AMC10数学竞赛是美国高中数学竞赛中的一项，是针对高中一年级及初中三年级学生的数学测试，该竞赛开始于2000年，分A赛和B赛，于每年的2月初和2月中举行，学生可任选参加一项即可。不管是对高校申请还是今后在数学领域的发展都极其有利！那么接下来跟随小编来看一下AMC10数学竞赛真题以及官方解答吧：Problem 13At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for of the babies born. There were four times as many sets of triplets as sets of quadruplets, and there was three times as many sets of twins as sets of triplets. How many of these babies were in sets of quadruplets?SolutionWe can set up a system of equations where is the sets of twins, is the sets of triplets, and is the sets of quadruplets.Solving for and in the second and third equations and substituting into the first equation yieldsSince we are trying to find the number of babies and NOT the number of sets of quadruplets, the solution is not , but rather . Therefore, we strategically use the second initial equation to realize that , leaving us with the number of babies born as quadruplets equal to .Alternative SolutionSay there are sets of twins, sets of triplets, and sets of quadruplets. That's twins, triplets, and quadruplets. A tenth of the babies are quadruplets and that's Problem 14How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line , the line and the line Solution 1The region is a right triangle which contains the following lattice points:Squares : Suppose that the top-right corner is , with . Then to include all other corners, we need . This produces squares.Squares : Here . To include all other corners, we need . This produces squares.Squares : Similarly this produces squares.No other squares will fit in the region. Therefore the answer is .Solution 2The vertical line is just to the right of , the horizontal line is just under , and the sloped line will always be above the value of . This means they will always miss being on a coordinate with integer coordinates so you just have to count the number of squares to the left, above, and under these lines. After counting the number of 1x1, 2x2, 3x3, squares and getting 30, 15, and 5 respectively, and we end up with .Solution by Wwang 以上就是小编对AMC10数学竞赛真题以及解析的介绍，希望对你有所帮助，如果想了解更多关于AMC数学竞赛报考点、南京AMC数学竞赛培训、美国数学竞赛AMC有用吗以及AMC学习资料等信息请持续关注AMC数学竞赛网。
AMC10数学竞赛是美国高中数学竞赛中的一项，是针对高中一年级及初中三年级学生的数学测试，该竞赛开始于2000年，分A赛和B赛，于每年的2月初和2月中举行，学生可任选参加一项即可。不管是对高校申请还是今后在数学领域的发展都极其有利！那么接下来跟随小编来看一下AMC10数学竞赛真题以及官方解答吧：Problem 11Carl decided to fence in his rectangular garden. He bought fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl’s garden?SolutionIf the dimensions are , then one side will have posts (including corners) and the other .The total number of posts is .Solve the system to get . Then the area is which is .Solution 2To do this problem, we have to draw a rectangle. We also have to keep track of the fence posts. Putting a post on each corner leaves us with only posts. Now there are twice as many posts on the longer side then the shorter side. From this we can see that we can put posts on the longer side and posts on the shorter side. On the shorter side, we have spaces between the posts. On the longer side, we have 7 spaces between the 8 fence posts. There are yards between each post. Therefore, the answer is . Problem 12Two different numbers are selected at random from and multiplied together. What is the probability that the product is even?SolutionThe product will be even if at least one selected number is even, and odd if none are. Using complementary counting, the chance that both numbers are odd is , so the answer is which is . 以上就是小编对AMC10数学竞赛真题以及解析的介绍，希望对你有所帮助，如果想了解更多关于AMC数学竞赛报考点、AMC美国大学生数学竞赛、美国数学竞赛AMC有用吗以及AMC学习资料等信息请持续关注AMC数学竞赛网。
AMC10数学竞赛是美国高中数学竞赛中的一项，是针对高中一年级及初中三年级学生的数学测试，该竞赛开始于2000年，分A赛和B赛，于每年的2月初和2月中举行，学生可任选参加一项即可。不管是对高校申请还是今后在数学领域的发展都极其有利！那么接下来跟随小编来看一下AMC10数学竞赛真题以及官方解答吧：Problem 9All three vertices of lie on the parabola defined by , with at the origin and parallel to the -axis. The area of the triangle is . What is the length of ?SolutionThe area of the triangle is , so , giving a total distance across the top of , which is answer . Problem 10A thin piece of wood of uniform density in the shape of an equilateral triangle with side length inches weighs ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has side length of inches. Which of the following is closest to the weight, in ounces, of the second piece?Solution 1We can solve this problem by using similar triangles, since two equilateral triangles are always similar. We can then use.We can then solve the equation to get which is closest to Solution 2Also recall that the area of an equilateral triangle is so we can give a ratio as follows: Cross multiplying and simplifying, we get Which is Solution by Solution 3Note that the ratio of the two triangle's weights is equal to the ratio of their areas, as the height is negligible. The ratio of their areas is equal to the square of the ratio of their sides. So if denotes the weight of the second triangle, we haveSolving gives us so the answer is . 以上就是小编对AMC10数学竞赛真题以及解析的介绍，希望对你有所帮助，如果想了解更多关于AMC数学竞赛报考点、AMC美国大学生数学竞赛、美国数学竞赛AMC有用吗以及AMC学习资料等信息请持续关注AMC数学竞赛网。
近几年，AMC美国数学竞赛在国内越来越火，逐渐被国内绝大多数爱好数学的同学所认识，其实AMC数学竞赛是一项在国际上非常重要的竞赛，当然对于一些刚刚接触AMC数学竞赛的小伙伴可能对AMC的一些相关问题还不是很了解，今天AMC数学竞赛网小编就来和大家一起看一下AMC数学竞赛常见问题解答（下），希望能够帮到大家：4. AMC 系列考试应该如何准备？考前18-12个月：开始接触目标年级的 AMC ，做近三年真题，记录易错的部分。考前12-4个月：用专用教材系统学习自己不足的章节，学习过程应有所偏重。考前3-2个月：做真题/模拟题训练，练习准确率/速度及独立思考能力，注意对于错题分析。考前1个月：给自己限定时间独立完成模考，找找适合自己的考试技巧。5. AMC 容易得奖吗？取决于学生的基础和学习方法，就得奖率而言，每年参加 AMC8 的学生中有 5% 能得荣誉奖，1% 能得最高荣誉奖。AMC 10/12 会选出前 2.5% 和 5% 的学生晋级更高等级的 AIME。如果在 AIME 甚至 USAMO 取得好成绩，不但能大大提升被美国排名前二十的大学录取几率，而且也会是中学生涯中一笔宝贵的财富。本人在中学年代，有幸参加全国高中数学联赛（CMO），见到了很多大神，确实能从他们身上学到很多，比如：专注、执着、一丝不苟；当然也有很多学不来的，比如天才般的思路：构造、论证、分析……6. AMC 学习和备考有没有什么教材？有官方出的系列教材，针对每个年级各一套，每套五册书共三十章，系统的覆盖了每个知识点。这个教材的优点是讲的非常详细，并且每道例题有对应的习题，缺点是很久没有更新，都是多年前的题。备考的话做历年真题尤为关键，真题能切实反映难度，建议考生在做真题的时候也给自己限定时间，“一口气”做完，模拟考场状态，不要被一些琐事打断。历年真题在官网上可以找到。7. 国内数学竞赛对 AMC 是否有帮助？数学是全世界的语言，全世界的数学竞赛都大同小异，客观的说，国内的课内数学和数学竞赛，都是略比国外的课内数学和数学竞赛难一点的，所以国内数学竞赛学的好的话，转 AMC 系列只需要提高一下自己的单词量，英语阅读理解能力。8. AMC 竞赛在每年什么时候？AMC8 在每年十一月中旬，AMC10 / 12 在每年二月中旬。以上就是小编为大家准备的AMC数学竞赛常见问题解答，如果想要了解更多资讯及AMC备考规划，请持续关注AMC数学竞赛网哦！
AMC10数学竞赛是美国高中数学竞赛中的一项，是针对高中一年级及初中三年级学生的数学测试，该竞赛开始于2000年，分A赛和B赛，于每年的2月初和2月中举行，学生可任选参加一项即可。不管是对高校申请还是今后在数学领域的发展都极其有利！那么接下来跟随小编来看一下AMC10数学竞赛真题以及官方解答吧：Problem 7The ratio of the measures of two acute angles is , and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree measures of the two angles?SolutionWe can set up a system of equations where and are the two acute angles. WLOG, assume that in order for the complement of to be greater than the complement of . Therefore, and . Solving for in the first equation and substituting into the second equation yieldsSubstituting this value back into the first equation yields , leaving equal to . Problem 8What is the tens digit of Solution 1Notice that, for , is congruent to when is even and when is odd. (Check for yourself). Since is even, and .So the answer is .Solution 2In a very similar fashion, we find that , which equals . Next, since every power (greater than ) of every number ending in will end in (which can easily be verified), we get . (In this way, we don't have to worry about the exponent very much.) Finally, , and thus , as above. 以上就是小编对AMC10数学竞赛真题以及解析的介绍，希望对你有所帮助，如果想了解更多关于AMC数学竞赛报考点、AMC美国大学生数学竞赛、美国数学竞赛AMC有用吗以及AMC学习资料等信息请持续关注AMC数学竞赛网。
AMC10数学竞赛是美国高中数学竞赛中的一项，是针对高中一年级及初中三年级学生的数学测试，该竞赛开始于2000年，分A赛和B赛，于每年的2月初和2月中举行，学生可任选参加一项即可。不管是对高校申请还是今后在数学领域的发展都极其有利！那么接下来跟随小编来看一下AMC10数学竞赛真题以及官方解答吧：Problem 5The mean age of Amanda's cousins is , and their median age is . What is the sum of the ages of Amanda's youngest and oldest cousins?SolutionThe sum of the ages of the cousins is times the mean, or . There are an even number of cousins, so there is no single median, so must be the median of the two in the middle. Therefore the sum of the ages of the two in the middle is . Subtracting from produces . Problem 6Laura added two three-digit positive integers. All six digits in these numbers are different. Laura's sum is a three-digit number . What is the smallest possible value for the sum of the digits of ?Solution 1Let the two three-digit numbers she added be and with and . The hundreds digits of these numbers must be at least and , so and .Say and ; then we just need with and having different digits which aren't or .There are many solutions, but and give which proves that is attainable.Solution 2For this problem, to find the -digit integer with the smallest sum of digits, one should make the units and tens digit add to . To do that, we need to make sure the digits are all distinct. For the units digit, we can have a variety of digits that work. works best for the top number which makes the bottom digit . The tens digits need to add to because of the that needs to be carried from the addition of the units digits. We see that and work the best as we can't use and . Finally, we use and for our hundreds place digits.Adding the numbers and , we get which means our answer is . 以上就是小编对AMC10数学竞赛真题以及解析的介绍，希望对你有所帮助，如果想了解更多关于AMC数学竞赛报考点、AMC美国大学生数学竞赛、美国数学竞赛AMC有用吗以及AMC学习资料等信息请持续关注AMC数学竞赛网。
AMC10数学竞赛是美国高中数学竞赛中的一项，是针对高中一年级及初中三年级学生的数学测试，该竞赛开始于2000年，分A赛和B赛，于每年的2月初和2月中举行，学生可任选参加一项即可。不管是对高校申请还是今后在数学领域的发展都极其有利！那么接下来跟随小编来看一下AMC10数学竞赛真题以及官方解答吧：Problem 3Let . What is the value of ?SolutionSubstituting carefully, becomes which is .Solution 2Solution by e_power_pi_times_iSubstitute into the equation. Now, it is . Since , it is a positive number, so . Now the equation is . This further simplifies to , so the answer is Problem 4Zoey read books, one at a time. The first book took her day to read, the second book took her days to read, the third book took her days to read, and so on, with each book taking her more day to read than the previous book. Zoey finished the first book on a Monday, and the second on a Wednesday. On what day the week did she finish her th book?SolutionThe process took days, so the last day was days after the first day. Since is divisible by , both must have been the same day of the week, so the answer is . 以上就是小编对AMC10数学竞赛真题以及解析的介绍，希望对你有所帮助，如果想了解更多关于AMC数学竞赛报考点、AMC美国大学生数学竞赛、美国数学竞赛AMC有用吗以及AMC学习资料等信息请持续关注AMC数学竞赛网。
AMC10数学竞赛是美国高中数学竞赛中的一项，是针对高中一年级及初中三年级学生的数学测试，该竞赛开始于2000年，分A赛和B赛，于每年的2月初和2月中举行，学生可任选参加一项即可。不管是对高校申请还是今后在数学领域的发展都极其有利！那么接下来跟随小编来看一下AMC10数学竞赛真题以及官方解答吧：Problem 1What is the value of when ?SolutionFactorizing the numerator, then becomes which is equal to which is .Problem 2If , what is ?Solution 1 which is .Solution 2We can replace and with and respectively. Then substituting with and we can get and substitute to get which is 以上就是小编对AMC10数学竞赛真题以及解析的介绍，希望对你有所帮助，更多学习资料请持续关注AMC数学竞赛网。