[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-444298-105":59,"doc-detail-444298-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","2024-amc-10a-problems","2024 AMC 10A Problems","","A collection of AMC 10A competition math problems from Problem 1 through Problem 25, covering a broad range of algebraic, number theory, combinatorics, geometry, and probability questions. Topics include modeling and solving with functions, prime decompositions, digit sums, modular-style operations, swaps and permutations, optimization among integer triples, team-forming arrangements, commuting transformations, and geometric configurations with tangency or similar figures. Each problem is presented as a standalone question with its specific data and constraints, designed for exam preparation and problem-solving practice.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/2024-amc-10a-problems/444298/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/2024-amc-10a-problems/444298.png","ImageObject",300,407,{"name":92,"@type":93},"Alex Sinclair","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-07","2026-09-29",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",24,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What kinds of topics are covered in the AMC 10A problems here?","Question",{"text":113,"@type":114},"The set includes algebra and modeling, number theory (primes, digit sums), combinatorics (arrangements and teams), geometry (triangles, similar rectangles, tangency), and probability (dice and movement on cubes).","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"Are the questions independent of each other?",{"text":118,"@type":114},"Yes. Each problem is written as a standalone prompt with its own given conditions, and solving one does not require results from another problem.",{"name":120,"@type":111,"acceptedAnswer":121},"What is Problem 6 about?",{"text":122,"@type":114},"It asks for the minimum number of successive swaps of adjacent letters needed to transform the string ABCDEF into FEDCBA.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},444298,1791089420,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":145,"read_time":146,"preview_page_count":4},1099523882182,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","2024 AMC 10A Problems  \nProblem 1  \nWhat is the value of  \nProblem 2  \nA model used to estimate the time it will take to hike to the top of the mountain on a trail is of the form  where  and  are constants,  is the time in minutes,  is the length of the trail in miles, and  is the altitude gain in feet. The model estimates that it will take  minutes to hike to the top if a trail is  miles long and ascends  feet, as well as if a trail is  miles long and ascends  feet. How many minutes does the model estimate it will take to hike to the top if the trail is  miles long and ascends  feet?  \nProblem 3  \nWhat is the sum of the digits of the smallest prime that can be written as a sum of  distinct primes?  \nProblem 4  \nThe number  is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write this sum?  \nProblem 5  \nWhat is the least value of  such that  is a multiple of ?  \nProblem 6  \nWhat is the minimum number of successive swaps of adjacent letters in the string ABCDEF that are needed to change the string to FEDCBA?  \n(For example, 3 swaps are required to change ABC to CBA; one such sequence of swaps is ABC  BAC  BCA  CBA.)  \nProblem 7  \nThe product of three integers is  . What is the least possible positive sum of the three integers?  \nProblem 8  \nAmy, Bomani, Charlie, and Daria work in a chocolate factory. On Monday Amy, Bomani, and Charlie started working at  and were able to pack  , , and  packages, respectively, every  minutes. At some later time, Daria joined the group, and Daria was able to pack  packages every  minutes. Together, they finished packing  packages at exactly  . At what time did Daria join the group?  \nProblem 9  \nIn how many ways can  juniors and  seniors form  disjoint teams of  people so that each team has  juniors and  seniors?  \nProblem 10  \nConsider the following operation. Given a positive integer  , if  is a multiple of , then you replace  by  . If  is not a multiple of  , then you replace  by . Then continue this process. For example, beginning with  , this procedure gives  . Suppose you start with  . What value results if you perform this operation exactly  times?  \nProblem 11  \nHow many ordered pairs of integers  satisfy ?  \nProblem 12  \nZelda played the Adventures of Math game on August 1 and scored  points. She continued to play daily over the next  days. The bar chart below shows the daily change in her score compared to the day before. (For example, Zelda's score on August 2 was  points.) What was Zelda's average score in points over the  days?  \nProblem 13  \nTwo transformations are said to commute if applying the first followed by the second gives the same result as applying the second followed by the first. Consider these four transformations ofthe coordinate plane:  \n• a translation  units to the right,  \n• a  -rotation counterclockwise about the origin,  \n• a reflection across the -axis, and  \n• a dilation centered at the origin with scale factor  .  \nOf the  pairs of distinct transformations from this list, how many commute?  \nProblem 14  \nOne side ofan equilateral triangle of height  lies on line  A circle of radius  is tangent to  and is externally tangent to the triangle. The area of the region exterior to the triangle and the circle and bounded by the triangle, the circle, and line  can be written as  where  and  are positive integers and  is not divisible by the square of any prime. What is   \nProblem 15  \nLet  be the greatest integer such that both  and  are perfect squares. What is the units digit of ?  \nProblem 16  \nAll of the rectangles in the figure below, which is drawn to scale, are similar to the enclosing rectangle. Each number represents the area of the rectangle. What is length ?  \nProblem 17  \nTwo teams are in a best-two-out-of-three playoff: the teams will play at most  games, and the winner of the playoff is the first team to win  games. The first game is played on Team A's home field, and the remaining ","cbCaigOSg11dXyjt","https://ap.wps.com/l/cbCaigOSg11dXyjt","pdf",384725,11,"English","# Problem Set\n## Problems 1-25","[{\"question\":\"What kinds of topics are covered in the AMC 10A problems here?\",\"answer\":\"The set includes algebra and modeling, number theory (primes, digit sums), combinatorics (arrangements and teams), geometry (triangles, similar rectangles, tangency), and probability (dice and movement on cubes).\"},{\"question\":\"Are the questions independent of each other?\",\"answer\":\"Yes. Each problem is written as a standalone prompt with its own given conditions, and solving one does not require results from another problem.\"},{\"question\":\"What is Problem 6 about?\",\"answer\":\"It asks for the minimum number of successive swaps of adjacent letters needed to transform the string ABCDEF into FEDCBA.\"}]","2024 AMC 10A Problems | PDF",1790707482,28]