2012年美国高中数学竞赛试题及详细解答

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2012年美国数学奥林匹克(USAMO)试题及其解答

杏坛孔门
2012 年美国数学奥林匹克试题（USAMO）及其解答
田开斌解答
1、求所有整数 n≥3，使得对于任意 n 个正实数a1 、a2 、a3 、 … … 、an ，如果满足 Max�a1 、a2 、a3 、 … … 、an � ≤ n · Min�a1 、a2 、a3 、 … … 、an �，则其中可以取出三个数，它们能够构成一个锐角三角形的三条边的长度。解：对于任意 n≤12，在序列 1、1、√2、√3、√5、√8、√13、√21、√34、√55、√89、12中取前 n 个数，都有 Max�a1 、a2 、a3 、 … … 、an � ≤ n · Min�a1 、a2 、a3 、 … … 、an �，但任意三个数都不能构成锐角三角形的三条边。所以满足条件的 n≥13。下面我们用反证法证明 n≥13 时，都满足条件。我们给 n 个正实数从小到大排序为a1 ≤ a2 ≤ a3 ≤ ⋯ ≤ an ，若其中任意三个数，都不能构成一个锐角三角形的三条边，则有a1 2 ≤ a2 2 ，ak 2 + ak+1 2 ≤ ak+2 2 ，其中 1≤k≤n-2。于是知an 2 ≥ fn · a1 2 ，即an ≥ �fn · a1 ，其中fn 为斐波那契数列的第 n 项。又当 n≥13 时，根据数学归纳法易知都有�fn ＞n，此时则有an ≥ �fn · a1 ＞na1 ，与 Max�a1 、a2 、a3 、 … … 、an � ≤ n · Min�a1 、a2 、a3 、 … … 、an �矛盾。所以当 n≥13 时，都满足条件。综上所述知，满足条件的 n 为所有不小于 13 的自然数。 2、一个圆被 432 个点等分为 432 段弧，将其中 108 个点染成红色，108 个点染成绿色， 108 个点染成蓝色，108 个点染成黄色。求证：可以在每种颜色的点中各选 3 个点，使得由同色点构成的四个三角形都全等。解：我们记 f(m)表示 m 除以 432 的余数，其中 0≤f(m)≤431。我们从某点开始，按顺时针方向依次给 432 个点排序为 0、1、2、3、……431。设 108 个红点所在位置依次为a1 、a2 、a3 、 … … 、a108 ，108 个绿点依次为b1 、b2 、b3 、 … … 、b108 ， 108 个蓝点依次为c1 、c2 、c3 、 … … 、c108 ，108 个黄点依次为d1 、d2 、d3 、 … … 、d108 。记Xi = �f(a1 + i)、f(a2 + i)、f(a3 + i)、 … … 、f(a108 + i)� ∩ �b1 、b2 、b3 、 … … 、b108 � b1、b2、b3、……、b108=108（j=1、2、3、……108），所以

美国数学学会中学生数学竞赛真题和答案解析2012AMC8-solutions

Solutions Pamphlet
American Mathematics Competitions
28th Annual
AMC 8
American Mathematics Contest 8 Tuesday, November 13, 2012
This Solutions Pamphlet gives at least one solution for each problem on this year’s exam and shows that all the problems can be solved using material normally associated with the mathematics curriculum for students in eighth grade or below. These solutions are by no means the only ones possible, nor are they necessarily superior to others the reader may devise. We hope that teachers will share these solutions with their students. However, the publication, reproduction, or communication of the problems or solutions of the AMC 8 during the period when students are eligible to participate seriously jeopardizes the integrity of the results. Dissemination at any time via copier, telephone, email, internet or media of any type is a violation of the competition rules.

2012美国数学竞赛AMC12B

2012 AMC 12B ProblemsProblem 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 3For a science project, Sammy observed a chipmunk and 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 4Suppose that the euro is worth 1.30 dollars. If Diana has 500 dollars and Etienne has 400 euros, by what percent is the value of Etienne's money greater that the value of Diana's money?SolutionProblem 5Two integers have a sum of 26. when two more integers are added to the first two, the sum is 41. Finally, when two more integers are added to the sum of the previous 4 integers, the sum is 57. What is the minimum number of even integers among the 6 integers?SolutionProblem 6In order to estimate the value of where and are real numbers with , Xiaoli rounded up by a small amount, rounded down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct?SolutionProblem 7Small lights are hung on a string 6 inches apart in the order red, red, green, green, green, red, red, green, green, green, and so on continuing this pattern of 2 red lights followed by 3 green lights. How many feet separate the 3rd red light and the 21st red light?Note: 1 foot is equal to 12 inches.SolutionProblem 8A 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 9It takes Clea 60 seconds to walk down an escalator when it is not moving, and 24 seconds when it is moving. How seconds would it take Clea to ride the escalator down when she is not walking?SolutionProblem 10What is the area of the polygon whose vertices are the points of intersection of the curves and ?SolutionProblem 11In the equation below, and are consecutive positive integers, and , , and represent number bases: What is ?SolutionProblem 12How many sequences of zeros and ones of length 20 have all the zeros consecutive, or all the ones consecutive, or both?SolutionProblem 13Two parabolas have equations and , where , , , and are integers, eachchosen independently by rolling a fair six-sided die. What is the probability that the parabolas will have a least one point in common?SolutionProblem 14Bernardo 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 addes 50 to it and passes the result to Bernardo. The winner is the last person who produces a number less than 1000. Let N be the smallest initial number that results in a win for Bernardo. What is the sum of the digits of N?SolutionProblem 15Jesse 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 16Amy, 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 17Square lies in the first quadrant. Points and lie on lines and , respectively. What is the sum of the coordinates of the center of the square ?SolutionProblem 18Let be a list of the first 10 positive integers such that for each either oror both appear somewhere before in the list. How many such lists are there?SolutionProblem 19A unit cube has vertices and . Vertices , , and are adjacent to , and forvertices and are opposite to each other. A regular octahedron has one vertex in each of the segments , , , , , and . What is the octahedron's side length?SolutionProblem 20A trapezoid has side lengths 3, 5, 7, and 11. The sums of all the possible areas of the trapezoid can be written in the form of , where , , and are rational numbers and and are positive integers not divisible by the square of any prime. What is the greatest integer less than or equal to ?SolutionProblem 21Square is inscribed in equiangular hexagon with on , on , and on . Suppose that , and . What is the side-length of the square?SolutionProblem 22A bug travels from to along the segments in the hexagonal lattice pictured below. 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?SolutionProblem 23Consider all polynomials of a complex variable, , where and are integers, , and the polynomial has a zero with What is the sum of all values over all the polynomials with these properties?SolutionProblem 24\item Define the function on the positive integers by setting and if is the prime factorization of , then For every , let. For how many in the range is the sequenceunbounded?Note: A sequence of positive numbers is unbounded if for every integer , there is a member of the sequence greater than .Problem 25\item Let . Let be the set of all right triangles whose vertices are in . For every right triangle with vertices , , and in counter-clockwise order and right angle at , let . What is。

美国高中生数学试题及答案

美国高中生数学试题及答案一、选择题（每题2分，共20分）1. 下列哪个数是无理数？A. 3.14159B. πC. 0.33333...D. √22. 如果一个函数f(x) = 2x^2 + 3x - 5，那么f(-2)的值是多少？A. -1B. 1C. 3D. 53. 一个直角三角形的两条直角边分别为3和4，斜边的长度是多少？A. 5B. 6C. 7D. 84. 以下哪个方程没有实数解？A. x^2 + 4x + 4 = 0B. x^2 - 4x + 4 = 0C. x^2 + 4x - 5 = 0D. x^2 - 9 = 05. 一个圆的半径是5，那么它的面积是多少？A. 25πC. 75πD. 100π6. 以下哪个是二次方程的根？A. x = 2B. x = -2C. x = 3D. x = -37. 如果一个数列是等差数列，且前三项为2, 5, 8，那么第10项是多少？A. 23B. 24C. 25D. 268. 一个函数g(x) = 3x - 2，当x = 4时，g(x)的值是多少？A. 10B. 12C. 14D. 169. 以下哪个是线性方程的解？A. x = 0B. x = 1C. x = 2D. x = 310. 一个正方体的体积是27立方单位，它的边长是多少？A. 3C. 9D. 12二、填空题（每题3分，共15分）11. 一个圆的周长是2πr，其中r是______。

12. 一个二次方程ax^2 + bx + c = 0的判别式是______。

13. 一个数的平方根是4，那么这个数是______。

14. 如果一个数列是等比数列，且首项为2，公比为3，那么第5项是______。

15. 一个函数h(x) = kx + b，当k不等于0时，这个函数是______函数。

三、解答题（每题5分，共25分）16. 解方程：3x + 5 = 14。

17. 证明：如果一个三角形的两边长分别为a和b，且a + b > c，那么这个三角形是存在的。

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。

美国imo数学竞赛试题及答案

美国imo数学竞赛试题及答案问题1：代数问题设$ a, b, c $ 是正实数，满足 $ a + b + c = 1 $。

证明：\[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \geq 9 \]问题2：几何问题在三角形 $ ABC $ 中，点 $ D $ 和 $ E $ 分别是边 $ BC $ 和 $ AC $ 上的点，使得 $ AD $ 平行于 $ BE $。

如果$ \angle A = 60^\circ $，证明 $ \angle ADB = \angle BEC $。

问题3：数论问题给定一个正整数 $ n $，证明对于所有 $ n $ 的倍数 $ k $，$ k $ 除以 $ n $ 的余数等于 $ k $ 除以 $ n+1 $ 的余数。

问题4：组合问题有 $ 2n $ 个不同的球和 $ n $ 个相同的盒子。

证明至少有一个盒子包含至少 $ 3 $ 个球。

问题5：不等式问题证明对于所有正实数 $ x $ 和 $ y $，以下不等式成立：\[ \sqrt{x^2 + y^2} + \sqrt{2xy} \geq x + y \]答案问题1：代数问题由柯西不等式，我们知道：\[ (a + b + c)\left(\frac{1}{a} + \frac{1}{b} +\frac{1}{c}\right) \geq (1 + 1 + 1)^2 \]因为 $ a + b + c = 1 $，所以：\[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \geq 9 \]问题2：几何问题由于 $ AD $ 平行于 $ BE $，根据相似三角形的性质，我们有$ \triangle ABD \sim \triangle CBE $。

2012年美国高中生数学建模竞赛特等奖论文

题目：How Much Gas Should I Buy This Week?题目来源：2012年第十五届美国高中生数学建模竞赛（HiMCM）B题获奖等级：特等奖，并授予INFORMS奖论文作者：深圳中学2014届毕业生李依琛、王喆沛、林桂兴、李卓尔指导老师：深圳中学张文涛AbstractGasoline is the bleed that surges incessantly within the muscular ground of city; gasoline is the feast that lures the appetite of drivers. “To fill or not fill?” That is the question flustering thousands of car owners. This paper will guide you to predict the gasoline prices of the coming week with the currently available data with respect to swift changes of oil prices. Do you hold any interest in what pattern of filling up the gas tank can lead to a lower cost in total?By applying the Time series analysis method, this paper infers the price in the imminent week. Furthermore, we innovatively utilize the average prices of the continuous two weeks to predict the next two week’s average price; similarly, employ the four-week-long average prices to forecast the average price of four weeks later. By adopting the data obtained from 2011and the comparison in different aspects, we can obtain the gas price prediction model ：G t+1=0.0398+1.6002g t+−0.7842g t−1+0.1207g t−2+ 0.4147g t−0.5107g t−1+0.1703g t−2+ε .This predicted result of 2012 according to this model is fairly ideal. Based on the prediction model,We also establish the model for how to fill gasoline. With these models, we had calculated the lowest cost of filling up in 2012 when traveling 100 miles a week is 637.24 dollars with the help of MATLAB, while the lowest cost when traveling 200 miles a week is 1283.5 dollars. These two values are very close to the ideal value of cost on the basis of the historical figure, which are 635.24 dollars and 1253.5 dollars respectively. Also, we have come up with the scheme of gas fulfillment respectively. By analyzing the schemes of gas filling, we can discover that when you predict the future gasoline price going up, the best strategy is to fill the tank as soon as possible, in order to lower the gas fare. On the contrary, when the predicted price tends to decrease, it is wiser and more economic for people to postpone the filling, which encourages people to purchase a half tank of gasoline only if the tank is almost empty.For other different pattern for every week’s “mileage driven”, we calculate the changing point of strategies-changed is 133.33 miles.Eventually, we will apply the models -to the analysis of the New York City. The result of prediction is good enough to match the actual data approximately. However, the total gas cost of New York is a little higher than that of the average cost nationally, which might be related to the higher consumer price index in the city. Due to the limit of time, we are not able to investigate further the particular factors.Keywords: gasoline price Time series analysis forecast lowest cost MATLABAbstract ---------------------------------------------------------------------------------------1 Restatement --------------------------------------------------------------------------------------21. Assumption----------------------------------------------------------------------------------42. Definitions of Variables and Models-----------------------------------------------------4 2.1 Models for the prediction of gasoline price in the subsequent week------------4 2.2 The Model of oil price next two weeks and four weeks--------------------------5 2.3 Model for refuel decision-------------------------------------------------------------52.3.1 Decision Model for consumer who drives 100 miles per week-------------62.3.2 Decision Model for consumer who drives 200 miles per week-------------73. Train and Test Model by 2011 dataset---------------------------------------------------8 3.1 Determine the all the parameters in Equation ② from the 2011 dataset-------8 3.2 Test the Forecast Model of gasoline price by the dataset of gasoline price in2012-------------------------------------------------------------------------------------10 3.3 Calculating ε --------------------------------------------------------------------------12 3.4 Test Decision Models of buying gasoline by dataset of 2012-------------------143.4.1 100 miles per week---------------------------------------------------------------143.4.2 200 miles per week---------------------------------------------------------------143.4.3 Second Test for the Decision of buying gasoline-----------------------------154. The upper bound will change the Decision of buying gasoline---------------------155. An analysis of New York City-----------------------------------------------------------16 5.1 The main factor that will affect the gasoline price in New York City----------16 5.2 Test Models with New York data----------------------------------------------------185.3 The analysis of result------------------------------------------------------------------196. Summery& Advantage and disadvantage-----------------------------------------------197. Report----------------------------------------------------------------------------------------208. Appendix------------------------------------------------------------------------------------21 Appendix 1(main MATLAB programs) ------------------------------------------------21 Appendix 2(outcome and graph) --------------------------------------------------------34The world market is fluctuating swiftly now. As the most important limited energy, oil is much accounted of cars owners and dealer. We are required to make a gas-buying plan which relates to the price of gasoline, the volume of tank, the distance that consumer drives per week, the data from EIA and the influence of other events in order to help drivers to save money.We should use the data of 2011 to build up two models that discuss two situations: 100miles/week or 200miles/week and use the data of 2012 to test the models to prove the model is applicable. In the model, consumer only has three choices to purchase gas each week, including no gas, half a tank and full tank. At the end, we should not only build two models but also write a simple but educational report that can attract consumer to follow this model.1.Assumptiona)Assume the consumer always buy gasoline according to the rule of minimumcost.b)Ignore the difference of the gasoline weight.c)Ignore the oil wear on the way to gas stations.d)Assume the tank is empty at the beginning of the following models.e)Apply the past data of crude oil price to predict the future price ofgasoline.(The crude oil price can affect the gasoline price and we ignore thehysteresis effect on prices of crude oil towards prices of gasoline.)2.Definitions of Variables and Modelst stands for the sequence number of week in any time.(t stands for the current week. (t-1) stands for the last week. (t+1) stands for the next week.c t: Price of crude oil of the current week.g t: Price of gasoline of the t th week.P t: The volume of oil of the t th week.G t+1: Predicted price of gasoline of the (t+1)th week.α,β: The coefficient of the g t and c t in the model.d: The variable of decision of buying gasoline.(d=1/2 stands for buying a half tank gasoline)2.1 Model for the prediction of gasoline price in the subsequent weekWhether to buy half a tank oil or full tank oil depends on the short-term forecast about the gasoline prices. Time series analysis is a frequently-used method to expect the gasoline price trend. It can be expressed as:G t+1=α1g t+α2g t−1+α3g t−2+α4g t−3+…αn+1g t−n+ε ----Equation ①ε is a parameter that reflects the influence towards the trend of gasoline price in relation to several aspects such as weather data, economic data, world events and so on.Due to the prices of crude oil can influence the future prices of gasoline; we will adopt the past prices of crude oil into the model for gasoline price forecast.G t+1=(α1g t+α2g t−1+α3g t−2+α4g t−3+⋯αn+1g t−n)+(β1g t+β2g t−1+β3g t−2+β4g t−3+⋯βn+1g t−n)+ε----Equation ②We will use the 2011 data set to calculate the all coefficients and the best delay periods n.2.2 The Model of oil price next two weeks and four weeksWe mainly depend on the prediction of change of gasoline price in order to make decision that the consumer should buy half a tank or full tank gas. When consumer drives 100miles/week, he can drive whether 400miles most if he buys full tank gas or 200miles most if he buys half a tank gas. When consumer drives 200miles/week, full tank gas can be used two weeks most or half a tank can be used one week most. Thus, we should consider the gasoline price trend in four weeks in future.Equation ②can also be rewritten asG t+1=(α1g t+β1g t)+(α2g t−1+β2g t−1)+(α3g t−2+β3g t−2)+⋯+(αn+1g t−n+βn+1g t−n)+ε ----Equation ③If we define y t=α1g t+β1g t,y t−1=α2g t−1+β2g t−1, y t−2=α3g t−2+β3g t−2……, and so on.Equation ③can change toG t+1=y t+y t−1+y t−2+⋯+y t−n+ε ----Equation ④We use y(t−1,t)denote the average price from week (t-1) to week (t), which is.y(t−1,t)=y t−1+y t2Accordingly, the average price from week (t-3) to week (t) isy(t−3,t)=y t−3+y t−2+y t−1+y t.4Apply Time series analysis, we can get the average price from week (t+1) to week (t+2) by Equation ④,G(t+1,t+2)=y(t−1,t)+y(t−3,t−2)+y(t−5,t−4), ----Equation ⑤As well, the average price from week (t+1) to week (t+4) isG(t+1,t+4)=y(t−3,t)+y(t−7,t−4)+y(t−11,t−8). ----Equation ⑥2.3 Model for refuel decisionBy comparing the present gasoline price with the future price, we can decide whether to fill half or full tank.The process for decision can be shown through the following flow chart.Chart 1For the consumer, the best decision is to get gasoline with the lowest prices. Because a tank of gasoline can run 2 or 4 week, so we should choose a time point that the price is lowest by comparison of the gas prices at present, 2 weeks and 4 weeks later separately. The refuel decision also depends on how many free spaces in the tank because we can only choose half or full tank each time. If the free spaces are less than 1/2, we can refuel nothing even if we think the price is the lowest at that time.2.3.1 Decision Model for consumer who drives 100 miles per week.We assume the oil tank is empty at the beginning time(t=0). There are four cases for a consumer to choose a best refuel time when the tank is empty.i.g t>G t+4and g t>G t+2, which means the present gasoline price is higherthan that either two weeks or four weeks later. It is economic to fill halftank under such condition. ii. g t <Gt +4 and g t <G t +2, which means the present gasoline price is lower than that either two weeks or four weeks later. It is economic to fill fulltank under such condition. iii. Gt +4>g t >G t +2, which means the present gasoline price is higher than that two weeks later but lower than that four weeks later. It is economic to fillhalf tank under such condition. iv. Gt +4<g t <G t +2, which means the present gasoline price is higher than that four weeks later but lower than that two weeks later. It is economic to fillfull tank under such condition.If other time, we should consider both the gasoline price and the oil volume in the tank to pick up a best refuel time. In summary, the decision model for running 100 miles a week ist 2t 4t 2t 4t 2t 4t 2t 4t 11111411111ˆˆ(1)1((1)&max(,))24442011111ˆˆˆˆ1/2((1)&G G G (&))(0(1G G )&)4424411ˆˆˆ(1)0&(G 4G G (G &)t i t i t t t t i t i t t t t t t i t t d t or d t g d d t g or d t g d t g or ++++----+++-++<--<<--<>⎧⎪=<--<<<--<<<⎨⎪⎩--=><∑∑∑∑∑t 2G ˆ)t g +<----Equation ⑦d i is the decision variable, d i =1 means we fill full tank, d i =1/2 means we fill half tank. 11(1)4t i tdt ---∑represents the residual gasoline volume in the tank. The method of prices comparison was analyzed in the beginning part of 2.3.1.2.3.2 Decision Model for consumer who drives 200 miles per week.Because even full tank can run only two weeks, the consumer must refuel during every two weeks. There are two cases to decide whether to buy half or full tank when the tank is empty. This situation is much simpler than that of 100 miles a week. The process for decision can also be shown through the following flow chart.Chart 2The two cases for deciding buy half or full tank are: i. g t >Gt +1, which means the present gasoline price is higher than the next week. We will buy half tank because we can buy the cheaper gasoline inthe next week. ii. g t <Gt +1, which means the present gasoline price is lower than the next week. To buy full tank is economic under such situation.But we should consider both gasoline prices and free tank volume to decide our refueling plan. The Model is111t 11t 111(1)1220111ˆ1/20(1)((1)0&)22411ˆ(1&G )0G 2t i t t i t i t t t t t i t t d t d d t or d t g d t g ----++<--<⎧⎪=<--<--=>⎨⎪⎩--=<∑∑∑∑ ----Equation ⑧3. Train and Test Model by the 2011 datasetChart 33.1 Determine all the parameters in Equation ② from the 2011 dataset.Using the weekly gas data from the website and the weekly crude price data from , we can determine the best delay periods n and calculate all the parameters in Equation ②. For there are two crude oil price dataset (Weekly Cushing OK WTI Spot Price FOB and Weekly Europe Brent SpotPrice FOB), we use the average value as the crude oil price without loss of generality. We tried n =3, 4 and 5 respectively with 2011 dataset and received comparison graph of predicted value and actual value, including corresponding coefficient.（A ） n =3(the hysteretic period is 3)Graph 1 The fitted price and real price of gasoline in 2011(n=3)We find that the nearby effect coefficient of the price of crude oil and gasoline. This result is same as our anticipation.（B）n=4(the hysteretic period is 4)Graph 2 The fitted price and real price of gasoline in 2011（n=4）(C) n=5(the hysteretic period is 5)Graph 3 The fitted price and real price of gasoline in 2011（n=5）Via comparing the three figures above, we can easily found that the predictive validity of n=3(the hysteretic period is 3) is slightly better than that of n=4(the hysteretic period is 4) and n=5(the hysteretic period is 5) so we choose the model of n=3 to be the prediction model of gasoline price.G t+1=0.0398+1.6002g t+−0.7842g t−1+0.1207g t−2+ 0.4147g t−0.5107g t−1+0.1703g t−2+ε----Equation ⑨3.2 Test the Forecast Model of gasoline price by the dataset of gasoline price in 2012Next, we apply models in terms of different hysteretic periods(n=3,4,5 respectively), which are shown in Equation ②,to forecast the gasoline price which can be acquired currently in 2012 and get the graph of the forecast price and real price of gasoline:Graph 4 The real price and forecast price in 2012（n=3）Graph 5 The real price and forecast price in 2012（n=4）Graph 6 The real price and forecast price in 2012（n=5）Conserving the error of observation, predictive validity is best when n is 3, but the differences are not obvious when n=4 and n=5. However, a serious problem should be drawn to concerns: consumers determines how to fill the tank by using the trend of oil price. If the trend prediction is wrong (like predicting oil price will rise when it actually falls), consumers will lose. We use MATLAB software to calculate the amount of error time when we use the model of Equation ⑨to predict the price of gasoline in 2012. The graph below shows the result.It’s not difficult to find the prediction effect is the best when n is 3. Therefore, we determined to use Equation ⑨as the prediction model of oil price in 2012.G t+1=0.0398+1.6002g t+−0.7842g t−1+0.1207g t−2+ 0.4147g t−0.5107g t−1+0.1703g t−2+ε3.3 Calculating εSince political occurences, economic events and climatic changes can affect gasoline price, it is undeniable that a ε exists between predicted prices and real prices. We can use Equation ②to predict gasoline prices in 2011 and then compare them with real data. Through the difference between predicted data and real data, we can estimate the value of ε .The estimating process can be shown through the following flow chartChart 4We divide the international events into three types: extra serious event, major event and ordinary event according to the criteria of influence on gas prices. Then we evaluate the value: extra serious event is 3a, major event is 2a, and ordinary event is a. With inference to the comparison of the forecast price and real price in 2011, we find that large deviation of data exists at three time points: May 16,2011, Aug 08,2011 andOct 10,2011. After searching, we find that some important international events happened nearly at the three time points. We believe that these events which occurred by chance affect the international prices of gasoline so the predicted prices deviate from the actual prices. The table of events and the calculation of the value of a areTherefore, by generalizing several sets of particular data and events, we can estimate the value of a:a=26.84 ----Equation ⑩The calculating process is shown as the following graph.Since now we have obtained the approximate value of a, we can evaluate the future prices according to currently known gasoline prices and crude oil prices. To improve our model, we can look for factors resulting in some major turning point in the graph of gasoline prices. On the ground that the most influential factors on prices in 2012 are respectively graded, the difference between fact and prediction can be calculated.3.4 Test Decision Models of buying gasoline by the dataset of 2012First, we use Equation ⑨to calculate the gasoline price of next week and use Equation ⑤and Equation ⑥to calculate the gasoline price trend of next two to four weeks. On the basis above, we calculate the total cost, and thus receive schemes of buying gasoline of 100miles per week according to Equation ⑦and Equation ⑧. Using the same method, we can easily obtain the pattern when driving 200 miles per week. The result is presented below.We collect the important events which will affect the gasoline price in 2012 as well. Therefore, we calculate and adjust the predicted price of gasoline by Equation ⑩. We calculate the scheme of buying gasoline again. The result is below:3.4.1 100 miles per weekT2012 = 637.2400 （If the consumer drives 100 miles per week, the total cost inTable 53.4.2 200 miles per weekT2012 = 1283.5 （If the consumer drives 200 miles per week, the total cost in 2012 is 1283.5 USD）. The scheme calculated by software is below：Table 6According to the result of calculating the buying-gasoline scheme from the model, we can know: when the gasoline price goes up, we should fill up the tank first and fill up again immediately after using half of gasoline. It is economical to always keep the tank full and also to fill the tank in advance in order to spend least on gasoline fee. However, when gasoline price goes down, we have to use up gasoline first and then fill up the tank. In another words, we need to delay the time of filling the tank in order to pay for the lowest price. In retrospect to our model, it is very easy to discover that the situation is consistent with life experience. However, there is a difference. The result is based on the calculation from the model, while experience is just a kind of intuition.3.4.3 Second Test for the Decision of buying gasolineSince the data in 2012 is historical data now, we use artificial calculation to get the optimal value of buying gasoline. The minimum fee of driving 100 miles per week is 635.7440 USD. The result of calculating the model is 637.44 USD. The minimum fee of driving 200 miles per week is 1253.5 USD. The result of calculating the model is 1283.5 USD. The values we calculate is close to the result of the model we build. It means our model prediction effect is good. (we mention the decision people made every week and the gas price in the future is unknown. We can only predict. It’s normal to have deviation. The buying-gasoline fee which is based on predicted calculation must be higher than the minimum buying-gasoline fee which is calculated when all the gas price data are known.)We use MATLAB again to calculate the total buying-gasoline fee when n=4 and n=5. When n=4,the total fee of driving 100 miles per week is 639.4560 USD and the total fee of driving 200 miles per week is 1285 USD. When n=5, the total fee of driving 100 miles per week is 639.5840 USD and the total fee of driving 200 miles per week is 1285.9 USD. The total fee are all higher the fee when n=3. It means it is best for us to take the average prediction model of 3 phases.4. The upper bound will change the Decision of buying gasoline.Assume the consumer has a mileage driven of x1miles per week. Then, we can use 200to indicate the period of consumption, for half of a tank can supply 200-mile x1driving. Here are two situations:<1.5①200x1>1.5②200x1In situation①, the consumer is more likely to apply the decision of 200-mile consumer’s; otherwise, it is wiser to adopt the decision of 100-mile consumer’s. Therefore, x1is a critical value that changes the decision if200=1.5x1x1=133.3.Thus, the mileage driven of 133.3 miles per week changes the buying decision.Then, we consider the full-tank buyers likewise. The 100-mile consumer buys half a tank once in four weeks; the 200-mile consumer buys half a tank once in two weeks. The midpoint of buying period is 3 weeks.Assume the consumer has a mileage driven of x2miles per week. Then, we can to illustrate the buying period, since a full tank contains 400 gallons. There use 400x2are still two situations:<3③400x2>3④400x2In situation③, the consumer needs the decision of 200-mile consumer’s to prevent the gasoline from running out; in the latter situation, it is wiser to tend to the decision of 100-mile consumer’s. Therefore, x2is a critical value that changes the decision if400=3x2x2=133.3We can find that x2=x1=133.3.To wrap up, there exists an upper bound on “mileage driven”, that 133.3 miles per week is the value to switch the decision for buying weekly gasoline. The following picture simplifies the process.Chart 45. An analysis of New Y ork City5.1 The main factors that will affect the gasoline price in New York CityBased on the models above, we decide to estimate the price of gasoline according to the data collected and real circumstances in several cities. Specifically, we choose New York City as a representative one.New York City stands in the North East in the United States, with the largest population throughout the country as 8.2 million. The total area of New York City is around 1300 km2, with the land area as 785.6 km2(303.3 mi2). One of the largest trading centers in the world, New York City has a high level of resident’s consumption. As a result, the level of the price of gasoline in New York City is higher than the average regular oil price of the United States. The price level of gasoline and its fluctuation are the main factors of buying decision.Another reasonable factor we expect is the distribution of gas stations. According to the latest report, there are approximately 1670 gas stations in the city area (However, after the impact of hurricane Sandy, about 90 gas stations have been temporarily out of use because of the devastation of Sandy, and there is still around 1580 stations remaining). From the information above, we can calculate the density of gas stations thatD(gasoline station)= t e amount of gas stationstotal land area =1670 stations303.3 mi2=5.506 stations per mi2This is a respectively high value compared with several other cities the United States. It also indicates that the average distance between gas stations is relatively small. The fact that we can neglect the distance for the cars to get to the station highlights the role of the fluctuation of the price of gasoline in New York City.Also, there are approximately 1.8 million residents of New York City hold the driving license. Because the exact amount of cars in New York City is hard to determine, we choose to analyze the distribution of possible consumers. Thus, we can directly estimate the density of consumers in New York City in a similar way as that of gas stations:D(gasoline consumers)= t e amount of consumerstotal land area = 1.8 million consumers303.3 mi2=5817consumers per mi2Chart 5In addition, we expect that the fluctuation of the price of crude oil plays a critical role of the buying decision. The media in New York City is well developed, so it is convenient for citizens to look for the data of the instant price of crude oil, then to estimate the price of gasoline for the coming week if the result of our model conforms to the assumption. We will include all of these considerations in our modification of the model, which we will discuss in the next few steps.For the analysis of New York City, we apply two different models to estimate the price and help consumers make the decision.5.2 Test Models with New York dataAmong the cities in US, we pick up New York as an typical example. The gas price data is downloaded from the website () and is used in the model described in Section 2 and 3.The gas price curves between the observed data and prediction data are compared in next Figure.Figure 6The gas price between the observed data and predicted data of New York is very similar to Figure 3 in US case.Since there is little difference between the National case and New York case, the purchase strategy is same. Following the same procedure, we can compare the gas cost between the historical result and predicted result.For the case of 100 miles per week, the total cost of observed data from Feb to Oct of 2012 in New York is 636.26USD, while the total cost of predicted data in the same period is 638.78USD, which is very close. It proves that our prediction model is good. For the case of 200 miles per week, the total cost of observed data from Feb to Oct of 2012 in New York is 1271.2USD, while the total cost of predicted data in the same period is 1277.6USD, which is very close. It proves that our prediction model is good also.5.3 The analysis of resultBy comparing, though density of gas stations and density of consumers of New York is a little higher than other places but it can’t lower the total buying-gas fee. Inanother words, density of gas stations and density of consumers are not the actual factors of affecting buying-gas fee.On the other hand, we find the gas fee in New York is a bit higher than the average fee in US. We can only analyze preliminary it is because of the higher goods price in New York. We need to add price factor into prediction model. We can’t improve deeper because of the limited time. The average CPI table of New York City and USA is below:Datas Statistics website(/xg_shells/ro2xg01.htm)6. Summery& Advantage and disadvantageTo reach the solution, we make graphs of crude oil and gasoline respectively and find the similarity between them. Since the conditions are limited that consumers can only drive 100miles per week or 200miles per week, we separate the problem into two parts according to the limitation. we use Time series analysis Method to predict the gasoline price of a future period by the data of several periods in the past. Then we take the influence of international events, economic events and weather changes and so on into consideration by adding a parameter. We give each factor a weight consequently and find the rules of the solution of 100miles per week and 200miles per week. Then we discuss the upper bound and clarify the definition of upper bound to solve the problem.According to comparison from many different aspects, we confirm that the model expressed byEquation ⑨is the best. On the basis of historical data and the decision model of buying gasoline(Equation ⑦and Equation ⑧), we calculate that the actual least cost of buying gasoline is 635.7440 USD if the consumer drives 100 miles per week (the result of our model is 637.24 USD) and the actual least cost of buying gasoline is 1253.5 USD(the result of our model is 1283.5 USD) if the consumer drives 100 miles per week. The result we predicted is similar to the actual result so the predictive validity of our model is finer.Disadvantages:1.The events which we predicted are difficult to quantize accurately. The turningpoint is difficult for us to predict accurately as well.2.We only choose two kinds of train of thought to develop models so we cannotevaluate other methods that we did not discuss in this paper. Other models which are built up by other train of thought are possible to be the optimal solution.。

2012年 AMC8 美国数学竞赛试题+答案(英文版)

(A) 5 (B) 6 (C) 7 (D) 11 (E) 12
12. What is the units digit of 132012? (A) 1 (B) 3 (C) 5 (D) 7
(E) 9
13. Jamar bought some pencils costing more than a penny each at the school bookstore and paid $1.43. Sharona bought some of the same pencils and paid $1.87. How many more pencils did Sharona buy than Jamar?
(A) 5 : 10pm (B) 5 : 21pm (C) 5 : 41pm (D) 5 : 57pm (E) 6 : 03pm
4. Peter’s family ordered a 12-slice pizza for dinner. Peter ate one slice and shared another slice equally with his brother Paul. What fraction of the pizza did Peter eat?
(A) 3 (B) 4 (C) 5 (D) 6 (E) 7
18. What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50?
(A) 3127 (B) 3133 (C) 3137 (D) 3139 (E) 3149
22. Let R be a set of nine distinct integers. Six of the elements are 2, 3, 4, 6, 9, and 14. What is the number of possible values of the median of R ?

AMC12美国数学竞赛2012-2014

AMC12 2014AProblem 1What isSolutionAt the theater children get in for half price. The price for adult tickets and child tickets is . How much would adult tickets and child tickets cost?SolutionWalking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible?SolutionSuppose that cows give gallons of milk in days. At this rate, how many gallons of milk will cows give in days?SolutionOn an algebra quiz, of the studentsscored points, scored points, scored points, and the rest scored points. What is the difference between the mean and median score of the students' scores on this quiz?SolutionThe difference between a two-digit number and the number obtained by reversing its digits is times the sum of the digits of either number. What is the sum of the two digit number and its reverse?SolutionThe first three terms of a geometric progression are , , and . What is the fourth term?SolutionA customer who intends to purchase an appliance has three coupons, only one of which may be used:Coupon 1: off the listed price if the listed price is at leastCoupon 2: dollars off the listed price if the listed price is at leastCoupon 3: off the amount by which the listed price exceedsFor which of the following listed prices will coupon offer a greater price reduction than either coupon or coupon ?Five positive consecutive integers starting with have average . What is the average of consecutive integers that start with ?SolutionThree congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length . The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. What is the length of one of the two congruent sides of one of the isosceles triangles?SolutionDavid drives from his home to the airport to catch a flight. He drives miles in the first hour, but realizes that he will be hour late if he continues at this speed. He increases his speed by miles per hour for the rest of the way to the airport and arrives minutes early. How many miles is the airport from his home?SolutionTwo circles intersect at points and . The minor arcs measure on one circle and on the other circle. What is the ratio of the area of the larger circle to the area of the smaller circle?A fancy bed and breakfast inn has rooms, each with a distinctive color-coded decor. One day friends arrive to spend the night. There are no other guests that night. The friends can room in any combination they wish, but with no morethan friends per room. In how many ways can the innkeeper assign the guests to the rooms?SolutionLet be three integers such that is an arithmetic progressionand is a geometric progression. What is the smallest possible value of ?SolutionA five-digit palindrome is a positive integer with respective digits , where is non-zero. Let be the sum of all five-digit palindromes. What is the sum of the digits of .SolutionThe product , where the second factor has digits, is an integer whose digits have a sum of . What is ?SolutionA rectangular box contains a sphere of radius and eight smaller spheres of radius . The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is ?SolutionThe domain of the function is an interval of length , where and are relatively prime positive integers. What is ?SolutionThere are exactly distinct rational numbers such that andhas at least one integer solution for . What is ?SolutionIn , , , and . Points and lieon and respectively. What is the minimum possible valueof ?SolutionFor every real number , let denote the greatest integer not exceeding , and let The set of all numbers suchthat and is a union of disjoint intervals. What is the sum of the lengths of those intervals?SolutionThe number is between and . How many pairs ofintegers are there such that andSolutionThe fraction where is the length of the period of the repeating decimal expansion. What is the sum ?SolutionLet , and for , let . For how many values of is ?The parabola has focus and goes through the points and . For how many points with integer coefficients is it truethat ?AMC 12 2013AProblem 1Square has side length . Point is on , and the areaof is . What is ?SolutionA softball team played ten games, scoring , and runs. They lost by one run in exactly five games. In each of the other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?SolutionA flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?SolutionWhat is the value ofSolutionTom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $, Dorothy paid $, and Sammy paid $. In order to share the costs equally, Tom gave Sammy dollars, and Dorothy gave Sammy dollars. What is ?SolutionIn a recent basketball game, Shenille attempted only three-point shots andtwo-point shots. She was successful on of her three-point shots and of her two-point shots. Shenille attempted shots. How many points did she score?SolutionThe sequence has the property that every term beginning with the third is the sum of the previous two. That is,Suppose that and . What is ?SolutionGiven that and are distinct nonzero real numbers such that , what is ?SolutionIn , and . Points and are onsides , , and , respectively, such that and are parallelto and , respectively. What is the perimeter of parallelogram ?SolutionLet be the set of positive integers for which has the repeating decimal representation with and different digits. What is the sum of the elements of ?SolutionTriangle is equilateral with . Points and are on and points and are on such that both and are parallel to . Furthermore, triangle and trapezoids and all have the same perimeter. What is ?SolutionThe angles in a particular triangle are in arithmetic progression, and the side lengths are . The sum of the possible values of x equals where , and are positive integers. What is ?SolutionLet points and .Quadrilateral is cut into equal area pieces by a line passing through . This line intersects at point , where these fractions are in lowest terms. What is ?SolutionThe sequence, , , ,is an arithmetic progression. What is ?SolutionRabbits Peter and Pauline have three offspring—Flopsie, Mopsie, and Cotton-tail. These five rabbits are to be distributed to four different pet stores so that no store gets both a parent and a child. It is not required that every store gets a rabbit. In how many different ways can this be done?Solution, , are three piles of rocks. The mean weight of the rocks in is pounds, the mean weight of the rocks in is pounds, the mean weight of the rocks in the combined piles and is pounds, and the mean weight of the rocks in the combined piles and is pounds. What is the greatest possible integer value for the mean in pounds of the rocks in the combined piles and ?SolutionA group of pirates agree to divide a treasure chest of gold coins among themselves as follows. The pirate to take a share takes of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the pirate receive?SolutionSix spheres of radius are positioned so that their centers are at the vertices of a regular hexagon of side length . The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere?SolutionIn , , and . A circle with center andradius intersects at points and . Moreover and have integer lengths. What is ?SolutionLet be the set . For , define to mean thateither or . How many ordered triples of elements of have the property that , , and ?SolutionConsider. Which of the following intervals contains ?SolutionA palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base 10 with no leading zeros. A 6-digit palindrome is chosen uniformly at random. What is the probability that is also a palindrome?Solutionis a square of side length . Point is on such that . The square region bounded by is rotated counterclockwise with center , sweeping out a region whose area is , where , , and are positive integers and . What is ?SolutionThree distinct segments are chosen at random among the segments whoseend-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area?SolutionLet be defined by . How many complexnumbers are there such that and both the real and the imaginary parts of are integers with absolute value at most ?AMC12 2012AProblem 1A bug crawls along a number line, starting at . It crawls to , then turns around and crawls to . How many units does the bug crawl altogether?SolutionCagney can frost a cupcake every seconds and Lacey can frost a cupcake every seconds. Working together, how many cupcakes can they frostin minutes?SolutionA box centimeters high, centimeters wide, and centimeters long canhold grams of clay. A second box with twice the height, three times the width, and the same length as the first box can hold grams of clay. What is ?SolutionIn a bag of marbles, 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?SolutionA fruit salad consists of blueberries, raspberries, grapes, and cherries. The fruit salad has a total of pieces of fruit. There are twice as many raspberries as blueberries, three times as many grapes as cherries, and four times as many cherries as raspberries. How many cherries are there in the fruit salad?SolutionThe sums of three whole numbers taken in pairs are , , and . What is the middle number?SolutionMary divides a circle into 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?SolutionAn iterative average of the numbers , , , , and 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?SolutionA year is a leap year if and only if the year number is divisible by (such as ) or is divisible by but not by (such as ). The anniversary of the birth of novelist Charles Dickens was celebrated on February , , a Tuesday. On what day of the week was Dickens born?SolutionA triangle has area , one side of length , and the median to that side of length . Let be the acute angle formed by that side and the median. Whatis ?SolutionAlex, Mel, and Chelsea play a game that has rounds. In each round there is a single winner, and the outcomes of the rounds are independent. For each round the probability that Alex wins is , and Mel is twice as likely to win as Chelsea. What is the probability that Alex wins three rounds, Mel wins two rounds, and Chelsea wins one round?SolutionA square region is externally tangent to the circle withequation at the point on the side . Vertices and are on the circle with equation . What is the side length of this square?SolutionPaula the painter and her two helpers each paint at constant, but different, rates. They always start at , and all three always take the same amount of time to eat lunch. On Monday the three of them painted of a house, quittingat . On Tuesday, when Paula wasn't there, the two helpers paintedonly of the house and quit at . On Wednesday Paula worked by herself and finished the house by working until . How long, in minutes, was each day's lunch break?SolutionThe closed curve in the figure is made up of congruent circular arcs each of length , where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side . What is the area enclosed by the curve?SolutionA square is partitioned into 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 the rotated 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?SolutionCircle has its center lying on circle . The two circles meet at and . Point in the exterior of lies on circle and , ,and . What is the radius of circle ?SolutionLet be a subset of with the property that no pair of distinct elements in has a sum divisible by . What is the largest possible size of ?SolutionTriangle has , , and . Let denote the intersection of the internal angle bisectors of . What is ?SolutionAdam, 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?SolutionConsider the polynomialThe coefficient of is equal to . What is ?SolutionLet , , and be positive integers with such thatWhat is ?SolutionDistinct planes intersect the interior of a cube . Let be the unionof the faces of and let . The intersection of and consists of the union of all segments joining the midpoints of every pair of edges belonging to the same face of . What is the difference between the maximum and minimum possible values of ?SolutionLet be the square one of whose diagonals hasendpoints and . A point is chosen uniformly at random over all pairs of real numbers and suchthat and . Let be a translated copyof centered at . What is the probability that the square region determinedby contains exactly two points with integer coefficients in its interior?SolutionLet be the sequence of real numbers definedby , and in general,Rearranging the numbers in the sequence in decreasing order produces anew sequence . What is the sum of all integers , , such thatSolutionLet where denotes the fractional part of . Thenumber is the smallest positive integer such that the equationhas at least real solutions. What is ? Note: the fractional part of is a real number such that and is an integer.2014A1.C2.B3.B4.A5.C6.D7.A8.C9.B10.B11.C12.D13.B14.C15.B16.D17.A18.C19.E20.D21.A22.B23.B24.C25.B 2013A1. E2. C3. E4. C5. B6. B7. C8. D9. C10. D11. C12. A13. B14. B15. D16. E17. D18. B19. D20. B21. A22. E23. C24. E25. A2012A1. E2. D3. D4. C5. D6. D7. C8. C9. A10. D11. B12. D13. D14. E15. A16. E17. B18. A19. B20. B21. E22. C23. C24. C25. C。

赏析几道美国AMC12数学竞赛题

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2012年美国数学建模题目

2007_2012AMC8中文试题和答案解析

2007年美国AMC8(2007年11月日时间40分钟)1.如果希瑞莎能够持续6周，平均每周花10小时帮忙照顾房子，她的父母就帮她买她喜爱乐团的入场券。

在前五周她分别花了 & 11、7、12及10小时照顾房子。

在最后一周，她必须要花多少小时去照顾房子才能获得入场券？ (A) 9 (B) 10 (C) 11 (D) 12 (E) 13 。

2.调查650位学生对面食种类的偏好。

选项包含：卤味面、起司肉燥面、水饺、意大利面，调查结果如长条图所示。

试问偏好意大利面的学生数与偏好起司肉燥面的学生数之比值为多少？2 15 5 5 (A) 2 (B) 1 (C) 5 (D) 5 (E) 5。

524323. 250的最小两个质因子之和为多少？ (A) 2 (B) 5 (C) 7 (D) 10 (E) 12。

4. 某间鬼屋有六个窗子。

小精灵乔治丛一个窗子进入屋内，而从不同的另一个窗子出来的方法共有多少种？ (A) 12 (B) 15 (C) 18 (D) 30 (E) 36。

5.姜德想买一辆价值美金500元的越野脚踏车。

在他生日时，祖父母给他美金50元，姑姑给他美金35元，表哥给他美金15元。

他送报纸每周可赚美金16元。

若用他生日得到的所有礼金及送报纸所有赚得的钱去买越野脚踏车，他需要送几周的报纸才能有足够的钱？(A) 24 (B) 25 (C) 26 (D) 27 (E) 28。

6. 在1985年美国的长途电话费是每分钟 41分钱，在2005年的长途电话费是每分钟7分钱＜试求每分钟长途电话费下降的百分率最接近下列哪一项？(A) 7 (B) 17 (C) 34 (D) 41(E) 807. 房间内5个人的平均年龄为30岁。

若其中一位18岁的人离开了房间，则剩下四个人的平均年龄是几岁？ (A) 25 (B) 26 (C) 29 (D) 33 (E) 36。

8.在梯形 ABCD 中，AD 垂直 DC ，AD = AB =3，DC =6。

2012 MCM Contest Problems(2012年美国赛题目—中英文版)

2012 MCM Contest Problems (2012年美国赛题目—中英文版) PROBLEM A:The Leaves of a Tree "How much do the leaves on a tree weigh?" How might one estimate the actual weight of the leaves (or for that matter any other parts of the tree)? How might one classify leaves? Build a mathematical model to describe and classify leaves. Consider and answer the following:?6?1 Why do leaves have the various shapes that they have??6?1 Do the shapes “minimize” overlapping individual shadows that are cast, so as to maximize exposure? Does the distribution of leaves within the “volume” of the tree and its branches effect the shape??6?1 Speaking of profiles, is leaf shape (general characteristics) related to tree profile/branching structure??6?1 How would you estimate the leaf mass of a tree? Is there a correlation between the leaf mass and the size characteristics of the tree (height, mass, volume defined by the profile)?In addition to your one page summary sheet prepare a one page letter to an editor of a scientific journal outlining your key findings.PROBLEM B:Camping along the Big Long River Visitors to the Big Long River (225 miles) can enjoy scenic views and exciting white water rapids. The river is inaccessible to hikers, so the only way to enjoy it is to take a river trip that requires several days of camping. River trips all start at First Launch and exit the river at Final Exit, 225 miles downstream. Passengers take either oar- powered rubber rafts, which travel on average 4 mph or motorized boats, which travel on average 8 mph. The trips range from 6 to 18 nights of camping on the river, start to finish.. The government agency responsible for managing this river wants every trip to enjoy a wilderness experience, with minimal contact with other groups of boats on the river. Currently, X trips travel down the Big Long River each year during a six month period (the rest of the year it is too cold for river trips). There are Y camp sites on the Big Long River, distributed fairly uniformly throughout the river corridor. Given the rise in popularity of river rafting, the park managers have been asked to allow more trips to travel down the river. They want to determine how they might schedule an optimal mix of trips, of varying duration (measured in nights on the river) and propulsion (motor or oar) that will utilize the campsites in the best way possible. In other words, how many more boat trips could be added to the Big Long River’s rafting season? The river managers have hired you to advise them on ways in which to develop the best schedule and on ways in which to determine the carrying capacity of the river, remembering that no two sets of campers can occupy the same site at the same time. In addition to your one page summary sheet, prepare a one page memo to the managers of the river describing your key findings.A“多少钱，树上的叶子重？”你如何估计的实际重量的叶（或对任何其他部分的树）？你如何分类的叶子？建立一个数学模型，描述和分类的叶子。

数学建模美国赛2012题目

2012Contest ProblemsMCM PROBLEMSPROBLEM A:The Leaves of a Tree"How much do the leaves on a tree weigh?"How might one estimate the actual weight of the leaves(or for that matter any other parts of the tree)?How might one classify leaves?Build a mathematical model to describe and classify leaves.Consider and answer the following:•Why do leaves have the various shapes that they have?•Do the shapes“minimize”overlapping individual shadows that are cast,so as to maximize exposure?Does the distribution of leaves within the“volume”of the tree and its branches effect the shape?•Speaking of profiles,is leaf shape(general characteristics)related to tree profile/branching structure?•How would you estimate the leaf mass of a tree?Is there a correlation between the leaf mass and the size characteristics of the tree(height,mass,volume defined by the profile)?In addition to your one page summary sheet prepare a one page letter to an editor of a scientific journal outlining your key findings.PROBLEM B:Camping along the Big Long RiverVisitors to the Big Long River(225miles)can enjoy scenic views and exciting white water rapids. The river is inaccessible to hikers,so the only way to enjoy it is to take a river trip that requires several days of camping.River trips all start at First Launch and exit the river at Final Exit,225 miles downstream.Passengers take either oar-powered rubber rafts,which travel on average4 mph or motorized boats,which travel on average8mph.The trips range from6to18nights of camping on the river,start to finish..The government agency responsible for managing this river wants every trip to enjoy a wilderness experience,with minimal contact with other groups of boats on the river.Currently,X trips travel down the Big Long River each year during a six month period(the rest of the year it is too cold for river trips).There are Y camp sites on the Big Long River,distributed fairly uniformly throughout the river corridor.Given the rise in popularity ofriver rafting,the park managers have been asked to allow more trips to travel down the river.They want to determine how they might schedule an optimal mix of trips,of varying duration(measured in nights on the river)and propulsion(motor or oar)that will utilize the campsites in the best way possible.In other words,how many more boat trips could be added to the Big Long River’s rafting season?The river managers have hired you to advise them on ways in which to develop the best schedule and on ways in which to determine the carrying capacity of the river,remembering that no two sets of campers can occupy the same site at the same time.In addition to your one page summary sheet,prepare a one page memo to the managers of the river describing your key findings.ICM PROBLEMPROBLEM C:Modeling for Crime BustingClick the title below to download a ZIP file containing the2012ICM Problem.Your ICM submission should consist of a1page Summary Sheet and your solution cannot exceed20pages for a maximum of21pages.2012 ICM ProblemModeling for Crime BustingYour organization, the Intergalactic Crime Modelers (ICM), is investigating a conspiracy to commit a criminal act. The investigators are highly confident they know several members of the conspiracy, but hope to identify the other members and the leaders before they make arrests. The conspirators and the possible suspected conspirators all work for the same company in a large office complex. The company has been growing fast and making a name for itself in developing and marketing computer software for banks and credit card companies. ICM has recently found a small set of messages from a group of 82 workers in the company that they believe will help them find the most likely candidates for the unidentified co‐conspirators and unknown leaders. Since the message traffic is for all the office workers in the company, it is very likely that some (maybe many) of the identified communicators in the message traffic are not involved in the conspiracy. In fact, they are certain that they know some people who are not in the conspiracy. The goal of the modeling effort will be to identify people in the office complex who are the most likely conspirators. A priority list would be ideal so ICM could investigate, place under surveillance, and/or interrogate the most likely candidates. A discriminate line separating conspirators from non‐conspirators would also be helpful to distinctly categorize the people in each group. It would also be helpful to the DA’s office if the model nominated the conspiracy leaders.Before the data of the current case are given to your crime modeling team, your supervisor gives you the following scenario (called Investigation EZ) that she worked on a few years ago in another city. Even though she is very proud of her work on the EZ case, she says it is just a very small, simple example, but it may help you understand your task. Her data follow:The ten people she was considering as conspirators were: Anne#, Bob, Carol, Dave*, Ellen, Fred, George*, Harry, Inez, and Jaye#. (* indicates prior known conspirators, # indicate prior known non‐conspirators)Chronology of the 28 messages that she had for her case with a code number for the topic of each message that she assigned based on her analysis of the message:Anne to Bob: Why were you late today? (1)Bob to Carol: That darn Anne always watches me. I wasn't late. (1)Carol to Dave: Anne and Bob are fighting again over Bob's tardiness. (1)Dave to Ellen: I need to see you this morning. When can you come by? Bring the budget files. (2)Dave to Fred: I can come by and see you anytime today. Let me know when it is a good time. Should I bring the budget files? (2)Dave to George: I will see you later ‐‐‐ lots to talk about. I hope the others are ready. It is important to get this right. (3)Harry to George: You seem stressed. What is going on? Our budget will be fine. (2) (4)Inez to George: I am real tired today. How are you doing? ( 5)Jaye to Inez: Not much going on today. Wanna go to lunch today? (5)Inez to Jaye: Good thing it is quiet. I am exhausted. Can't do lunch today ‐‐‐ sorry! (5)George to Dave: Time to talk ‐‐‐ now! (3)Jaye to Anne: Can you go to lunch today? (5)Dave to George: I can't. On my way to see Fred. (3)George to Dave: Get here after that. (3)Anne to Carol: Who is supposed to watch Bob? He is goofing off all the time. (1)Carol to Anne: Leave him alone. He is working well with George and Dave. (1)George to Dave: This is important. Darn Fred. How about Ellen? (3)Ellen to George: Have you talked with Dave? (3)George to Ellen: Not yet. Did you? (3)Bob to Anne: I wasn't late. And just so you know ‐‐‐ I am working through lunch. (1)Bob to Dave: Tell them I wasn't late. You know me. (1)Ellen to Carol: Get with Anne and figure out the budget meeting schedule for next week and help me calm George. (2)Harry to Dave: Did you notice that George is stressed out again today? (4)Dave to George: Darn Harry thinks you are stressed. Don't get him worried or he will be nosing around.(4)George to Harry: Just working late and having problems at home. I will be fine. (4)Ellen to Harry: Would it be OK, if I miss the meeting today? Fred will be there and he knows the budget better than I do. (2)Harry to Fred: I think next year's budget is stressing out a few people. Maybe we should take time to reassure people today. (2) (4)Fred to Harry: I think our budget is pretty healthy. I don't see anything to stress over. (2)END of MESSAGE TRAFFICYour supervisor points outs that she assigned and coded only 5 different topics of messages: 1) Bob's tardiness, 2) the budget, 3) important unknown issue but assumed to be part of conspiracy, 4) George's stress, and 5) lunch and other social issues. As seen in the message coding, some messages had two topics assigned because of the content of the messages.The way your supervisor analyzed her situation was with a network that showed the communication links and the types of messages. The following figure is a model of the message network that resulted with the code for the types of messages annotated on the network graph.Figure 1: Network of messages from EZ CaseYour supervisor points out that in addition to known conspirators George and Dave, Ellen and Carol were indicted for the conspiracy based on your supervisor's analysis and later Bob self‐admitted his involvement in a plea bargain for a reduced sentence, but the charges against Carol were later dropped. Your supervisor is still pretty sure Inez was involved, but the case against her was never established. Your supervisor's advice to your team is identify the guilty parties so that people like Inez don't get off, people like Carol are not falsely accused, and ICM gets the credit so people like Bob do not have the opportunity to get reduced sentences.The current case:Your supervisor has put together a network‐like database for the current case, which has the same structure, but is a bit larger in scope. The investigators have some indications that a conspiracy is taking place to embezzle funds from the company and use internet fraud to steal funds from credit cards of people who do business with the company. The small example she showed you for case EZ had only 10 people (nodes), 27 links (messages), 5 topics, 1 suspicious/conspiracy topic, 2 known conspirators, and 2known non‐conspirators. So far, the new case has 83 nodes, 400 links (some involving more than 1 topic), over 21,000 words of message traffic, 15 topics (3 have been deemed to be suspicious), 7 known conspirators, and 8 known non‐conspirators. These data are given in the attached spreadsheet files: Names.xls, Topics.xls, and Messages.xls. Names.xls contains the key of node number to the office workers' names. Topics.xls contains the code for the 15 topic numbers to a short description of the topics. Because of security and privacy issues, your team will not have direct transcripts of all the message traffic. Messages.xls provides the links of the nodes that transmitted messages and the topic code numbers that the messages contained. Several messages contained up to three topics. To help visualize the message traffic, a network model of the people and message links is provided in Figure 2. In this case, the topics of the messages are not shown in the figure as they were in Figure 1. These topic numbers are given in the file Messages.xls and described in Topics.xls.Figure 2: Visual of the network model of the 83 people (nodes) and 400 messages between thesepeople (links).Requirements:Requirement 1: So far, it is known that Jean, Alex, Elsie, Paul, Ulf, Yao, and Harvey are conspirators. Also, it is known that Darlene, Tran, Jia, Ellin, Gard, Chris, Paige, and Este are not conspirators. The three known suspicious message topics are 7, 11, and 13. There is more detail about the topics in file Topics.xls. Build a model and algorithm to prioritize the 83 nodes by likelihood of being part of the conspiracy and explain your model and metrics. Jerome, Delores, and Gretchen are the senior managers of the company. It would be very helpful to know if any of them are involved in the conspiracy.Requirement 2: How would the priority list change if new information comes to light that Topic 1 is also connected to the conspiracy and that Chris is one of the conspirators?Requirement 3: A powerful technique to obtain and understand text information similar to this message traffic is called semantic network analysis; as a methodology in artificial intelligence and computational linguistics, it provides a structure and process for reasoning about knowledge or language. Another computational linguistics capability in natural language processing is text analysis. For our crime busting scenario, explain how semantic and text analyses of the content and context of the message traffic (if you could obtain the original messages) could empower your team to develop even better models and categorizations of the office personnel. Did you use any of these capabilities on the topic descriptions in file Topics.xls to enhance your model?Requirement 4: Your complete report will eventually go to the DA so it must be detailed and clearly state your assumptions and methodology, but cannot exceed 20 pages of write up. You may include your programs as appendices in separate files that do not count in your page restriction, but including these programs is not necessary. Your supervisor wants ICM to be the world's best in solving white‐collar, high‐tech conspiracy crimes and hopes your methodology will contribute to solving important cases around the world, especially those with very large databases of message traffic (thousands of people with tens of thousands of messages and possibly millions of words). She specifically asked you to include a discussion on how more thorough network, semantic, and text analyses of the message contents could help with your model and recommendations. As part of your report to her, explain the network modeling techniques you have used and why and how they can be used to identify, prioritize, and categorize similar nodes in a network database of any type, not just crime conspiracies and message data. For instance, could your method find the infected or diseased cells in a biological network where you had various kinds of image or chemical data for the nodes indicating infection probabilities and already identified some infected nodes?*Your ICM submission should consist of a 1 page Summary Sheet and your solution cannot exceed 20 pages for a maximum of 21 pages.。

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