2018年美国数学竞赛 AMC 试题
amc数学竞赛试题
AMC数学竞赛试题AMC数学竞赛是美国数学协会（AMC）主办的一项全球性数学竞赛活动，旨在促进学生对数学的兴趣和能力的培养。
本文将介绍一道AMC数学竞赛试题，并进行详细解析。
题目描述某AMC数学竞赛试题为：在正方形ABCD中，点E是边BC的中点，点F是边CD上的一点，且DF=CF。
连接线段AD与线段CF交于点G。
如果角ADG为45度，且面积△AFG为24平方单位，求正方形ABCD的面积。
解题思路我们可以通过观察题目中的信息，并利用几何图形的性质来解答这道题目。
首先，假设正方形ABCD的边长为a个单位。
由于点E是边BC的中点，所以线段AE与线段ED的长度相等，都等于a/2个单位。
同理，线段BF的长度也等于a/2个单位。
连接线段AD与线段CF，并设交点为G。
由于角ADG为45度，可以推导出△AFG为等腰直角三角形。
而已知△AFG的面积为24平方单位，所以可以得出AG和GF的长度都为√24=2√6个单位。
进一步观察，可以发现三角形△ADF和△GFC相似，因为∠ADF和∠GFC均为90度，且∠ADF=∠GFC=45度（根据题目中所给信息）。
根据相似三角形的性质，我们可以得到以下比例关系： AD/AF = GF/FC代入已知值和已推导出的值，可以得到：a/(a/2) = 2√6/(a/2) 解方程后可得：a^2 = 24 所以，正方形ABCD的面积为a^2 = 24个单位。
结论通过对题目所给信息的观察和利用几何图形的性质，《AMC数学竞赛试题》可以得到正方形ABCD的面积为24个单位。
这道题目考察了几何图形的相似性质和解方程的能力，对于参加数学竞赛的学生来说，是一道很好的训练题目。
AMC数学竞赛试题涵盖广泛的数学知识，并且注重培养学生的解题思路和创新能力。
通过参与AMC竞赛，学生可以锻炼自己的思维能力和解决问题的能力，在数学学科中取得更好的成绩。
amc8数学竞赛试题
amc8数学竞赛试题AMC 8数学竞赛试题[介绍]AMC 8数学竞赛是一项旨在培养学生数学兴趣和解决问题能力的国际性竞赛。
在这项竞赛中，学生将面对一系列挑战性的数学问题，要求运用自己的数学知识和技能进行解答。
本文将为您介绍一些典型的AMC 8数学竞赛试题，旨在帮助阅读者更好地了解这一竞赛的题型和难度。
[难度递增的问题]1. 题目：若a、b、c为正整数，并满足a+b+c=13，那么a、b、c的可能的取值有几个？解析：这是一道简单的组合问题。
因为a、b、c都是正整数，所以它们的取值范围在1到12之间。
我们可以采用穷举法来找出满足条件的取值。
2. 题目：在数列1，3，6，10，...中，每个数都是前一个数加上一个递增的正整数得到的，求第100个数是多少？解析：这是一道关于数列的问题。
通过观察数列的规律，我们可以发现每个数都是前一个数加上一个等差数列的公差得到的。
因此，我们可以通过求解等差数列的前100项和来得到结果。
[几何题]3. 题目：如图所示，已知正方形ABCD，E为边BC的中点，F为边CD上一点，且EF与边BC平行。
若AB的边长为6，求EF的长度。
解析：这是一道关于正方形的几何题。
根据题意，我们可以通过使用相似三角形的性质来求解。
首先，我们可以得到三角形ABF与三角形ECF相似，进而得到EF与AB的长度比例。
由此，我们可以计算出EF的长度。
[排列组合题]4. 题目：某城市共有6个球队，今天有两场比赛要进行。
每场比赛中，两个球队随机配对，一共进行了3场比赛。
那么每个球队都参与比赛的可能性有多大？解析：这是一道排列组合问题。
根据题意，我们需要计算从6个球队中选择3个球队参与比赛的可能性。
利用组合数学的知识，我们得到了参与比赛的球队个数的可能情况，进而计算出所需概率。
[数学推理题]5. 题目：下面的等式成立吗？7 + 7 - 7 × 7 ÷ 7 = 0解析：这是一道数学推理题。
根据四则运算的优先级，我们需要按照从左到右的顺序逐步进行计算。
2018年美国“数学大联盟杯赛”(中国赛区)初赛四年级试卷(1)
2017-2018年度美国“数学大联盟杯赛”(中国赛区)初赛(四年级)(初赛时间：2017年11月26日，考试时间90分钟，总分200分)学生诚信协议：考试期间，我确定没有就所涉及的问题或结论，与任何人、用任何方式交流或讨论，我确定我所填写的答案均为我个人独立完成的成果，否则愿接受本次成绩无效的处罚。
请在装订线内签名表示你同意遵守以上规定。
考前注意事项：1. 本试卷是四年级试卷，请确保和你的参赛年级一致；2. 本试卷共4页(正反面都有试题)，请检查是否有空白页，页数是否齐全；3. 请确保你已经拿到以下材料：本试卷(共4页，正反面都有试题)、答题卡、答题卡使用说明、英文词汇手册、草稿纸。
考试完毕，请务必将英文词汇手册带回家，上面有如何查询初赛成绩、及如何参加复赛的说明。
其他材料均不能带走，请留在原地。
选择题：每小题5分，答对加5分，答错不扣分，共200分，答案请填涂在答题卡上。
1.Which of the following is the smallest?A) 2.018 B) 20.18 C) 0.218 D) 20182.What is the least common multiple of 20 and 18?A) 90 B) 180 C) 240 D) 3603.The sum of the degree-measures of the exterior angles of a triangle is?A) 180 B) 360 C) 540 D) 7204.In the figure on the right, please put the numbers 1 – 11 in the elevencircles so that the three numbers in every straight line add up to 18.What is the number in the middle circle? Note: There are 5 straightlines in total in this figure.A) 6 B) 7 C) 8 D) 95.I am a lovely cat. When I multiply the digits of a whole numberand the product I get is 8, I put that whole number on my list offavorite numbers. Of the whole numbers from 1000 to 9999,how many would I put on my list of favorite numbers?A) 10 B) 12 C) 16 D) 206.Two planes take off at the same time from the same point to race to apoint and back. Place A travels at 180 miles per hour on the way out and240 miles per hour on the return trip. Plane B covers the entire distance at an averagespeed of 210 miles per hour. Which plane wins the race, or is it a tie?A) plane A wins B) plane B winsC) a tie D) non-deterministic 7.52 × 88 = 44 ×?A) 102 B) 96 C) 104 D) 1248.What is the smallest whole number that leaves a remainder of 4, 5, 6 when divided byeach of 5, 6, 7?A) 29 B) 209 C) 210 D) 20099.In △ABC, m∠A + m∠C = m∠B. What is the degree measure of ∠B?A) 80 B) 90 C) 100 D) 18010.I bought a toy for $10, sold it for $20, rebought it for $30, and resold it for $40. My totalprofit on the 4 transactions was ?A) 10 B) 20 C) 30 D) 4011.What is the greatest number of integers I can choose from the first ten positive integers sothat any 3 of the chosen integers could be the lengths of the three sides of a triangle?A) 4 B) 5 C) 6 D) 712.How many whole numbers between 200 and 400 have all their digits increasing in valuewhen read from left to right?A) 30 B) 36 C) 42 D) 4813.What is the value of 1% of 10% of 100?A) 0.01 B) 0.1 C) 1 D) 1014.If three cats can eat three bowls of food in three minutes, how many minutes will it take100 cats to eat 100 bowls of food?A) 1 B) 3C) 100 D) None of the above15.There are three squares. The area of the smallest one is 2. The side-length of the secondsquare is twice the side-length of the smallest one. And the side-length of the third square is three-times the side-length of the smallest one. The total area of the three squares isA) 12 B) 28 C) 36 D) 7216.A man, who had been married for three years, spent25of his yearly income on his family,14on business, and110on personal travel. If he saved $45000 during those three years, what was his annual income?A) $45000 B) $50000C) $65000 D) None of the above17.Given four different integers, at most how many different sums can be formed bychoosing two, three, or four of them and finding each sum?A) 8 B) 9 C) 10 D) 1118. Max places 100 eggs in 10 baskets, with each basket receiving at least1 egg, but no2 baskets receiving the same number of eggs. What is the greatest number of eggs that may be placed in a basket?A) 45 B) 47 C) 55 D) 6519. 2 + 3 × 4 – 5 =A) 0 B) 6 C) 9 D) 15 20. What is the highest power of 2 that divides 2 × 4 × 6 × 8 × 10? A) 25 B) 27 C) 28 D) 215 21. Which of the following is a prime number?A) 2017B) 2018C) 2015D) 201622. What is the greatest possible number of acute angles in a figure consisting of a triangleand a line passing through two sides of the triangle?A) 5B) 6C) 7D) 823. Amy can solve 5 questions every 3 minutes. Kate can solve 3 questions every 5 minutes.How many more questions Amy can solve than Kate in one hour?A) 15B) 32C) 60D) 6424. Using 3 Ts and 2 Js, in how many different orders can the five letters be arranged? Forexample, TTTJJ and TTJJT are two such different orders.A) 2B) 10C) 20D) 6025. Coastal Coconuts can divide all their coconuts evenly among 8, 9, or10 customers, with 1 coconut left over each time. If Coastal Coconuts has more than 1 coconut, what is the least number of coconuts they could have?A) 561 B) 721C) 831 D) None of the above 26. 35 ÷ 32 =A) 3 B) 9 C) 27 D) 81 27. If the sum of three prime numbers is 30, what is the least prime number?A) 2B) 3C) 5D) 728. Juxtaposing two identical squares to form a rectangle, the perimeter of the rectangle is 12less than the sum of the perimeter of the two squares. What is the side-length of the original square?A) 3B) 6C) 9D) 1229. It takes Mike 2 hours to finish a task. It takes 4 hours for Tom to finish the same task.Mike and Tom worked together on this task for one hour before Mike had to leave. How long will it take Tom to finish the rest of the task?A) 1 B) 2 C) 3 D) 4 30. The number of triangles in the figure on the right isA) 9 B) 10 C) 11 D) 12 31. What is the thousands digit of the product 1234560 × 2345670 × 3456780?A) 8B) 6C) 5D) 032. The sum of nine consecutive positive integers is always divisible byA) 2B) 5C) 7D) 933. You can put as many as 96 books in 6 backpacks. How many backpacks are necessary for144 books?A) 7B) 8C) 9D) 1034. The number of nickels I have is twice the number of dimes I have, and together thesecoins are worth more than $1. The least number of dimes that I can have isA) 5B) 6C) 8D) 1035. The ages of four kids are four consecutive positive integers. The product of their ages is3024. How old is the oldest kid?A) 8B) 9C) 10D) 1136. In the Game of Life, you earn 3 points for flipping a coin to “heads”, and 5 points forflipping a coin to “tails”. In all, how many positive whole number scores are IMPOSSIBLE to get after flipping it one or more times?A) 4B) 5C) 7D) 1137. Four monkeys can eat four bags of peanuts in three minutes. How many monkeys will ittake to eat 100 bags of peanuts in one hour?A) 4 B) 5 C) 20 D) 100 38. The tens digit of the product of the first 100 positive integers isA) 2B) 4C) 8D) 039. Someone put three dimes into my pile of quarters. If I add up the value of these coins,including the dimes, the sum could beA) $6.25B) $7.75C) $8.05D) $9.5040. Brooke's empty tub fills in 20 minutes with the drain plugged, andher full tub drains in 10 minutes with the water off. How manyminutes would it take the full tub to drain while the water is on?A) 12B) 15 C) 20 D) 30。
2018年度国外数学竞赛试题翻译汇编
(升级版)
赵力 2019 - 06 - 19
时间，就像高铁，一眨眼，就过站了……
人生很简单 总有一些风景，注定要错过 与其执着，不如随缘 只要懂得“珍惜、知足、感恩”就可以了！
笑看世事繁华，淡定人生心态 不索不可取，不求不可得 学会感恩，做人做事，无憾我心 不再奢望浮华之梦，不再……
v
纯属公益, 免费使用分享, 只送不卖
2018 年亚太地区数学奥林匹克试题
时间，一点不像高铁，过了站，居然买不到回来的车票！
生命，不就如一场雨吗 你曾无知地在其间雀跃，曾痴迷地在其间沉吟 但更多时候 你得忍受那些寒冷与潮湿，那些无奈与寂寞 并且以晴日的幻想来度日
当你握紧双手，里面什么也没有 当你打开双手，世界就在你手中
纯属公益, 免费使用分享, 只送不卖
目录
2018 年亚太地区数学奥林匹克 ……………………………………… 1 2018 年波罗的海地区数学奥林匹克 ………………………………… 2 2018 年第 10 届 Benelux 数学奥林匹克 ……………………………… 5 2018 年巴尔干地区数学奥林匹克 …………………………………… 6 2018 年巴尔干地区数学奥林匹克预选题…………………………… 7 2018 年巴尔干地区初中数学奥林匹克 ……………………………… 10 2018 年高加索地区数学奥林匹克 …………………………………… 11 2018 年中美洲及加勒比地区数学奥林匹克 ………………………… 13 2018 年 Cono Sur 数学奥林匹克 ……………………………………… 14 2018 年捷克-波兰-斯洛伐克联合数学竞赛 ………………………… 15 2018 年捷克和斯洛伐克数学奥林匹克 ……………………………… 16 2018 年多瑙河地区数学奥林匹克 …………………………………… 17 2018 年欧洲女子数学奥林匹克 ……………………………………… 19 2018 年欧洲数学杯奥林匹克 ………………………………………… 21 2018 年拉丁美洲数学奥林匹克 ……………………………………… 23 2018 年国际大都市数学竞赛(IOM) ………………………………… 24 2018 年第 2 届 IMO 复仇赛 …………………………………………… 25 2018 年第 5 届伊朗几何奥林匹克 …………………………………… 26 2018 年第 17 届基辅数学节竞赛 …………………………………… 30 2018 年地中海地区数学竞赛 ………………………………………… 32 2018 年中欧数学奥林匹克 …………………………………………… 33 2018 年北欧数学奥林匹克 …………………………………………… 35 2018 年泛非数学奥林匹克 …………………………………………… 36 2018 年泛非数学奥林匹克预选题 …………………………………… 38 2018 年罗马尼亚大师杯数学奥林匹克 ……………………………… 42
AIME(美国数学邀请赛)(2000-2018)答案
II
题号
1
2
3
4
5
6
7
8
答案
108
107
265
180
182
275
840
728
题号
9
10
11
12
13
14
15
答案
262
043
749
732
371
450
863
2017
I
题号
1
2
3
4
5
6
7
8
答案
390
062
069
803
321
048
564
041
题号
9
10
11
12
13
14
15
答案
045
056
360
252
059
4
5
6
7
8
答案
334
076
720
447
420
167
021
254
题号
9
10
11
12
13
14
15
答案
581
147
056
399
028
077
149
2015
I
题号
1
2
3
4
5
6
7
8
答案
722
139
307
507
341
058
539
695
题号
9
10
11
12
13
14
15
美国数学竞赛AMC题目及答案
2.3.What is the value of ?4.Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra $2.50 to cover her portion of the total bill. What was the total bill?5.Hammie is in the grade and weighs 106 pounds. His quadruplet sisters are tiny babies and weigh 5, 5, 6, and 8 pounds. Which is greater, the average (mean) weight of these five children or the median weight, and by how many pounds?6.The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, . What is the missing number in the top row?7.Trey and his mom stopped at a railroad crossing to let a train pass. As the train began to pass, Trey counted 6 cars in the first 10 seconds. It took the train 2 minutes and 45 seconds to clear the crossing at a constant speed. Which of the following was the most likely number of cars in the train?8.A fair coin is tossed 3 times. What is the probability of at least two consecutive heads?9.The Incredible Hulk can double the distance he jumps with each succeeding jump. If his first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then on which jump will he first be able to jump more than 1 kilometer?10.What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594?11. Ted's grandfather used his treadmill on 3 days this week. He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour. He walked at the rate of 3 miles per hour on Wednesday and at 4 miles per hour on Friday. If Grandfather had always walked at 4 miles per hour, he would have spent less time on the treadmill. How many minutes less?12. At the 2013 Winnebago County Fair a vendor is offering a "fair special" on sandals. If you buy one pair of sandals at the regular price of $50, you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals. What percentage of the $150 regular price did he save?13. When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of the following might her incorrect sum have differed from the correct one?14. Let the two digits be and .The correct score was . Clara misinterpreted it as . The difference between the two is which factors into . Therefore, since the difference is a multiple of 9, the only answerchoice that is a multiple of 9 is .15. If , , and , what is the product of , , and ?16. A number of students from Fibonacci Middle School are taking part in a community service project. The ratio of -graders to -graders is , and the the ratio of -graders to -graders is . What is the smallest number of students that could be participating in the project?17. The sum of six consecutive positive integers is 2013. What is the largest of these six integers?18. Isabella uses one-foot cubical blocks to build a rectangular fort that is 12 feet long, 10 feet wide, and 5 feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?--Arpanliku 16:22, 27 November 2013 (EST) Courtesy of Lord.of.AMC19. Bridget, Cassie, and Hannah are discussing the results of their last math test. Hannah shows Bridget and Cassie her test, but Bridget and Cassie don't show theirs to anyone. Cassie says, 'I didn't get the lowest score in our class,' and Bridget adds, 'I didn't get the highest score.' What is the ranking of the three girls from highest to lowest?20. A rectangle is inscribed in a semicircle with longer side on the diameter. What is the area of the semicircle?21. Samantha lives 2 blocks west and 1 block south of the southwest corner of City Park. Her school is 2 blocks east and 2 blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the northeast corner, and then bikes on streets to school. If her route is as short as possible, how many different routes can she take?22. Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicks are used altogether?23. Angle of is a right angle. The sides of are the diameters of semicircles as shown. The area of the semicircle on equals , and the arc of the semicircle on has length .What is the radius of the semicircle on ?24. Squares , , and are equal in area. Points and are the midpoints of sidesand , respectively. What is the ratio of the area of the shaded pentagon to the sum of the areas of the three squares?25. A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are inches, inches, and inches, respectively. The ball always remains in contact with the track and does not slip. What is the distance the center of theball travels over the course from A to B?1.2.The 50% off price of half a pound of fish is $3, so the 100%, or the regular price, of a half pound of fish is $6. Consequently, if half a pound of fish costs $6, then a whole pound of fish is dollars.3.Notice that we can pair up every two numbers to make a sum of 1:Therefore, the answer is .4.Each of her seven friends paid to cover Judi's portion. Therefore, Judi's portion must be . Since Judi was supposed to pay of the total bill, the total bill must be .5.The median here is obviously less than the mean, so option (A) and (B) are out.Lining up the numbers (5, 5, 6, 8, 106), we see that the median weight is 6 pounds.The average weight of the five kids is .Therefore, the average weight is bigger, by pounds, making the answer.6.Solution 1: Working BackwardsLet the value in the empty box in the middle row be , and the value in the empty box in the top row be . is the answer we're looking for.We see that , making .It follows that , so .Solution 2: Jumping Back to the StartAnother way to do this problem is to realize what makes up the bottommost number. This method doesn't work quite as well for this problem, but in a larger tree, it might be faster. (In this case, Solution 1 would be faster since there's only two missing numbers.)Again, let the value in the empty box in the middle row be , and the value in the empty box in the top row be . is the answer we're looking for.We can write some equations:Now we can substitute into the first equation using the two others:7.If Trey saw , then he saw .2 minutes and 45 seconds can also be expressed as seconds.Trey's rate of seeing cars, , can be multiplied by on the top and bottom (and preserve the same rate):. It follows that the most likely number of cars is . Solution 2minutes and seconds is equal to .Since Trey probably counts around cars every seconds, there are groups of cars that Trey most likely counts. Since , the closest answer choice is .8.First, there are ways to flip the coins, in order.The ways to get two consecutive heads are HHT and THH.The way to get three consecutive heads is HHH.Therefore, the probability of flipping at least two consecutive heads is .9.This is a geometric sequence in which the common ratio is 2. To find the jump that would be over a 1000 meters, we note that .However, because the first term is and not , the solution to the problem is10. To find either the LCM or the GCF of two numbers, always prime factorize first.The prime factorization of .The prime factorization of .Then, find the greatest power of all the numbers there are; if one number is one but not the other, use it (this is ). Multiply all of these to get 5940.For the GCF of 180 and 594, use the least power of all of the numbers that are in both factorizations and multiply. = 18.Thus the answer = = .We start off with a similar approach as the original solution. From the prime factorizations, the GCF is .It is a well known fact that . So we have,.Dividing by yields .Therefore, .11. We use that fact that . Let d= distance, r= rate or speed, and t=time. In this case, let represent the time.On Monday, he was at a rate of . So, .For Wednesday, he walked at a rate of . Therefore, .On Friday, he walked at a rate of . So, .Adding up the hours yields + + = .We now find the amount of time Grandfather would have taken if he walked at per day. Set up the equation, .To find the amount of time saved, subtract the two amounts: - = . To convert this to minutes, we multiply by .Thus, the solution to this problem is12. First, find the amount of money one will pay for three sandals without the discount. We have.Then, find the amount of money using the discount: .Finding the percentage yields .To find the percent saved, we have13. Let the two digits be and .The correct score was . Clara misinterpreted it as . The difference between the two is which factors into . Therefore, since the difference is a multiple of 9, the only answerchoice that is a multiple of 9 is .14. The probability that both show a green bean is . The probability that both show a red bean is . Therefore the probability is15.Therefore, .Therefore, .To most people, it would not be immediately evident that , so we can multiply 6's until we get the desired number:, so .Therefore the answer is .16. Solution 1: AlgebraWe multiply the first ratio by 8 on both sides, and the second ratio by 5 to get the same number for 8th graders, in order that we can put the two ratios together:Therefore, the ratio of 8th graders to 7th graders to 6th graders is . Since the ratio is in lowest terms, the smallest number of students participating in the project is .Solution 2: FakesolvingThe number of 8th graders has to be a multiple of 8 and 5, so assume it is 40 (the smallest possibility). Then there are 6th graders and 7th graders. The numbers of students is17. Solution 1The mean of these numbers is . Therefore the numbers are, so the answer isSolution 2Let the number be . Then our desired number is .Our integers are , so we have that.Solution 3Let the first term be . Our integers are . We have,18. Solution 1There are cubes on the base of the box. Then, for each of the 4 layers above the bottom (as since each cube is 1 foot by 1 foot by 1 foot and the box is 5 feet tall, there are 4 feet left), there arecubes. Hence, the answer is .Solution 2We can just calculate the volume of the prism that was cut out of the original box. Each interior side of the fort will be feet shorter than each side of the outside. Since the floor is foot, the height will be feet. So the volume of the interior box is .The volume of the original box is . Therefore, the number of blocks contained inthe fort is .19. If Hannah did better than Cassie, there would be no way she could know for sure that she didn't get the lowest score in the class. Therefore, Hannah did worse than Cassie. Similarly, if Hannah did worse than Bridget, there is no way Bridget could have known that she didn't get the highest in the class.Therefore, Hannah did better than Bridget, so our order is .20.A semicircle has symmetry, so the center is exactly at the midpoint of the 2 side on the rectangle, making the radius, by the Pythagorean Theorem, . The area is .21.The number of ways to get from Samantha's house to City Park is , and the number of ways toget from City Park to school is . Since there's one way to go through City Park (just walking straight through), the number of different ways to go from Samantha's house to City Park to school .22. There are vertical columns with a length of toothpicks, and there are horizontal rows with a length of toothpicks. An effective way to verify this is to try a small case, i.e. a grid of toothpicks.Thus, our answer is .23. Solution 1If the semicircle on AB were a full circle, the area would be 16pi. Therefore the diameter of the first circle is 8. The arc of the largest semicircle would normally have a complete diameter of 17. The Pythagoreantheorem says that the other side has length 15, so the radius is .Solution 2We go as in Solution 1, finding the diameter of the circle on AC and AB. Then, an extended version of the theorem says that the sum of the semicircles on the left is equal to the biggest one, so the area of thelargest is , and the middle one is , so the radius is .24.First let (where is the side length of the squares) for simplicity. We can extend until it hits theextension of . Call this point . The area of triangle then is The area of rectangle is . Thus, our desired area is . Now, the ratio of the shaded area to thecombined area of the three squares is .Let the side length of each square be .Let the intersection of and be .Since , . Since and are vertical angles, they are congruent. We also have by definition.So we have by congruence. Therefore, .Since and are midpoints of sides, . This combined with yields.The area of trapezoid is .The area of triangle is .So the area of the pentagon is .The area of the squares is .Therefore, .Let the intersection of and be .Now we have and .Because both triangles has a side on congruent squares therefore .Because and are vertical angles .Also both and are right angles so .Therefore by AAS(Angle, Angle, Side) .Then translating/rotating the shaded into the position ofSo the shaded area now completely covers the squareSet the area of a square asTherefore, .25. Solution 1The radius of the ball is 2 inches. If you think about the ball rolling or draw a path for the ball (see figure below), you see that in A and C it loses inches, and it gains inches on B.So, the departure from the length of the track means that the answer is . Solution 2The total length of all of the arcs is . Since we want the path from the center, the actual distance will be shorter. Therefore, the only answer choice less than is . Thissolution may be invalid because the actual distance can be longer if the path the center travels is on the outside of the curve, as it is in the middle bump.古希腊哲学大师亚里士多德说：人有两种，一种即“吃饭是为了活着”,一种是“活着是为了吃饭”.一个人之所以伟大，首先是因为他有超于常人的心。
美国数学竞赛AMC题目及答案
2.3.Whatisthevalueof4.5.Hammieisinthegradeandweighs106pounds.Hisquadrupletsistersaretinybabiesandweigh5,5,6,and8pounds.W hichisgreater,theaverage(mean)weightofthesefivechildrenorthemedianweight,andbyhowman ypounds6.Thenumberineachboxbelowistheproductofthenumbersinthetwoboxesthattouchitintherowab ove.Forexample,.Whatisthemissingnumberinthetoprow7.8.Afaircoinistossed3times.Whatistheprobabilityofatleasttwoconsecutiveheads9.TheIncredibleHulkcandoublethedistancehejumpswitheachsucceedingjump.Ifhisfirstjumpis1 meter,thesecondjumpis2meters,thethirdjumpis4meters,andsoon,thenonwhichjumpwillhefirst beabletojumpmorethan1kilometer10.Whatistheratiooftheleastcommonmultipleof180and594tothegreatestcommonfactorof180a nd59411.12.Atthe2013WinnebagoCountyFairavendorisofferinga"fairspecial"onsandals.Ifyoubuyonepai rofsandalsattheregularpriceof$50,yougetasecondpairata40%discount,andathirdpairathalfthe regularprice.Javiertookadvantageofthe"fairspecial"tobuythreepairsofsandals.Whatpercentag eofthe$150regularpricedidhesave13.WhenClaratotaledherscores,sheinadvertentlyreversedtheunitsdigitandthetensdigitofones core.Bywhichofthefollowingmightherincorrectsumhavedifferedfromthecorrectone14.Letthetwodigitsbe and.Thecorrectscorewas.Claramisinterpreteditas.Thedifferencebetweenthetwoiswhichfactorsinto.Therefore,sincethedifferenceisamultipleof9,theonlyanswerchoicethatisamultipleof9 is.15.If,,and,whatistheproductof,,and16.AnumberofstudentsfromFibonacciMiddleSchoolaretakingpartinacommunityserviceproject.Theratioof-gradersto-gradersis,andthetheratioof-gradersto-gradersis .Whatisthesmallestnumberofstudentsthatcouldbeparticipatingintheproject17.Thesumofsixconsecutivepositiveintegersis2013.Whatisthelargestofthesesixintegers18.--Arpanliku16:22,27November2013(EST)Courtesyof19.Bridget,Cassie,andHannaharediscussingtheresultsoftheirlastmathtest.HannahshowsBridg etandCassiehertest,butBridgetandCassiedon'tshowtheirstoanyone.Cassiesays,'Ididn'tgetthel owestscoreinourclass,'andBridgetadds,'Ididn'tgetthehighestscore.'Whatistherankingofthethr eegirlsfromhighesttolowest20.Arectangleisinscribedinasemicirclewithlongersideonthediameter.Whatistheareaofthesemicircle21.22.Toothpicksareusedtomakeagridthatis60toothpickslongand32toothpickswide.Howmanytoo thpicksareusedaltogether23.Angle of isarightangle.Thesidesofarethediametersofsemicirclesasshown.Theareaofthesemicircleon equals,andthearcofthesemicircleon haslength.Whatistheradiusofthesemicircleon24.Squares,,and areequalinarea.Points andarethemidpointsofsides and,respectively.Whatistheratiooftheareaoftheshadedpentagontothesumoftheareasofthethreesquares25.Aballwithdiameter4inchesstartsatpointAtorollalongthetrackshown.Thetrackiscomprisedof3semicirculararcswhoseradiiare inches,inches,and1.2.The50%offpriceofhalfapoundoffishis$3,sothe100%,ortheregularprice,ofahalfpoundoffishis$6.Consequently,ifhalfapoundoffishcosts$6,thenawholepoundoffishis dollars.3.Noticethatwecanpairupeverytwonumberstomakeasumof1:Therefore,theansweris.4.Eachofhersevenfriendspaid tocoverJudi'sportion.Therefore,Judi'sportionmustbe.SinceJudiwassupposedtopay ofthetotalbill,thetotalbillmustbe.5.Themedianhereisobviouslylessthanthemean,sooption(A)and(B)areout. Liningupthenumbers(5,5,6,8,106),weseethatthemedianweightis6pounds.Theaverageweightofthefivekidsis.Therefore,theaverageweightisbigger,by pounds,makingtheanswer.6.Solution1:WorkingBackwardsLetthevalueintheemptyboxinthemiddlerowbe,andthevalueintheemptyboxinthetoprowbe. istheanswerwe'relookingfor.Weseethat,making.Itfollowsthat,so.Solution2:JumpingBacktotheStartAnotherwaytodothisproblemistorealizewhatmakesupthebottommostnumber.Thismethoddoes n'tworkquiteaswellforthisproblem,butinalargertree,itmightbefaster.(Inthiscase,Solution1wou ldbefastersincethere'sonlytwomissingnumbers.)Again,letthevalueintheemptyboxinthemiddlerowbe,andthevalueintheemptyboxinthetoprowbe.istheanswerwe'relookingfor. Wecanwritesomeequations: Nowwecansubstituteintothefirstequationusingthetwoothers:7.IfTreysaw,thenhesaw.2minutesand45secondscanalsobeexpressedas seconds.Trey'srateofseeingcars,,canbemultipliedbyonthetopandbottom(andpreservethesamerate):.Itfollowsthatthemostlikelynumberofcarsis. Solution2minutesand secondsisequalto.SinceTreyprobablycountsaround carsevery seconds,thereare groupsofcarsthatTreymostlikelycounts.Since,theclosestanswerchoiceis.8.First,thereare waystoflipthecoins,inorder. ThewaystogettwoconsecutiveheadsareHHT andTHH. ThewaytogetthreeconsecutiveheadsisHHH.Therefore,theprobabilityofflippingatleasttwoconsecutiveheadsis.9.Thisisageometricsequenceinwhichthecommonratiois2.Tofindthejumpthatwouldbeovera1000meters,wenotethat.However,becausethefirsttermis andnot,thesolutiontotheproblemis10.TofindeithertheLCMortheGCFoftwonumbers,alwaysprimefactorizefirst. Theprimefactorizationof.Theprimefactorizationof.Then,findthegreatestpowerofallthenumbersthereare;ifonenumberisonebutnottheother,useit(t hisis).Multiplyallofthesetoget5940.FortheGCFof180and594,usetheleastpowerofallofthenumbersthatareinbothfactorizationsandmultiply.=18.Thustheanswer==. Westartoffwithasimilarapproachastheoriginalsolution.Fromtheprimefactorizations,theGCFis .Itisawellknownfactthat.Sowehave,.Dividingby yields.Therefore,.11.Weusethatfactthat.Letd=distance,r=rateorspeed,andt=time.Inthiscase,letrepresentthetime.OnMonday,hewasatarateof.So,.ForWednesday,hewalkedatarateof.Therefore,.OnFriday,hewalkedatarateof.So,.Addingupthehoursyields++=.WenowfindtheamountoftimeGrandfatherwouldhavetakenifhewalkedatperday.Setuptheequation,.Tofindtheamountoftimesaved,subtractthetwoamounts:-=.Toconvertthistominutes,wemultiplyby.Thus,thesolutiontothisproblemis12.First,findtheamountofmoneyonewillpayforthreesandalswithoutthediscount.Wehave.Then,findtheamountofmoneyusingthediscount:.Findingthepercentageyields.Tofindthepercentsaved,wehave13.Letthetwodigitsbe and.Thecorrectscorewas.Claramisinterpreteditas.Thedifferencebetweenthetwoiswhichfactorsinto.Therefore,sincethedifferenceisamultipleof9,theonlyanswerchoicethatisamultipleof9is.14.Theprobabilitythatbothshowagreenbeanis.Theprobabilitythatbothshowaredbeanis.Thereforetheprobabilityis15.Therefore,.Therefore,.Tomostpeople,itwouldnotbeimmediatelyevidentthat,sowecanmultiply6'suntilwegetthedesirednumber:,so.Thereforetheansweris.16.Solution1:AlgebraWemultiplythefirstratioby8onbothsides,andthesecondratioby5togetthesamenumberfor8thgra ders,inorderthatwecanputthetworatiostogether:Therefore,theratioof8thgradersto7thgradersto6thgradersis.Sincetheratioisinlowestterms,thesmallestnumberofstudentsparticipatingintheprojectis.Solution2:FakesolvingThenumberof8thgradershastobeamultipleof8and5,soassumeitis40(thesmallestpossibility).Thenthereare6thgradersand7thgraders.Thenumbersofstudentsis17.Solution1Themeanofthesenumbersis.Thereforethenumbersare,sotheanswerisSolution2Letthe numberbe.Thenourdesirednumberis.Ourintegersare,sowehavethat.Solution3Letthefirsttermbe.Ourintegersare.Wehave,18.Solution1Therearecubesonthebaseofthebox.Then,foreachofthe4layersabovethebottom(assinceeachcubeis1foot by1footby1footandtheboxis5feettall,thereare4feetleft),therearecubes.Hence,theansweris.Solution2Wecanjustcalculatethevolumeoftheprismthatwascutoutoftheoriginalbox.Eachinteriorsideofthefortwillbe feetshorterthaneachsideoftheoutside.Sincethefloorisfoot,theheightwillbe feet.Sothevolumeoftheinteriorboxis.Thevolumeoftheoriginalboxis.Therefore,thenumberofblockscontainedinthefortis.19.IfHannahdidbetterthanCassie,therewouldbenowayshecouldknowforsurethatshedidn'tgett helowestscoreintheclass.Therefore,HannahdidworsethanCassie.Similarly,ifHannahdidworseth anBridget,thereisnowayBridgetcouldhaveknownthatshedidn'tgetthehighestintheclass.Therefore,HannahdidbetterthanBridget,soourorderis.20.Asemicirclehassymmetry,sothecenterisexactlyatthemidpointofthe2sideontherectangle,makingtheradius,bythePythagoreanTheorem,.Theareais.21.ThenumberofwaystogetfromSamantha'shousetoCityParkis,andthenumberofwaystogetfromCityParktoschoolis.Sincethere'sonewaytogothroughCityPark(justwalkingstraightthrough),thenumberofdifferentwaystogofromSamantha'shousetoCityParktoschool.22.Thereare verticalcolumnswithalengthof toothpicks,andthereare horizontalrowswithalengthof toothpicks.Aneffectivewaytoverifythisistotryasmallcase,i.e.agridoftoothpicks.Thus,ouransweris.23.Solution1.Solution2WegoasinSolution1,findingthediameterofthecircleonACandAB.Then,anextendedversionofthet heoremsaysthatthesumofthesemicirclesontheleftisequaltothebiggestone,sotheareaofthelargestis,andthemiddleoneis,sotheradiusis.24.Firstlet(where isthesidelengthofthesquares)forsimplicity.Wecanextenduntilithitstheextensionof.Callthispoint.Theareaoftriangle thenis Theareaofrectangle is.Thus,ourdesiredareais.Now,theratiooftheshadedareatothecombinedareaofthethreesquaresis.Solution2Letthesidelengthofeachsquarebe.Lettheintersectionof and be.Since,.Since andareverticalangles,theyarecongruent.Wealsohave bydefinition.Sowehave by congruence.Therefore,.Since and aremidpointsofsides,.Thiscombinedwith yields.Theareaoftrapezoid is.Theareaoftriangle is.Sotheareaofthepentagon is.Theareaofthe squaresis.Therefore,.Solution3Lettheintersectionof and be.Nowwehave and.Becausebothtriangleshasasideoncongruentsquarestherefore.Because and areverticalangles.Alsoboth and arerightanglesso.ThereforebyAAS(Angle,Angle,Side).Thentranslating/rotatingtheshaded intothepositionofSotheshadedareanowcompletelycoversthesquareSettheareaofasquareasTherefore,.25.Solution1Theradiusoftheballis2inches.Ifyouthinkabouttheballrollingordrawapathfortheball(seefigurebelow),youseethatinAandCitloses inches,anditgains inchesonB.So,thedeparturefromthelengthofthetrackmeansthattheansweris.Solution2Thetotallengthofallofthearcsis.Sincewewantthepathfromthecenter,theactualdistancewillbeshorter.Therefore,theonlyanswerchoicelessthan is.Thissolutionmaybeinvalidbecausetheactualdistancecanbelongerifthepaththecent ertravelsisontheoutsideofthecurve,asitisinthemiddlebump.。
amc 历年试题
amc 历年试题
AMC 历年试题指的是美国数学竞赛（American Mathematics Competition，简称AMC）过去举办过的所有考试的试题集合。
AMC是美国的一项数学竞赛，旨在发现和培养数学人才，提高数学教育水平。
以下是3道AMC历年试题的示例：
1.题目：如果一个四边形的周长是20，且其中一边的长度是5，那么这个四
边形的面积最大是多少？
2.题目：若对于所有x和y，都有|f(x)| = |g(y)|，并且f和g都有无穷积分。
如果对于某个固定的x值，都有f(x) = ng(n) (a)，那么对于所有的x值，都有f(x) = ng(n) (b)，其中n是一个整数。
3.题目：当0 < b < a < 1时，以下哪个不等式是正确的？
A. log(a + b) < log(2a)
B. log(a + b) < log(2b)
C. log(2b) < log(a + b) < log(2a)
D. log(2a) < log(a + b) < log(2b)
AMC历年试题是对考生数学能力和思维能力的全面检验，能够帮助考生提升自己的数学水平和应试能力。
这些试题也有助于发现和培养数学人才，为数学教育和研究做出贡献。
2018数学建模美赛题目
2018数学建模美赛题目
2018年数学建模美赛题目如下：
A题是多跳高频（HF）广播传播问题。
在高频（HF，定义为3-30 MHz）上，无线电波可以通过电离层和地球以外的多次反射传播很长的距离。
对于低于最大可用频率（MUF）的频率，来自地面源的HF无线电波从电离层反射回地球，在此它们可能再次反射回电离层，在此又可能反射回地球，依此类推。
随着每个连续的跃点走得更远。
除其他因素外，反射表面的特性决定了反射波的强度以及信号最终传播的距离，同时又保持了有用的信号完整性。
而且，MUF随季节、一天中的时间和太阳条件而变化。
MUF上方的频率不会被反射/折射，而是通过电离层进入太空。
在这个问题中，重点特别放在
海洋表面的反射上。
C题是能源生产问题。
能源生产和使用是任何经济结构的主要部分。
这个问题提供了一个数据集，包含了美国四个州在过去50年的能源生产和消费数据，以及一些人口和经济信息。
参赛者需要使用这些数据来回答以下问题：这四个州的能源生产和消费模式有何不同？这些模式如何随时间变化？这些模式如何与人口和经济活动相关联？如何预测未来的能源生产和消费趋势？如何优化能源生产和消费以减少对环境的影响？
以上内容仅供参考，建议查阅美赛官网获取更准确的信息。
2018年美国“数学大联盟杯赛”(中国赛区)初赛五年级试卷
2017-2018年度美国“数学大联盟杯赛”(中国赛区)初赛(五年级)(初赛时间：2017年11月26日，考试时间90分钟，总分200分)学生诚信协议：考试期间，我确定没有就所涉及的问题或结论，与任何人、用任何方式交流或讨论， 我确定我所填写的答案均为我个人独立完成的成果，否则愿接受本次成绩无效的处罚。
请在装订线内签名表示你同意遵守以上规定。
考前注意事项：1. 本试卷是五年级试卷，请确保和你的参赛年级一致；2. 本试卷共4页(正反面都有试题)，请检查是否有空白页，页数是否齐全；3. 请确保你已经拿到以下材料：本试卷(共4页，正反面都有试题)、答题卡、答题卡使用说明、英文词汇手册、 草稿纸。
考试完毕，请务必将英文词汇手册带回家，上面有如何查询初赛成绩、 及如何参加复赛的说明。
其他材料均不能带走，请留在原地。
选择题：每小题5分，答对加5分，答错不扣分，共200分，答案请填涂在答题卡上。
1. The smallest possible sum of two different prime numbers isA) 3B) 4C) 5D) 62. The greatest common factor of two numbers is3. The product of these two numbers mustbe divisible byA) 6 B) 9 C) 12 D) 18 3. The sum of 5 consecutive one-digit integers is at most A) 15 B) 25 C) 35 D) 45 4. How many two-digit multiples of 10 are also multiples of 12?A) 4B) 3C) 2D) 15. I have read exactly13of the total number of chapters in my 120-page book. If each chapter has the same whole number of pages, then the total number of chapters I have left could beA) 16 B) 24 C) 32 D) 50 6. What is the greatest odd factor of 44 × 55 × 66?A) 36 B) 55 C) 35 × 55 D) 36 × 55 7. What is the sum of the factors of the prime number 2017? A) 2016B) 2017C) 2018D) 20198. Lynn ran in 6 times as many races as the number of racesshe won. How many of her 126 races did Lynn not win?A) 21B) 90C) 96D) 1059. The least common multiple of 8 and 12 is the greatest common factor of 120 andA) 80B) 124C) 144D) 18010. January has the greatest possible number of Saturdays when January 1 occurs on any ofthe following days of the week exceptA) Thursday B) Friday C) Saturday D) Sunday 11. The number that is 10% of 1000 is 10 more than 10% ofA) 90B) 100C) 900D) 99012. The sum of 16 fours has the same value as the product of ? fours.A) 2 B) 3 C) 4 D) 16 13. Of the following, which is the sum of two consecutive integers?A) 111 111B) 222 222C) 444 444D) 888 88814. Abe drove for 2 hours at 30 km/hr. and for 3 hours at 50 km/hr. What was Abe’s averagespeed over the 5 hours?A) 35 km/hr.B) 40 km/hr.C) 42 km/hr.D) 45 km/hr.15. My broken watch runs twice as fast as it should. If my watch first broke at 6:15 P.M.,what time was displayed on my watch 65 minutes later?A) 7:20 P.M. B) 7:25 P.M.C) 8:20 P.M. D) 8:25 P.M.16. (2018 × 2017) + (2018 × 1) =A) 20172 B) 20182 C) 20183D) (2018 + 2017)217. A prized bird lays 2, 3, or 4 eggs each day. If the bird laid 17 eggs in 1 week,on at most how many days that week did the bird lay exactly 2 eggs?A) 2B) 3C) 4D) 518. Of the following, which could be the perimeter of a rectangle whoseside-lengths, in cm, are prime numbers?A) 10 cmB) 22 cmC) 34 cmD) 58 cm19. The average of all possible total values of a 4-coin stack of nickels and dimes (containingat least one of each coin) isA) 20¢B) 30¢C) 40¢D) 60¢20. The diameter of Ann’s drum i s 40 cm more than the radius. What is half the circumference of the drum?A) 120π cmB) 80π cmC) 60π cmD) 40π cm21. Of the following, which expression has the greatest number offactors that are multiples of 2018?A) 2018 × 12B) 20182C) 20192D) 20192019第1页，共4页 第2页，共4页22. When the sum of the factors of a prime number is divided by that prime number, theremainder isA) 0 B) 1 C) 2 D) 3 23. What is the sum of the digits of the greatest integer that has a square root less than 100? A) 18B) 36C) 99D) 10024. My favorite number has 6 different factors. If the product of all 6 factors is 123, what isthe sum of the factors of my favorite number?A) 24B) 28C) 32D) 3625. For how many different pairs of unequal positive integers less than 10 is the least commonmultiple of the numbers less than their product?A) 6B) 7C) 8D) 926. Exactly 12 of the students in my class have at least one brother, and 12 have at least onesister. If 13have no siblings, what fraction of the students in my class have at least onebrother and at least one sister?A) 16 B) 15 C) 14 D) 1327. Each day, Sal swims a lap 1 second faster than on the daybefore. If Sal swims a lap in 60 minutes on the 1st day, on what day does he swim a lap in 10% less time than the 1st day?A) 359th B) 360th C) 361st D) 362nd 28. 20172018 × 20172019 = 2017 ? × 20171009A) 1010B) 2010C) 3028D) 403829. Both arcs AB and AD are quarter circles of radius 5, figure on the right.Arc BCD is a semi-circle of radius 5. What is the area of the region ABCD ?A) 25 B) 10 + 5π C) 50D) 50 + 5π30. For every $5 I earn from my job, I save $2. For every $4 I save from my job, I am givenan additional $1 from my parents to add to my savings. How much must I earn in order to have $40 in savings?A) $160B) $120C) $100D) $8031. In the figure on the right, the side-length of the smaller squareis 4. The four arcs are four semi-circles. Each side of square ABCD is tangent to one of the semi-circles. The area of ABCD isA) 32B) 36C) 48D) 6432. A million is a large number, a “1” follo wed by 6 zeros. A googol is a large number, a “1”followed by one hundred zeros. A googolplex is a large number, a “1” followed by a googol of zeros. A googolplexian is a large number, a “1” fo llowed by a googolplex of zeros. A googolplexian isA) 10100 B) 1001010C) 100101010D) None of the above33. An integral triangle is a triangle with positive integral side-lengths and a positive area.Such a triangle can have a perimeter as small as 3. What is the next smallest possible perimeter of an integral triangle?A) 4B) 5C) 6D) 734. 2 liter of 2% fat milk + 3 liter of 3% fat milk = 5 liter of ? fat milkA) 2.5%B) 2.6%C) 5%D) 6%35. One day, a motorist came to a hill that was ten-mile drive up one side and a ten-mile drivedown the other. He drove up the hill at an average speed of 30 miles per hour. How fast will he have to drive down the other side to average 60 miles per hour for the entire 20-mile distance?A) 30 miles per hour B) 60 miles per hour C) 90 miles per hour D) None of the above 36. What is the weight of a fish if it weighs ten pounds plus half its weight?A) 10B) 15C) 20D) 2537. Without using pennies, how many different combinations of coins (nickels, dimes,quarters) will make 30 cents?A) 3B) 4C) 5D) 638. A man once bought a fine suit for which he paid $30 more than14of its price. How much did he pay for the suit? A) $30B) $35C) $40D) $4539. A father is five times as old as his son. In fifteen years he will be only twice as old. Howold is the father at present?A) 40B) 35C) 30D) 2540. It takes 30 minutes to completely fill a tank. If, however, a hole allows13of the water that is entering the tank to escape, how long will it then take to fill the tank?A) 40 B) 45 C) 60 D) 90第3页，共4页第4页，共4页。
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
2018 AIME I ProblemsProblem 1Let be the number of ordered pairs ofintegers with and such that thepolynomial can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find the remainder when is divided by .Problem 2The number can be written in base as , can be written inbase as , and can be written in base as , where . Find the base- representation of .Problem 3Kathy has red cards and green cards. She shuffles the cards and laysout of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGGR, or RRRRR will make Kathy happy,but RRRGR will not. The probability that Kathy will be happy is ,where and are relatively prime positive integers. Find . Problem 4In and . Point lies strictlybetween and on and point lies strictly between and on sothat . Then can be expressed in the form ,where and are relatively prime positive integers. Find .Problem 5For each ordered pair of real numbers satisfyingthere is a real number such thatFind the product of all possible values of .Problem 6Let be the number of complex numbers with the propertiesthat and is a real number. Find the remainder when is divided by .Problem 7A right hexagonal prism has height . The bases are regular hexagons with side length . Any of the vertices determine a triangle. Find the number of these triangles that are isosceles (including equilateral triangles).Problem 8Let be an equiangular hexagon suchthat , and . Denote the diameter of the largest circle that fits inside the hexagon. Find .Problem 9Find the number of four-element subsets of with the propertythat two distinct elements of a subset have a sum of , and two distinct elements of a subset have a sum of . Forexample, and are two such subsets.Problem 10The wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point . At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the inner circle the bug only walks in a counterclockwise direction, and along the outer circle the bug only walks in a clockwise direction. For example, the bug could travel along thepath , which has steps. Let be the number of paths with steps that begin and end at point . Find the remainder when is divided by .Problem 11Find the least positive integer such that when is written in base , its two right-most digits in base are .Problem 12For every subset of , let be the sum of the elements of , with defined to be . If is chosen at random among allsubsets of , the probability that is divisible by is , where and are relatively prime positive integers. Find .Problem 13Let have side lengths , , and .Point lies in the interior of , and points and are the incentersof and , respectively. Find the minimum possible areaof as varies along .Problem 14Let be a heptagon. A frog starts jumping at vertex . From any vertex of the heptagon except , the frog may jump to either of the two adjacentvertices. When it reaches vertex , the frog stops and stays there. Find the number of distinct sequences of jumps of no more than jumps that end at .Problem 15David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, , which can each be inscribed in a circle with radius . Let denote the measure of the acute angle made by the diagonals of quadrilateral , and define and similarly. Supposethat , , and . All three quadrilaterals have thesame area , which can be written in the form , where and are relatively prime positive integers. Find .2018 AMC 8 ProblemsProblem 1An amusement park has a collection of scale models, with ratio , of buildings and other sights from around the country. The height of the United States Capitol is 289 feet. What is the height in feet of its replica to the nearest whole number?Problem 2What is the value of the productProblem 3Students Arn, Bob, Cyd, Dan, Eve, and Fon are arranged in that order in a circle. They start counting: Arn first, then Bob, and so forth. When the number contains a 7 as a digit (such as 47) or is a multiple of 7 that person leaves the circle and the counting continues. Who is the last one present in the circle?Problem 4The twelve-sided figure shown has been drawn on graph paper. What is the area of the figure in ?Problem 5What is the valueof ?Problem 6On a trip to the beach, Anh traveled 50 miles on the highway and 10 miles on a coastal access road. He drove three times as fast on the highway as on the coastal road. If Anh spent 30 minutes driving on the coastal road, how many minutes did his entire trip take?Problem 7The -digit number is divisible by . What is the remainder when this number is divided by ?Problem 8Mr. Garcia asked the members of his health class how many days last week they exercised for at least 30 minutes. The results are summarized in the following bar graph, where the heights of the bars represent the number of students.What was the mean number of days of exercise last week, rounded to the nearest hundredth, reported by the students in Mr. Garcia's class?Problem 9Tyler is tiling the floor of his 12 foot by 16 foot living room. He plans to place one-foot by one-foot square tiles to form a border along the edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. How many tiles will he use?Problem 10The of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4?Problem 11Abby, Bridget, and four of their classmates will be seated in two rows of three for a group picture, as shown.If the seating positions are assigned randomly, what is the probability that Abby and Bridget are adjacent to each other in the same row or the same column?Problem 12The clock in Sri's car, which is not accurate, gains time at a constant rate. One day as he begins shopping he notes that his car clock and his watch (which is accurate) both say 12:00 noon. When he is done shopping, his watch says 12:30 and his car clock says 12:35. Later that day, Sri loses his watch. He looks at his car clock and it says 7:00. What is the actual time?Problem 13Laila took five math tests, each worth a maximum of 100 points. Laila's score on each test was an integer between 0 and 100, inclusive. Laila received the same score on the first four tests, and she received a higher score on the last test. Her average score on the five tests was 82. How many values are possible for Laila's score on the last test?Problem 14Let be the greatest five-digit number whose digits have a product of . What is the sum of the digits of ?Problem 15In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of square unit, then what is the area of the shaded region, in square units?Problem 16Professor Chang has nine different language books lined up on a bookshelf: two Arabic, three German, and four Spanish. How many ways are there to arrange the nine books on the shelf keeping the Arabic books together and keeping the Spanish books together?Problem 17Bella begins to walk from her house toward her friend Ella's house. At the same time, Ella begins to ride her bicycle toward Bella's house. They each maintain a constant speed, and Ella rides 5 times as fast as Bella walks. The distancebetween their houses is miles, which is feet, and Bella covers feet with each step. How many steps will Bella take by the time she meets Ella?Problem 18How many positive factors does have?Problem 19In a sign pyramid a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. How many possible ways are there to fill the four cells in the bottom row to produce a "+" at the top of the pyramid?Problem 20In a point is on with and Point ison so that and point is on so that What is the ratio of the area of to the area ofProblem 21How many positive three-digit integers have a remainder of 2 when divided by 6, a remainder of 5 when divided by 9, and a remainder of 7 when divided by 11?Problem 22Point is the midpoint of side in square and meets diagonal at The area of quadrilateral is What is the areaofProblem 23From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon?Problem 24In the cube with opposite vertices and and are the midpoints of edges and respectively. Let be the ratio of the area of the cross-section to the area of one of the faces of the cube. What isProblem 25How many perfect cubes lie between and , inclusive?2018 AMC 10A ProblemsProblem 1What is the value ofProblem 2Liliane has more soda than Jacqueline, and Alice has more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alice have?Liliane has more soda than Alice.Liliane has more soda than Alice.Liliane has more soda than Alice.Liliane has more soda than Alice.Liliane has more soda than Alice.Problem 3A unit of blood expires after seconds. Yasin donates a unit of blood at noon of January 1. On what day does his unit of blood expire?Problem 4How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)Problem 5Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least 6 miles away," Bob replied, "We are at most 5 miles away." Charlie then remarked, "Actually the nearest town is at most 4 miles away." It turned out that none of the three statements were true. Let be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of ?Problem 6Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of 0, and the score increases by 1 for each like vote and decreases by 1 for each dislike vote. At one point Sangho saw that his video had a score of 90, and that of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point?Problem 7For how many (not necessarily positive) integer values of is the value of an integer?Problem 8Joe has a collection of 23 coins, consisting of 5-cent coins, 10-cent coins, and 25-cent coins. He has 3 more 10-cent coins than 5-cent coins, and the total value of his collection is 320 cents. How many more 25-cent coins does Joe have than 5-cent coins?Problem 9All of the triangles in the diagram below are similar to iscoceles triangle , inwhich . Each of the 7 smallest triangles has area 1, and has area 40. What is the area of trapezoid ?Problem 10Suppose that real number satisfies. What is the valueof ?Problem 11When fair standard -sided die are thrown, the probability that the sum of the numbers on the top faces is can be written as, where is a positive integer. What is ?Problem 12How many ordered pairs of real numbers satisfy the following system ofequations?Problem 13A paper triangle with sides of lengths 3, 4, and 5 inches, as shown, is folded so that point falls on point . What is the length in inches of the crease?Problem 14What is the greatest integer less than or equal toProblem 15Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points and , as shown in the diagram. The distance can be written in the form , where and are relatively prime positive integers. What is ?Problem 16Right triangle has leg lengths and . Including and , how many line segments with integer length can be drawn from vertex to a point on hypotenuse ?Problem 17Let be a set of 6 integers taken from with the property that if and are elements of with , then is not a multiple of . What is the least possible values of an element inProblem 18How many nonnegative integers can be written in theformwhere for ?Problem 19A number is randomly selected from the set , and a number is randomly selected from . What is the probabilitythat has a units digit of ?Problem 20A scanning code consists of a grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of squares. A scanning code is called if its look does not change when the entire square is rotated by a multiple of counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes?Problem 21Which of the following describes the set of values of for which thecurves and in the real -plane intersect at exactly points?Problem 22Let and be positive integers suchthat , , ,and . Which of the following must be a divisor of ?Problem 23Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplantedsquare so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from to the hypotenuse is 2 units. What fraction of the field is planted?Problem 24Triangle with and has area . Let be the midpointof , and let be the midpoint of . The angle bisectorof intersects and at and , respectively. What is the area of quadrilateral ?Problem 25For a positive integer and nonzero digits , , and , let be the -digit integer each of whose digits is equal to ; let be the -digit integer each of whose digits is equal to , and let be the -digit (not -digit) integer each of whose digits is equal to . What is the greatest possiblevalue of for which there are at least two values of such that ?2018 AMC 10B ProblemsProblem 1Kate bakes a 20-inch by 18-inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches. How many pieces of cornbread does the pan contain?Problem 2Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles per hour), and his average speed during the second 30 minutes was 65 mph. What was his average speed, in mph, during the last 30 minutes?Problem 3In the expression each blank is to be filled in with one of the digits or with each digit being used once. How many different values can be obtained?Problem 4A three-dimensional rectangular box with dimensions , , and has faces whose surface areas are 24, 24, 48, 48, 72, and 72 square units. What is ?Problem 5How many subsets of contain at least one prime number?Problem 6A box contains 5 chips, numbered 1, 2, 3, 4, and 5. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds 4. What is the probability that 3 draws are required?Problem 7In the figure below, congruent semicircles are drawn along a diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let be the combined area of the small semicircles and be the area of the region inside the large semicircle but outside the small semicircles. The ratio is 1:18. What is ?Problem 8Sara makes a staircase out of toothpicks as shown:This is a 3-step staircase and uses 18 toothpicks. How many steps would be in a staircase that used 180 toothpicks?Problem 9The faces of each of 7 standard dice are labeled with the integers from 1 to 6. Let be the probability that when all 7 dice are rolled, the sum of the numbers on the top faces is 10. What other sum occurs with the same probability ?Problem 10In the rectangular parallelepiped shown, , , and . Point is the midpoint of . What is the volume of the rectangular pyramid with base and apex ?Problem 11Which of the following expressions is never a prime number when is a prime number?Problem 12Line segment is a diameter of a circle with . Point , not equal to or , lies on the circle. As point moves around the circle, the centroid (center of mass) of traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?Problem 13How many of the first numbers in the sequence are divisible by ?Problem 14A list of positive integers has a unique mode, which occurs exactly times. What is the least number of distinct values that can occur in the list?Problem 15A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point in the figure on the right. The box has base length and height . What is the area of the sheet of wrapping paper?Problem 16Let be a strictly increasing sequence of positive integers suchthat What is the remainderwhen is divided by ?Problem 17In rectangle , and . Points and lie on ,points and lie on , points and lie on , and points and lie on so that and the convex octagon is equilateral. The length of a side of this octagon can be expressed in the form , where , , and are integers and is not divisible by the square of any prime. What is ?Problem 18Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no child may sit directly in front of his or her sibling. How many seating arrangements are possible for this trip?Problem 19Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1 year old today. Today is the first of the 9 birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?Problem 20A function is defined recursivelyby and for allintegers . What is ?Problem 21Mary chose an even -digit number . She wrote down all the divisors of in increasing order fromleft to right: . At some moment Mary wrote as a divisor of . What is the smallest possible value of the next divisor written to the right of ?Problem 22Real numbers and are chosen independently and uniformly at random from the interval . Which of the following numbers is closest to the probability that and are the side lengths of an obtuse triangle?Problem 23How many ordered pairs of positive integers satisfy theequation where denotes the greatest common divisor of and , and denotes their least common multiple?Problem 24Let be a regular hexagon with side length . Denote by , , and the midpoints of sides , , and , respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of and ?Problem 25Let denote the greatest integer less than or equal to . How many real numbers satisfy the equation ?2018 AMC 12A ProblemsProblem 1A large urn contains balls, of which are red and the rest are blue. How many of the blue balls must be removed so that the percentage of red balls in the urn will be ? (No red balls are to be removed.)Problem 2While exploring a cave, Carl comes across a collection of -pound rocks worth each, -poundrocks worth each, and -pound rocks worth each. There are at least of each size. He can carry at most pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?Problem 3How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)Problem 4Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least 6 miles away," Bob replied, "We are at most 5 miles away." Charlie then remarked, "Actually the nearest town is at most 4 miles away." It turned out that none of the three statements were true. Let be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of ?Problem 5What is the sum of all possible values of for which the polynomials andhave a root in common?Problem 6For positive integers and such that , both the mean and the median ofthe set are equal to . What is ?Problem 7For how many (not necessarily positive) integer values of is the value of an integer?Problem 8All of the triangles in the diagram below are similar to iscoceles triangle , in which. Each of the 7 smallest triangles has area 1, and has area 40. What is the area of trapezoid ?Problem 9Which of the following describes the largest subset of values of within the closed interval forwhich for every between and , inclusive?How many ordered pairs of real numbers satisfy the following system of equations?Problem 11A paper triangle with sides of lengths 3,4, and 5 inches, as shown, is folded so that point falls on point . What is the length in inches of the crease?Problem 12Let be a set of 6 integers taken from with the property that if and are elements of with , then is not a multiple of . What is the least possible value of an element inProblem 13How many nonnegative integers can be written in the formwherefor ?Problem 14The solutions to the equation , where is a positive real number other thanor , can be written as where and are relatively prime positive integers. What is ?A scanning code consists of a grid of squares, with some of its squares colored black and therest colored white. There must be at least one square of each color in this grid of squares. A scanning code is called if its look does not change when the entire square is rotated by a multiple of counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes?Problem 16Which of the following describes the set of values of for which the curves andin the real -plane intersect at exactly points?Problem 17Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplanted squareso that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from to the hypotenuse is 2 units. What fraction of the field is planted?Triangle with and has area . Let be the midpoint of, and let be the midpoint of . The angle bisector of intersects and atand , respectively. What is the area of quadrilateral ?Problem 19Let be the set of positive integers that have no prime factors other than , , or . The infinite sumof the reciprocals of the elements of can be expressed as , where and are relatively primepositive integers. What is ?Problem 20Triangle is an isosceles right triangle with . Let be the midpoint ofhypotenuse . Points and lie on sides and , respectively, so thatand is a cyclic quadrilateral. Given that triangle has area , the length canbe written as , where , , and are positive integers and is not divisible by the square of any prime. What is the value of ?Problem 21Which of the following polynomials has the greatest real root?Problem 22The solutions to the equations and whereform the vertices of a parallelogram in the complex plane. The area of thisparallelogram can be written in the form where and are positive integersand neither nor is divisible by the square of any prime number. What isProblem 23In and Points and lie on sidesand respectively, so that Let and be the midpoints of segmentsand respectively. What is the degree measure of the acute angle formed by linesandProblem 24Alice, Bob, and Carol play a game in which each of them chooses a real number between 0 and 1. The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between 0 and 1, and Bob announces that he will choose his number uniformly at random from all thenumbers between and Armed with this information, what number should Carol choose to maximize her chance of winning?Problem 25For a positive integer and nonzero digits , , and , let be the -digit integer each of whosedigits is equal to ; let be the -digit integer each of whose digits is equal to , and let bethe -digit (not -digit) integer each of whose digits is equal to . What is the greatest possible value of for which there are at least two values of such that ?2018 AMC 12B ProblemsProblem 1Kate bakes 20-inch by 18-inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches. How many pieces of cornbread does the pan contain?Problem 2Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles per hour), and his average speed during the second 30 minutes was 65 mph. What was his average speed, in mph, during the last 30 minutes?Problem 3A line with slope 2 intersects a line with slope 6 at the point . What is the distance between the -intercepts of these two lines?Problem 4A circle has a chord of length , and the distance from the center of the circle to the chord is . What is the area of the circle?Problem 5How many subsets of contain at least one prime number?Suppose cans of soda can be purchased from a vending machine for quarters. Which of the following expressions describes the number of cans of soda that can be purchased for dollars, where 1 dollar is worth 4 quarters?Problem 7What is the value ofProblem 8Line segment is a diameter of a circle with . Point , not equal to or , lies on the circle. As point moves around the circle, the centroid (center of mass) of traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?Problem 9What isProblem 10A list of positive integers has a unique mode, which occurs exactly times. What is the least number of distinct values that can occur in the list?Problem 11。