Problem 16Nondegenerate has integer side lengths, is an angle bisector, , and . What is the smallest possible value of the perimeter?SolutionBy the Angle Bisector Theorem, we know that . If we use the lowest possible integer values for AB and BC (the measures of AD and DC, respectively), then , contradicting the Triangle Inequality. If we use the next lowest values (and ), the Triangle Inequality is satisfied. Therefore, our answer is , or choice .Problem 17A solid cube has side length inches. A -inch by -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?SolutionSolution 1Imagine making the cuts one at a time. The first cut removes a box . The second cut removes two boxes, each of dimensions, and the third cut does the same as the second cut, on the last two faces. Hence the total volume of all cuts is .Therefore the volume of the rest of the cube is.Solution 2We can use Principle of Inclusion-Exclusion to find the final volume of the cube.There are 3 "cuts" through the cube that go from one end to the other. Each of these "cuts" has cubic inches. However, we can not just sum their volumes, as the central cube is included in each of these three cuts. To get the correct result, we can take the sum of the volumes of the three cuts, and subtract the volume of the central cube twice.Hence the total volume of the cuts is.Therefore the volume of the rest of the cube is.Solution 3We can visualize the final figure and see a cubic frame. We can find the volume of the figure by adding up the volumes of the edges and corners.Each edge can be seen as a box, and each corner can be seen as a box..Problem 18Bernardo randomly picks 3 distinct numbers from the setand arranges them in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbers from the set 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?SolutionWe can solve this by breaking the problem down into cases and adding up the probabilities.Case : Bernardo picks . If Bernardo picks a then it is guaranteed that his number will be larger than Silvia's. The probability that he will pick a is .Case : Bernardo does not pick . Since the chance of Bernardo picking is , the probability of not picking is .If Bernardo does not pick 9, then he can pick any number from to . Since Bernardo is picking from the same set of numbers as Silvia, the probability that Bernardo's number is larger is equal to the probability that Silvia's number is larger.Ignoring the for now, the probability that they will pick the same number is the number of ways to pick Bernardo's 3 numbers divided by the number of ways to pick any 3 numbers.We get this probability to beProbability of Bernardo's number being greater isFactoring the fact that Bernardo could've picked a but didn't:Adding up the two cases we getProblem 19Equiangular hexagon has side lengthsand . The area of is of the area of the hexagon. What is the sum of all possible values of ?SolutionSolution 1It is clear that is an equilateral triangle. From the Law of Cosines, we get that . Therefore, the area of is .If we extend , and so that and meet at , and meet at , and and meet at , we find that hexagon is formed by taking equilateral triangle of side length and removing three equilateral triangles, , and , of side length . The area of is therefore.Based on the initial conditions,Simplifying this gives us . By Vieta's Formulas we know that the sum of the possible value of is .Solution 2As above, we find that the area of is .We also find by the sine triangle area formula that, and thusThis simplifies to.Problem 20A fly trapped inside a cubical box with side length 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?SolutionThe distance of an interior diagonal in this cube is and the distance of a diagonal on one of the square faces is . It would not make sense if the fly traveled an interior diagonal twice in a row, as it would return to the point it just came from, so at most the final sum can only have 4 as the coefficient of . The other 4 paths taken can be across a diagonal on one of the faces, so the maximum distance traveled is.Problem 21The polynomial has three positive integer zeros. What is the smallest possible value of ?SolutionBy Vieta's Formulas, we know that is the sum of the three roots of the polynomial . Also, 2010 factors into. But, since there are only three roots to the polynomial, two of the four prime factors must be multiplied so that we are left with three roots. To minimize , and should be multiplied, which means will be and the answer is .Problem 22Eight points are chosen on a circle, and chords are drawn connecting every pair of points. No three chords intersect in a single point insidethe circle. How many triangles with all three vertices in the interior of the circle are created?SolutionTo choose a chord, we know that two points must be chosen. This implies that for three chords to create a triangle and not intersect at a single point, six points need to be chosen. Therefore, the answer is which is equivalent to 28,Problem 23Each of 2010 boxes in a line contains a single red marble, and for , the box in the position also contains 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 be the probability that Isabella stops afterdrawing exactly marbles. What is the smallest value of for which ?SolutionThe probability of drawing a white marble from box is . Theprobability of drawing a red marble from box is .The probability of drawing a red marble at box is thereforeIt is then easy to see that the lowest integer value of that satisfies the inequality is .Problem 24The number obtained from the last two nonzero digits of is equal to . What is ?SolutionWe will use the fact that for any integer ,First, we find that the number of factors of in is equal to. Let . The we want is therefore the last two digits of , or . Since there is clearly an excess of factors of 2, we know that , so it remains to find .If we divide by by taking out all the factors of in , we canwrite as where where every multiple of 5 is replaced by the number with all its factors of 5 removed. Specifically, every number in the form is replaced by , and every number in the form is replaced by .The number can be grouped as follows:Using the identity at the beginning of the solution, we can reducetoUsing the fact that (or simply the fact that if you have your powers of 2 memorized), we can deduce that . Therefore.Finally, combining with the fact that yields.Problem 25Jim starts with a positive integer 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 , then his sequence contains numbers:Let be the smallest number for which Jim’s sequence has numbers. What is the units digit of ?SolutionWe can find the answer by working backwards. We begin with on the bottom row, then the goes to the right of the equal's sign in the row above. We find the smallest value for whichand , which is .We repeat the same procedure except with for the next row and for the row after that. However, at the fourth row, wesee that solving yields , in which case it would be incorrect since is not the greatest perfect square less than or equal to . So we make it a and solve . We continue on using this same method where we increase the perfect square until can be made bigger than it. When we repeat this until we have rows, we get:Hence the solution is the last digit of , which is .。

amc10真题与答案解析

amc10真题与答案解析AMC10真题与答案解析美国数学竞赛（AMC）是一项极具挑战性的数学竞赛，旨在推动学生的数学学习和解决问题的能力。

其中AMC10是为中学生设计的竞赛，是很多学生展现才华并锻炼数学素养的重要机会。

本文将对一道AMC10真题进行解析，帮助读者理解和掌握解题技巧。

以下是一道来自AMC10的真题："In the xy-plane, the graph of$\log_{16}x+\log_{16}y=\tfrac{3}{2}$ is drawn. The line$y=x$ intersects the graph at points $A$ and $B$. What is the length of segment $AB$?"这道题目涉及了对数的性质和直线与曲线的交点问题。

首先，我们先来理解题目所给出的等式。

$\log_{16}x$表示以16为底的对数，可以简化为$\log_2x^4$。

同样地，$\log_{16}y$可以简化为$\log_2y^4$。

所以等式可以写为$\log_2x^4 + \log_2y^4 =\frac{3}{2}$。

我们可以将等式进一步简化为$\log_2 (x^4y^4) =\frac{3}{2}$。

根据对数的性质，我们可以将等式转化为指数形式，得到$x^4y^4 = 2^{\frac{3}{2}}$。

然后我们来解决直线$y=x$与曲线$y=x^4$的交点问题。

我们可以将$x$代入曲线方程中，得到$y=x^4$。

因此，我们可以将交点问题转化为求解以下方程组：$x^4 =x$和$x^4y^4 = 2^{\frac{3}{2}}$。

首先，我们来解原方程$x^4 = x$。

很明显，$x=0$和$x=1$是方程的两个根。

我们可以进一步分析，当$x>1$时，$x^4$的增长速度比$x$快，所以方程没有其他解。

然后，我们来解方程$x^4y^4 = 2^{\frac{3}{2}}$。

2010AMC10美国数学竞赛B卷

2010 AMC10美国数学竞赛B卷1. What is 10010031001003（）（）?---(A) -20,000 (B) -10,000 (C) -297 (D) -6 (E) 02. 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?(A) 15 (B) 20 (C) 25 (D) 30 (E) 353. 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?(A) 3 (B) 4 (C) 5 (D) 8 (E) 94. For a real number x, define ♡(x) to be the average of x and x2. What is♡(1)+ ♡(2)+ ♡(3)?(A) 3 (B) 6 (C) 10 (D) 12 (E) 205. 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?(A) 2 (B) 3 (C) 4 (D) 5 (E) 66. A circle is centered at O, AB is a diameter and C is a point on the circle with∠COB=50°. What is the degree measure of ∠CAB?(A) 20 (B) 25 (C) 45 (D) 50 (E) 657. 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?(A) 16 (B) 24 (C) 28 (D) 32 (E) 368. 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?(A) 1 (B) 2 (C) 3 (D) 4 (E) 59. Lucky Larry’s teacher asked him to substitute numbers for a, b, c, d, and e in the expression ((()))---+and evaluate the result. Larry ignored the parent thesea b c d ebut 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?(A) -5 (B) -3 (C) 0 (D) 3 (E) 510. 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 rainin the evening, for a total of 16 miles in 40 minutes. How many minutes did she drive in the rain?(A) 18 (B) 21 (C) 24 (D) 27 (E) 3011. A shopper plans to purchase an item that has a listed price greater than $100 and can use any one of the 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?(A) 50 (B) 60 (C) 75 (D) 80 (E) 10012. 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 difference between the maximum and the minimum possible values of x?(A) 0 (B) 20 (C) 40 (D) 60 (E) 8013. What is the sum of all the solutions of 2602=--?x x x(A) 32 (B)60 (C)92 (D) 120 (E) 12414. The average of the numbers 1, 2, 3, …, 98, 99, and x is 100x. What is x? (A)49101 (B) 50101 (C) 12 (D) 51101 (E) 509915. 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?(A) 25(B) 27 (C) 29 (D) 31 (E) 3316. A square of side length 1 and a circle of radiusshare the same center. What is the area inside the circle, but outside the square?(A)13π- (B) 29π- (C) 18π (D) 14(E) 29π17. 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 here team. Andrea ’s teammates Beth and Carla placed 37th and 64th , respectively. How many schools are in the city?(A) 22(B) 23 (C) 24 (D) 25 (E) 2618. Positive integers a, b, and c are randomly and independently selected withreplacement from the set {1, 2, 3, …, 2010}. What is the probability that abc ab a++ is divisible by 3?(A) 13(B) 2981(C) 3181(D) 1127(E) 132719. A circle with center O has area 156π. Triangle ABC is equilateral, BC is a chordon the circle, OA=, and point O is outside △ABC. What is the side length of △ABC?(A)(B) 64 (C) (D) 12 (E) 1820. 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?(A) 18 (B) 27 (C) 36 (D) 81 (E) 10821. A palindrome between 1000 and 10,000 is chosen at random. What is the probability that it is divisible by 7?(A) 1/10 (B) 1/9 (C) 1/7 (D) 1/6 (E) 1/522. Seven distinct pieces of candy are to be distributed among three bags. The red bag must each receive at least one piece of candy; the white bag may remain empty. How many arrangements are possible?(A) 1930 (B) 1931 (C) 1932 (D) 1933 (E) 193423. The entries in a 3×3 array include all the digits from 1 through 9, arranged so thatthe entries in every row and column are in increasing order. How many such arrays are there?(A) 18(B) 24 (C) 36 (D) 42 (E) 6024. 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?(A) 30(B) 31 (C) 32 (D) 33 (E) 3425. Let a>0, and let P(x) be a polynomial with integer coefficients such that(1)(3)(5)(7),(2)(4)(6)(8).P P P P a and P P P P a ========- What is the smallest possible value of a?(A) 105(B) 315 (C) 945 (D) 7! (E) 8!。

美国AMC10中文版试题及答案

2000到20XX年AMC10美国数学竞赛0 0P 0 A 0 B 0 C 0D 0 全美中学数学分级能力测验(AMC 10)2000年第01届美国AMC10 (2000年2月日时间75分钟)1. 国际数学奥林匹亚将于在美国举办，假设I 、M 、O 分别表示不同的正整数，且满足I ⨯M ⨯O =2001，则试问I +M +O 之最大值为。

(A) 23 (B) 55 (C) 99 (D) 111 (E) 6712. 2000(20002000)为。

(A) 2.53 (B) 5.06 (C) 6.24 (D) 7.42 (E) 8.775. 如图M ，N 分别为PA 与PB 之中点，试问当P 在一条平行AB 的直在线移动时，下列各数值有项会变动。

(A) 0 (B) 4 (C) 6 (D) 7 (E) 97. 如图，矩形ABCD 中，AD =1，P 在AB 上，且DP 与DB 三等分∠ADC ，试问△BDP 之周长为。

2010 AMC10A试题(含答案)

2010 AMC10A ProblemsQ1. Mary’s top book shelf holds ﬁve books with the follo wing widths, in centimeters:6,12, 1, 2.5, and 10. What is the average book width, in centimeters?A) 1 B) 2 C) 3 D) 4 E)5Q2. 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?A)54B)43C)32D) 2 E)3Q3.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?A)3 B)13 C)18 D) 25 E)29 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?A)50.2 B)51.5 C)52.4 D) 53.8 E)55.2Q5.The area of a circle whose circumference is 24π is kπ. What is the value of k?A)6 B)12 C)24 D) 36 E)144Q6. For positive numbers x and y the operation ♠(x, y) is deﬁned as1(,)x y xy♠=-What is ♠(2, ♠(2, 2))?A)23B)1 C)43D)53E)2Q7.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?A)1 B) D)2 E)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?A)9 B)11 C)12 D) 13 E)14Q9.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 ?A)20 B)21 C)22 D) 23 E)24Q11.The length of the interval of solutions of the inequality 23a x b ≤+≤is 10.What is b a -？A)6 B)10 C)15 D) 20 E)30Q12.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?A)0.04 B)0.4π C)0.4 D) 4πE)4Q13.Angelina drove at an average rate of 80 kmh and then stopped 20 minutes for gas. After the stop, she drove at an average rate of 100 kmh. 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? A)880100()2503t t +-= B)80250t = C)100250t = D)90250t = E)880()1002503t t -+=Q14.Triangle ABC has 2AB 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 ?A)60° B)75° C)90° D) 105° E)120°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 amphibians are frogs?A)0 B)1 C)2 D) 3 E)4Q16.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?A)30 B)33 C)35 D) 36 E)37 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?A)7 B)8 C)10 D) 12 E)15Q18.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?A)4772B)3756C)23D)4972E)3956Q19.Equiangular hexagon ABCDEF has side lengths AB = CD = EF = 1 and BC = DE = F A = r. The area of △ACE is 70% of the area of the hexagon. What is the sum of all possible values of r?A)3B)103C)4 D)174E) 6Q20.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?A)4+B)2+C)2+D)E)Q21.The polynomial 322010-+-has three positive integer roots. What isx ax bxthe smallest possible value of a?A)78 B)88 C)98 D)108 E)118Q22.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?A) 28 B) 56 C) 70 D) 84 E) 140Q23.Each of 2010 boxes in a line contains a single red marble, and for 1 ≤k≤2010, the box in the k th 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 which1()P n<?2010A) 45 B) 63 C) 64 D) 201 E) 1005Q24.The number obtained from the last two nonzero digits of 90! is equal to n. What is n?A) 12 B) 32 C) 48 D) 52 E) 68Q25.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:Let N be the smallest number for which Jim’s sequence has 8 numbers. What is the units digit of N?A) 1 B) 3 C) 5 D) 7 E) 9AMC10A 2010中文解析Q1. Mary’s top book shelf holds ﬁve books with the follo wing widths, in centimeters:6,12, 1, 2.5, and 10. What is the average book width, in centimeters?A) 1 B) 2 C) 3 D) 4 E)5翻译：Mary最上面的书架上有五本书，宽度分别为: 6厘米、1厘米、2.5厘米和10厘米。

-年amc 10a和b竞赛真题及答案(英文版)

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2010-2015年AMC 10A和B竞赛真题及答案（英文版）
你好，目前，只分享了2010-2013年AMC 10A和B竞赛真题及答案（英文版），2014-2015年的暂时还没有上传，等采纳后再私信我吧！需要的话，我可以将Word文档中的原图文件一同发给你，求采纳，毕竟我花了差不多一个下午才整理完，谢谢！(@_@)
2010年AMC 10Aห้องสมุดไป่ตู้赛真题及答案（英文版）
2010年AMC 10B竞赛真题及答案（英文版）
2011年AMC 10A竞赛真题及答案（英文版）
2011年AMC 10B竞赛真题及答案（英文版）
Problem 4
2012年AMC 10A竞赛真题及答案（英文版）
go去wentgone
2012年AMC 10B竞赛真题及答案（英文版）
get得到gotgot
sink下沉sank / sunksunk / sunken
give给gavegiven
hit打hithit
不规则动词表
2013年AMC 10A竞赛真题及答案（英文版）
mistake误认mistookmistaken
grow成长grewgrown
think思考thoughtthought

高中一年级美国数学竞赛试题(简称AMC10)2012年B卷39617197

Problem 1Each third-grade classroom at Pearl Creek Elementary has 18 students and 2 pet rabbits. How many more students than rabbits are there in all 4 of the third-grade classrooms?SolutionProblem 2A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle?SolutionProblem 3The point in the xy-plane with coordinates (1000, 2012) is reflected across the line y=2000. What are the coordinates of the reflected point?SolutionProblem 4When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many marbles will be left over?SolutionProblem 5Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is 10%. She leaves a 15% tip on the price of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of 27.50 dollars for dinner. What is the cost of her dinner without tax or tip in dollars?SolutionProblem 6In order to estimate the value of x-y where x and y are real numbers with x > y > 0, Xiaoli rounded x up by a small amount, rounded y down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct?A) Her estimate is larger than x-y B) Her estimate is smaller than x-y C) Her estimate equals x-y D) Her estimate equals y - x E) Her estimate is 0SolutionProblem 7For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide?SolutionProblem 8What is the sum of all integer solutions to ?SolutionProblem 9Two integers have a sum of 26. When two more integers are added to the first two integers the sum is 41. Finally when two more integers are added to the sum of the previous four integers the sum is 57. What is the minimum number of even integers among the 6 integers?SolutionProblem 10How many ordered pairs of positive integers (M,N) satisfy the equation =SolutionProblem 11A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible?SolutionProblem 12Point B is due east of point A. Point C is due north of point B. The distance betweenpoints A and C is , and = 45 degrees. Point D is 20 meters due Northof point C. The distance AD is between which two integers?SolutionProblem 13It takes Clea 60 seconds to walk down an escalator when it is not operating, and only 24 seconds to walk down the escalator when it is operating. How many seconds does it take Clea to ride down the operating escalator when she just stands on it?SolutionProblem 14Two equilateral triangles are contained in square whose side length is . The bases of these triangles are the opposite side of the square, and their intersection is a rhombus. What is the area of the rhombus?SolutionProblem 15In a round-robin tournament with 6 teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that could be tied for the most wins at the end on the tournament?SolutionProblem 16Three circles with radius 2 are mutually tangent. What is the total area of the circles and the region bounded by them, as shown in the figure?SolutionProblem 17Jesse cuts a circular paper disk of radius 12 along two radii to form two sectors, the smaller having a central angle of 120 degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger?SolutionProblem 18Suppose that one of every 500 people in a certain population has a particular disease, which displays no symptoms. A blood test is available for screening for this disease. For a person who has this disease, the test always turns out positive. For aperson who does not have the disease, however, there is a false positive rate--inother words, for such people, of the time the test will turn out negative, butof the time the test will turn out positive and will incorrectly indicate that the person has the disease. Let be the probability that a person who is chosen at random from this population and gets a positive test result actually has the disease. Which of the following is closest to ?SolutionProblem 19In rectangle , , , and is the midpoint of . Segmentis extended 2 units beyond to point , and is the intersection of and. What is the area of ?SolutionProblem 20Bernardo and Silvia play the following game. An integer between 0 and 999, inclusive, is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds 50 to it and passes the result to Bernardo. The winner is the last person who produces a number less than 1000. Let be the smallest initial number that resultsin a win for Bernardo. What is the sum of the digits of ?SolutionProblem 21Four distinct points are arranged on a plane so that the segments connecting them have lengths , , , , , and . What is the ratio of to ?SolutionProblem 22Let be a list of the first 10 positive integers such that for eacheither or or both appear somewhere before in the list. How many such lists are there?SolutionProblem 23A solid tetrahedron is sliced off a wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object?SolutionProblem 23A solid tetrahedron is sliced off a wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object?SolutionProblem 24Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those girls but disliked by the third. In how many different ways is this possible?SolutionProblem 25A bug travels from to along the segments in the hexagonal lattice picturedbelow. The segments marked with an arrow can be traveled only in the direction of the arrow, and the bug never travels the same segment more than once. How many different paths are there?Retrieved from"/Wiki/index.php/2012_AMC_10B_Problems"。

amc10公式范文

amc10公式范文AMC 10 (也被称为美国数学竞赛10级) 是由美国数学学会 (the Mathematical Association of America, MAA) 主办的一项年度数学竞赛，面向的是9-10年级的学生。

本文将介绍一些与AMC 10相关的公式和概念。

一、代数公式：1.因式分解公式：a^2-b^2=(a+b)(a-b)这个公式用于将一个二次差分形式的代数式因式分解为两个因数的乘积。

2. 二次方程解的公式：对于二次方程ax^2 + bx + c = 0，其解为：x = (-b ± √(b^2 - 4ac)) / (2a)这个公式被称为二次方程求根公式，用于求解二次方程的解。

3. 一元二次不等式解的公式：对于一元二次不等式ax^2 + bx + c > 0，其解的范围取决于判别式Δ = b^2 - 4ac 的正负性：当Δ>0时，解的范围为x<(-b-√Δ)/(2a)或x>(-b+√Δ)/(2a)当Δ=0时，解的范围为x=-b/(2a)当Δ<0时，无解。

二、几何公式：1.长方形面积公式：面积=长×宽2.正方形面积公式：面积=边长×边长3.三角形面积公式：面积=底×高/24.直角三角形勾股定理：a^2+b^2=c^2根据直角三角形的两条直角边边长可用该公式求得斜边的长度。

5.直线斜率公式：设点A(x1,y1)和点B(x2,y2)在直线上，直线的斜率m=(y2-y1)/(x2-x1)三、组合数学公式：1.阶乘公式：n!=n×(n-1)×(n-2)×...×2×1阶乘表示将一个正整数n连乘至1的结果。

2.组合公式：C(n,r)=n!/(r!×(n-r)!)组合公式计算了从n个元素中选取r个不同元素的不计次序的所有方法数。

3.排列公式：P(n,r)=n!/(n-r)!排列公式计算了从n个元素中选取r个不同元素的次序不同的所有方法数。

AMC10-2010

2010A 1Mary’s top book shelf holds ﬁve books with the following widths,in centimeters:6,12,1,2.5,and 10.What is the average book width,in centimeters?(A)1(B)2(C)3(D)4(E)52Four 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?[asy]unitsize(8mm);defaultpen(linewidth(.8pt));draw(scale(4)*unitsquare);draw((0,3)–(4,3));draw((1,3)–(1,4));draw((2,3)–(2,4));draw((3,3)–(3,4));[/asy](A)54(B)43(C)32(D)2(E)33Tyrone had 97marbles and Eric had 11marbles.Tyrone then gave some of his marbles ot Eric so that Tyrone ended with twice as many marbles as Eric.How many marbles did Tyrone give to Eric?(A)3(B)13(C)18(D)25(E)294A book that is to be recorded onto compact discs takes 412minutes to read aloud.Each disc can hold up to 56minutes 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?(A)50.2(B)51.5(C)52.4(D)53.8(E)55.25The area of a circle whose circumference is 24πis kπ.What is the value of k ?(A)6(B)12(C)24(D)36(E)1446For positive numbers x and y the operation ♠(x,y )is deﬁned as ♠(x,y )=x −1y What is ♠(2,♠(2,2))?(A)23(B)1(C)43(D)53(E)27Crystal 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?(A)1(B)√2(C)√3(D)2(E)2√2This ﬁle was downloaded from the AoPS −MathLinks Math Olympiad Resources Page Page 1http://www.mathlinks.ro/20108Tony works 2hours a day and is paid $0.50per hour for each full year of his age.During a six month period Tony worked 50days and earned $630.How old was Tony at the end of the six month period?(A)9(B)11(C)12(D)13(E)149A palindrome ,such as 83438,is a number that remains the same when its digits are reversed.The numbers x and x +32are three-digit and four-digit palindromes,respectively.What is the sum of the digits of x?(A)20(B)21(C)22(D)23(E)2410Marvin had a birthday on Tuesday,May 27in the leap year 2008.In what year will hisbirthday next fall on a Saturday?(A)2011(B)2012(C)2013(D)2015(E)201711The length of the interval of solutions of the inequality a ≤2x +3≤b is 10.What is b −a ?(A)6(B)10(C)15(D)20(E)3012Logan is constructing a scaled model of his town.The city’s water tower stands 40metershigh,and the top portion is a sphere that holes 100,000liters of water.Logan’s miniature water tower holds 0.1liters.How tall,in meters,should Logan make his tower?(A)0.04(B)0.4π(C)0.4(D)4π(E)413Angelina drove at an average rate of 80kph and then stopped 20minutes for gas.After thestop,she drove at an average rate of 100kph.Altogether she drove 250km in a total trip time of 3hours including the stop.Which equation could be used to solve for the time t in hours that she drove before her stop?(A)80t +100(8/3−t )=250(B)80t =250(C)100t =250(D)90t =250(E)80(8/3−t )+100t =25014Triangle 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 CF E is equilateral.What is ∠ACB ?(A)60◦(B)75◦(C)90◦(D)105◦(E)120◦15In a magical swamp there are two species of talking amphibians:toads,whose statements arealways true,and frogs,whose statements are always false.Four amphibians,Brian,Chris,LeRoy,and Mike live together in the 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.”2010How many of these amphibians are frogs?(A)0(B)1(C)2(D)3(E)416Nondegenerate ABC has integer side lengths,BD is an angle bisector,AD =3,andDC =8.What is the smallest possible value of the perimeter?(A)30(B)33(C)35(D)36(E)3717A solid cube has side length 3inches.A 2-inch by 2-inch square hole is cut into the center ofeach 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?(A)7(B)8(C)10(D)12(E)1518Bernardo randomly picks 3distinct numbers from the set {1,2,3,4,5,6,7,8,9}and arrangesthem in descending order to form a 3-digit number.Silvia randomly picks 3distinct 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?(A)4772(B)3756(C)23(D)4972(E)395619Equiangular hexagon ABCDEF has side lengths AB =CD =EF =1and BC =DE =F A =r .The area of ACE is 70(A)4√33(B)103(C)4(D)174(E)620A ﬂy trapped inside a cubical box with side length 1meter decides to relieve its boredom byvisiting 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?(A)4+4√(B)2+4√+2√(C)2+3√+3√(D)4√+4√(E)3√2+5√321The polynomial x 3−ax 2+bx −2010has three positive integer zeros.What is the smallestpossible value of a ?(A)78(B)88(C)98(D)108(E)11822Eight points are chosen on a circle,and chords are drawn connecting every pair of points.Nothree chords intersect in a single point inside the circle.How many triangles with all three vertices in the interior of the circle are created?(A)28(B)56(C)70(D)84(E)140201023Each of 2010boxes in a line contains a single red marble,and for 1≤k ≤2010,the box in thekth 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 )<12010?(A)45(B)63(C)64(D)201(E)100524The number obtained from the last two nonzero digits of 90!is equal to n .What is n ?(A)12(B)32(C)48(D)52(E)6825Jim starts with a positive integer n and creates a sequence of numbers.Each successivenumber 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 5numbers:5555−72=66−22=22−12=11−12=0Let N be the smallest number for which Jim’s sequence has 8numbers.What is the units digit of N ?(A)1(B)3(C)5(D)7(E)92010B 1What is 100(100−3)−(100·100−3)?(A)−20,000(B)−10,000(C)−297(D)−6(E)02Makayla attended two meetings during her 9-hour work day.The ﬁrst meeting took 45minutes and the second meeting took twice as long.What percent of her work day was spent attending meetings?(A)15(B)20(C)25(D)30(E)353A drawer contains red,green,blue,and white socks with at least 2of each color.What is the minimum number of socks that must be pulled from the drawer to guarantee a matching pair?(A)3(B)4(C)5(D)8(E)94For a real number x ,deﬁne ♥(x )to be the average of x and x 2.What is ♥(1)+♥(2)+♥(3)?(A)3(B)6(C)10(D)12(E)20[Thanks PowerOfPi,that’s exactly how the heart looks like.]5A month with 31days 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?(A)2(B)3(C)4(D)5(E)66A circle is centered at O ,AB is a diameter and C is a point on the circle with ∠COB =50◦.What is the degree measure of ∠CAB ?(A)20(B)25(C)45(D)50(E)657A triangle has side lengths 10,10,and 12.A rectangle has width 4and area equal to the area of the rectangle.What is the perimeter of this rectangle?(A)16(B)24(C)28(D)32(E)368A 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 of x are possible?(A)1(B)2(C)3(D)4(E)520109Lucky 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 rry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincedence.The numbers Larry substituted for a ,b ,c ,and d were 1,2,3,and 4,respectively.What number did Larry substitute for e ?(A)−5(B)−3(C)0(D)3(E)510Shelby drives her scooter at a speed of 30miles per hour if it is not raining,and 20milesper 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 16miles in 40minutes.How many minutes did she drive in the rain?(A)18(B)21(C)24(D)27(E)3011A shopper plans to purchase an item that has a listed price greater than $100and can useany one of the three coupns.Coupon A gives 15%oﬀthe listed price,Coupon B gives $30the listed price,and Coupon C gives 25%oﬀthe 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 ?(A)50(B)60(C)75(D)80(E)10012At the beginning of the school year,50%of all students in Mr.Well’s 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 ?(A)0(B)20(C)40(D)60(E)8013What is the sum of all the solutions of x =|2x −|60−2x ||?(A)32(B)60(C)92(D)120(E)12414The average of the numbers 1,2,3,...,98,99,and x is 100x .What is x ?(A)49101(B)50101(C)12(D)51101(E)509915On a 50-question multiple choice math contest,students receive 4points for a correct answer,0points for an answer left blank,and -1point 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?(A)25(B)27(C)29(D)31(E)33201016A square of side length 1and a circle of radius √3/3share the same center.What is the areainside the circle,but outside the square?(A)π3−1(B)2π9−√33(C)π18(D)14(E)2π/917Every high school in the city of Euclid sent a team of 3students to a math contest.Eachparticipant 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 37th and 64th,respectively.How many schools are in the city?(A)22(B)23(C)24(D)25(E)2618Positive integers a,b,and c are randomly and independently selected with replacement fromthe set {1,2,3,...,2010}.What is the probability that abc +ab +a is divisible by 3?(A)13(B)2981(C)3181(D)1127(E)132719A circle with center O has area 156π.Triangle ABC is equilateral,BC is a chord on the circle,OA =4√3,and point O is outside ABC .What is the side length of ABC ?(A)2√(B)6(C)4√(D)12(E)1820Two circles lie outside regular hexagon ABCDEF .The ﬁrst is tangent to AB ,and the secondis tangent to DE .Both are tangent to lines BC and F A .What is the ratio of the area of the second circle to that of the ﬁrst circle?(A)18(B)27(C)36(D)81(E)10821A palindrome between 1000and 10,000is chosen at random.What is the probability that itis divisible by 7?(A)110(B)19(C)17(D)16(E)1522Seven distinct pieces of candy are to be distributed among three bags.The red bag and theblue bag must each receive at least one piece of candy;the white bag may remain empty.How many arrangements are possible?(A)1930(B)1931(C)1932(D)1933(E)193423The entries in a 3×3array include all the digits from 1through 9,arranged so that theentries in every row and column are in increasing order.How many such arrays are there?(A)18(B)24(C)36(D)42(E)60201024A high school basketball game between the Raiders and Wildcats was tied at the end of theﬁrst 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 100points.What was the total number of points scored by the two teams in the ﬁrst half?(A)30(B)31(C)32(D)33(E)3425Let a >0,and let P (x )be a polynomial with integer coeﬃcients such thatP (1)=P (3)=P (5)=P (7)=a ,andP (2)=P (4)=P (6)=P (8)=−a .What is the smallest possible value of a ?(A)105(B)315(C)945(D)7!(E)8!。

2012年-AMC10-A卷

2012 AMC 10A ProblemsInstructions1.This is a 25-question, multiple choice test. Each question is followed by answers markedA, B, C, D and E. Only one of these is correct.2.You will receive 6 points for each correct answer, 1.5 points for each problem leftunanswered, and 0 points for each incorrect answer.3.No aids are permitted other than scratch paper, graph paper, ruler, compass, protractorand erasers (No problems on the test will require the use of a calculator).4.Figures are not necessarily drawn to scale.5.You will have 75 minutes working time to complete the test.1.Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30seconds. Working together, how many cupcakes can they frost in 5 minutes?2.A square with side length 8 is cut in half, creating two congruent rectangles. What are thedimensions of one of these rectangles?3.A bug crawls along a number line, starting at -2. It crawls to -6, then turns around and crawlsto 5. How many units does the bug crawl altogether?4.Let and . What is the smallest possible degree measure forangle CBD?st year 100 adult cats, half of whom were female, were brought into the Smallville AnimalShelter. 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?6.The product of two positive numbers is 9. The reciprocal of one of these numbers is 4 timesthe reciprocal of the other number. What is the sum of the two numbers?7.In a bag of marbles, of the marbles are blue and the rest are red. If the number of redmarbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red?8.The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middlenumber?9.A pair of six-sided 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 of each 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?10.Mary divides a circle into 12 sectors. The central angles of these sectors, measured indegrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?11.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?12.A year is a leap year if and only if the year number is divisible by 400 (such as 2000) or isdivisible by 4 but not 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?13.An iterative average of the numbers 1, 2, 3, 4, and 5 is computed the following way. Arrangethe 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?14.Chubby makes nonstandard checkerboards that have squares on each side. Thecheckerboards 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?15.Three unit squares and two line segments connecting two pairs of vertices are shown. Whatis the area of ?16.Three runners start running simultaneously from the same point on a 500-meter circulartrack. 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?17.Let and be relatively prime integers with and = . What is ?18.The closed curve in the figure is made up of 9 congruent circular arcs each of length ,where each of the centers of the corresponding circles is among the vertices of a regularhexagon of side 2. What is the area enclosed by the curve?19.Paula the painter and her two helpers each paint at constant, but different, rates. Theyalways 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 P.M. How long, in minutes, was each day's lunch break?20.A x square is partitioned into unit squares. Each unit square is painted either white orblack with each color being equally likely, chosen independently and at random. The square is then rotated clockwise about its center, and every white square in a position formerlyoccupied by a black square is painted black. The colors of all other squares are leftunchanged. What is the probability the grid is now entirely black?21.Let points = , = , = , and = . Points , , , andare midpoints of line segments and respectively. What is the area of?22.The sum of the first positive odd integers is 212 more than the sum of the first positiveeven integers. What is the sum of all possible values of ?23.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?24.Let , , and be positive integers with such that and. What is ?25.Real numbers , , and are chosen independently and at random from the intervalfor some positive integer . The probability that no two of , , and are within 1 unitof each other is greater than . What is the smallest possible value of ?1-5: DEECB 6-10: DCDDC 11-15: DACBB 16-20: CCEDA21-25: CABED。

AMC10美国数学竞赛A卷附中文翻译和答案

AMC10 美国数学竞赛 A 卷附中文翻译和答案简介AMC10 是美国的一项全国性数学竞赛，主要面向高中学生。

此文档提供了 AMC10 美国数学竞赛 A 卷的问题、答案及中文翻译，帮助考生更好地理解和准备竞赛。

问题和答案问题 1让我们从一个整数开始，每一步都按照以下规则进行操作：如果当前的数是偶数，将它除以 2；如果当前的数是奇数，将它乘以 3 并加 1。

通过这样的操作，我们可以生成一个数列，例如，从 9 开始的数列如下所示：9，28，14，7，22，11，34，...。

显然，这个数列最终会包含两个连续的 1。

以下哪个数开始操作后会生成包含两个连续的 1？$\\textbf{(A)}\\ 111 \\qquad \\textbf{(B)}\\ 120 \\qquad \\textbf{(C)}\\ 125 \\qquad \\textbf{(D)}\\ 130 \\qquad\\textbf{(E)}\\ 139$$\\textbf{(D)}\\ 130$问题 2如果 $2^x \\times 5^y = 5000000$，那么x+x的值是多少？$\\textbf{(A)}\\ 6 \\qquad \\textbf{(B)}\\ 7 \\qquad\\textbf{(C)}\\ 10 \\qquad \\textbf{(D)}\\ 12 \\qquad\\textbf{(E)}\\ 15$答案 2$\\textbf{(C)}\\ 10$问题 3数轴上三个数x、x、x的平均值是 6。

给定x−x=8，x−x=12，那么x的值是多少？$\\textbf{(A)}\\ -10 \\qquad \\textbf{(B)}\\ -6 \\qquad \\textbf{(C)}\\ 0 \\qquad \\textbf{(D)}\\ 6 \\qquad\\textbf{(E)}\\ 10$答案 3$\\textbf{(B)}\\ -6$某菜市场的一个南瓜重 100 磅，其中 99% 的重量是水分。

amc10 立体几何题目

amc10 立体几何题目【最新版】目录1.AMC10 立体几何题目概述2.立体几何的基本概念和解题方法3.AMC10 立体几何题目的解题技巧和策略4.总结和建议正文【AMC10 立体几何题目概述】AMC10（美国数学竞赛 10 年级）是针对 10 年级学生的一项重要数学竞赛，其涉及的立体几何题目旨在考查学生的空间想象能力、逻辑思维能力以及数学应用能力。

立体几何题目通常以线线、线面、面面等关系为载体，要求学生运用相关定理和公式进行分析和求解。

【立体几何的基本概念和解题方法】立体几何是研究空间中点、线、面及其相关性质的数学分支。

在解决立体几何问题时，通常需要运用以下基本概念和方法：1.点、线、面的基本性质：了解点、线、面的概念以及它们之间的关系，如共线、共面等。

2.空间直线与平面的位置关系：主要包括直线与平面相交、平行和重合三种情况。

3.空间几何中的距离和角：学会计算空间中两点之间的距离、直线与平面之间的夹角等。

4.空间几何中的投影：了解正射投影、轴投影等投影方式，学会利用投影解决问题。

5.空间几何中的变换：学会运用平移、旋转等变换方法解决空间几何问题。

【AMC10 立体几何题目的解题技巧和策略】在解答 AMC10 立体几何题目时，可以采用以下技巧和策略：1.仔细阅读题目，理解题意，画出题目中涉及的图形，有助于建立直观的图形信息。

2.善于运用空间想象能力，将问题转化为平面几何问题，降低解题难度。

3.运用相关定理和公式，如线线平行、线面垂直等，进行分析和求解。

4.遇到复杂问题时，可以尝试将问题拆解为多个简单的子问题，逐步解决。

5.做好归纳总结，积累解题经验，提高解题速度和准确率。

【总结和建议】AMC10 立体几何题目对学生的空间想象能力、逻辑思维能力以及数学应用能力都有较高要求。

要解答好这类题目，需要熟练掌握立体几何的基本概念和解题方法，并灵活运用各种解题技巧和策略。

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2010AMC 10 答案： 1. (D) 2. (B) 3. (D) 4. (B) 5. (E) 6. (C) 7. (C) 8. (D) 9. (E) 10. (E) 11. (D) 12. (C) 13. (A) 14. (C) 15. (D) 16. (B) 17. (A) 18. (B) 19. (E) 20. (D) 21. (A) 22. (A) 23. (A) 24. (A) 25. (B)
25. 小甄由正整数 n 开始，造一个数列。每一项是由前一项减去最大可能的平方数，此平方数小于或等于前一项，直到等于 0 停止。例如，小甄从 n=55 开始，则此数列有 5 项如下： 55 2 557 =6 622=2 212=1 112=0 设小甄从 N 开始所造的数列有 8 项。试问最小的 N 其个位数字为何？ (A) 1 (B) 3 (C) 5 (D) 7 (E) 9 。
2010 年第 11 届美国 AMC10 (2010 年 2 月
1. 小华书架上放了五本书，各书的厚度分别为 6，厚度是多少公分？ (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 。 2. 四个一样大小的正方形与一个长方形拼成一个大正方形，如右图所示。试问长方形中长是宽的多少倍？ (A)
1 的最小 n 是多少？ (A) 45 (B) 63 (C) 64 (D) 201 (E) 1005 。 2010
24. 设 90！乘开后最后不为 0 的两个数字所形成的二位数为 n。试问 n 为下列哪一个数？ (注：k！=k(k1)(k2)…21) (A) 12 (B) 32 (C) 48 (D) 52 (E) 68 。
2 (C)
3 (D) 2 (E) 2 2 。
8. 东尼一天工作 2 小时，每小时的工资是美金 0.5 元乘以他工作时的年龄，他在六个月的期间一共工作了 50 天，总共赚得美金 630 元。试问他在六个月工作结束时的年龄为几岁？ (A) 9 (B) 11 (C) 12 (D) 13 (E) 14 。 9. 如果一个数由左至右念与由右至左念都相同，我们就称它为回文数，例如 83438 就是一个回文数。已知 x 与 x32 分别为三位数与四位数的回文数。试问 x 各位数字的和为多少？ (A) 20 (B) 21 (C) 22 (D) 23 (E) 24 。 10. 小华的生日是 5 月 27 日，在闰年 2008 年的生日是星期二。试问未来他的生日在星期六的最近一年是哪一年？ (A) 2011 (B) 2012 (C) 2013 (D) 2015 (E) 2017 。 11. 设不等式 a2x3b 解的范围之长度为 10。试问 ba=？ (A) 6 (B) 10 (C) 15 (D) 20 (E) 30 。 12. 小罗要按比例缩小尺寸制作他所住城镇的模型。他所住城镇的水塔高 40 公尺，顶端是一个球形的储水槽，可储存 100,000 公升的水。若小罗模型水塔的储水槽可储存 0.1 公升的水，则小罗模型水塔的高应是多少公尺？ (A) 0.04 (B)
1. 美国某通讯公司手机每个月的基本费为 20 美元，每传送一则简讯加收 5 美分(1 美元=100 美分)。若通话超过 30 小时，超过的时间每分钟加收 10 美分。已知小美一月份共传送了 100 则简讯及通话 30.5 小时，则她需要付多少美元？ (A) 24.00 (B) 24.50 (C) 25.50 (D) 28.00 (E) 30.00 。 2. 小瓶装有 35 毫升的洗发精，大瓶可装 500 毫升的洗发精。小华至少要买多少瓶小瓶的洗发精，才能装满一瓶大瓶的洗发精？ (A) 11 (B) 12 (C) 13 (D) 14 (E) 15 。 3. 若以<a, b>表示 a, b 两数的平均数，以【a, b, c】表示 a, b, c 三数的平均数，则【【1, 1, 0】,<0, 1>, 0】之值为何？ (A)
(A) 60 (B) 75 (C) 90 (D) 105 (E) 120。 15. 在一魔法沼泽中有两种会说话的两栖动物：一种是蟾蛛，它们一直都说实话，另一种是青蛙，它们一直都说谎话。四只两栖动物布莱恩、克芮斯、赖瑞及麦克住在此沼泽中生活在一起，它们有下列的对话。布莱恩说：麦克与我是不同的种类。克芮斯说：赖瑞是青蛙。赖瑞说：克芮斯是青蛙。麦克说：我们四个之中，至少有两个是蟾蛛。试问这四只两栖动物中有几只青蛙？ (A) 0 (B) 1 (C) 2 (D) 3 (E) 4 。 16. 在△ABC 中各边的边长为整数，D 为 AC 边上的一点， BD 为角平分线， AD =3 且 DC =8。试问此三角形的周长最小可能是多少？ (A) 30 (B) 33 (C) 35 (D) 36 (E) 37 。 17. 将边长为 3 公寸的正立方体，以每个面的中心为中心挖一个 2 公寸2 公寸的正方形洞穿通到对面，且这些正方形洞的各边均平行于大立方体的边。试问剩下立体的体积是多少立方公寸？ (A) 7 (B) 8 (C) 10 (D) 12 (E) 15 。 18. 小南从{1，2，3，4，5，6，7，8，9}中随意取出 3 个相异数，将它们以由大而小的方式排成一个三位数，希薇从{1，2，3，4，5，6，7，8}中随意取出 3 个相异数，也将它们以由大而小的方式排成一个三位数。试问小南所排成的数比希薇所排成的数大之机率为多少？ (A)
(A) 尺？ (A) 44 2 (B) 24 2 2 3 (C) 23 2 3 3 (D) 4 2 4 3 (E) 3 2 5 3 。 21. 方程式 x3ax2bx2010=0 有三个正整数解。试问 a 最小可能之值为何？ (A) 78 (B) 88 (C) 98 (D) 108 (E) 118 。 22. 一圆周上有八个点，每两点连接一条弦，且任意三条弦在圆内部都没有共同的交点。试问以这些弦在圆内部的交点为顶点且弦为边所形成的三角形总共有多少个？ (A) 28 (B) 56 (C) 70 (D) 84 (E) 140 。 23. 排成一列的 2010 个盒子，每个盒子中都刚好有一颗红弹珠，且对于 1k2010，第 k 个盒子中还有 k 颗白弹珠。小伊从第一个盒子开始，依序在每个盒子中任意取出一颗弹珠。当她取到第一颗红弹珠时就停止，并令 P(n)表示小伊恰好取出 n 颗弹珠时停止的机率。试问满足 P(n)<
2 47 37 49 39 (B) (C) (D) (E) 。 3 72 56 72 56
19. 在六边形 ABCDEF 中，各内角相等， AB = CD = EF =1 且 BC = DE = FA =r。设△ACE 的面积是六边形面积的 70%，试问所有可能的 r 之和为多少？
4 3 10 17 (B) (C) 4 (D) (E) 6 。 3 3 4 20. 一只苍蝇在一个边长 1 公尺的正立方盒子内，它为了排遣无聊决定要到盒子的每一个顶点。假设它从某一个顶点开始，经过每一个顶点恰好一次，最后回到出发的顶点。从一个顶点到另一个顶点的行进方式可以直线飞行或直线爬行。试问它行进的路线长最大可能是多少公
37 88 (C) (D) 13 (E) 14 。 3 7
6. 已知集合 A 中有 20 个元素，集合 B 中有 15 个元素。A∪B 是集合 A 与集合 B 的联集，它是由集合 A 与集合 B 中所有元素所形成的集合，则集合 A∪B 中至少有多少个元素？ (A) 5 (B) 15 (C) 20 (D) 35 (E) 300 。 7. 下列哪一个方程式没有解？ (A) (x7)2=0 (B) | 3x |5=0 (C)
8 8 t)=250 (B) 80t=250 (C) 100t=250 (D) 90t=250 (E) 80( t)100t=250 。 3 3
14. 三角形 ABC 中， AB =2 AC 。点 D 与点 E 分别在 AB 与 BC 边上使得BAE=ACD。设 AE 与
CD 交于 F 点，且△CFE 为正三角形，试问ACB=？
日时间 75 分钟)
1 ，1，2.5，及 10 公分。试问这些书的平均 2
5 4 3 (B) (C) (D) 2 (E) 3 。 4 3 2
3. 小明有 97 颗弹珠，小忠有 11 颗弹珠。小明给小忠一些弹珠后，小明的弹珠恰为小忠弹珠的两倍。试问小明给小忠多少颗弹珠？ (A) 3 (B) 13 (C) 18 (D) 25 (E) 29 。 4. 某本书转录成有声书需要报读 412 分钟，现在要将它储存在光盘片里。已知每片光盘最多可以储存报读时间 56 分钟。假设用最少片的光盘，使得每片光盘所储存报读的时间都相同。试问每片光盘应该储存报读的时间是多少分钟？ (A) 50.2 (B) 51.5 (C) 52.4 (D) 53.8 (E) 55.2 。 5. 若圆周长为 24之圆的面积为 k，则 k 之值为何？ (A) 6 (B) 12 (C) 24 (D) 36 (E) 144 。 6. 对于正数 x 及 y，定义运算(x，y)如下：(x，y)=x
第 12 届 AMC10 (2011 年 2 月 9 日

时间:10:35~11:50 75 分钟)
2011 年第 12 届 AMC10 考试须知
1. 未经监考人员宣布打开测验卷之前，不可先行打开试卷作答。 2. 本测验为选择题共有 25 题，每一题各有 A、B、C、D、E 五个选项，其中祇有一个选项是正确的答案。 3. 请将正确答案用 2B 铅笔在「答案栏」上适当的圆圈内涂黑，请检查所圈选的答案是否正确，并将错误及模糊不清部分擦拭干净。请注意，祇有将答案圈选清楚在答案卡上才得以计分。 4. 计分方式:每一题答对可得 6 分，不作答得 1.5 分，答错得 0 分。 5. 除了考试所准许使用的人、圆规、量角器、橡皮擦、方格纸及计算纸外，请勿携带任何辅助工具(包含计算器)进入考场，考卷上所有的题目均不需使用计算器便可作答。 6. 试卷内的图形可能未依比例绘制。 7. 考试之前，监考人员会指示你填写一些基本资料于答案卡上，待监考人员给予指示后开始作答，你作答时间共 75 分钟。 8. 当你完成作答后，请在答案卡的签名空格内签名。 9. AMC10 考生得 120 分以上，或者成绩名列前 2.5%者，将会受邀参加 2011 年 3 月 19 日星期六所举行的第 29 届 American Invitational Mathematics Examination(AIME)考试，本会预计于 2 月 25 日寄发 AIME 考试邀请通知书。