[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-444295-105":59,"doc-detail-444295-en":130},{"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":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","2023-amc-10a-problems","2023 AMC 10A Problems","","2023 AMC 10A Problems provides a set of 24 concise multiple-step mathematics problems covering number theory, geometry, probability, and algebraic reasoning. Tasks include distance and rate calculations, pizza and divisor weight relations, divisibility by specified primes, optimization of integer-sided polygons, counting digit appearances in date formats, and evaluations involving rotations and transformations. The set also includes nested-figure area bounds, tournament scoring with hand-preference probabilities, polynomial root constraints, and circle tangency radius determination.",{"@graph":69,"@context":122},[70,84,105],{"@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/2023-amc-10a-problems/444295/",{"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/2023-amc-10a-problems/444295.png","ImageObject",300,407,{"name":92,"@type":93},"Maya Linwood","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-06","2026-09-29",true,{"@type":102,"interactionType":103,"userInteractionCount":36},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What topics are covered across the 2023 AMC 10A Problems set?","Question",{"text":112,"@type":113},"The set spans number theory, geometry, probability, and algebraic/polynomial reasoning, with many questions requiring divisibility, counting, and optimization steps.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"Does the document include problems with probability and geometry together?",{"text":117,"@type":113},"Yes. For example, it contains die-based probability problems and geometry problems involving rotations, polygons, and tangency of circles.",{"name":119,"@type":110,"acceptedAnswer":120},"How many problems are included in the document text provided?",{"text":121,"@type":113},"The provided content lists Problems 1 through 24, indicating 24 questions in total.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},444295,1791132217,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":36,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145,"preview_page_count":4},962084928432,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","2023 AMC 10A Problems  \nProblem 1  \nCities 􀜣 and 􀜤 are 45 miles apart. Alice and Barbara start biking from 􀜣 and 􀜤 at speeds of 18 mph and 12 mph, respectively. How far away from city 􀜣 will they be when they meet?  \nProblem 2  \nThe weight of 13 of a large pizza together with 3 12 cups of orange slices is the same as the weight of 34 of a large pizza together with 12 cup of orange slices. A cup of orange slices weighs 14 of a pound. What is the weight, in pounds, of a large pizza?  \n(A) 1 45 (B) 2 (C) 2 25 (D) 3 (E) 3 35  \nProblem 3  \nHow many positive perfect squares less than 2023 are divisible by 5?  \nProblem 4  \nA quadrilateral has all integer sides lengths, a perimeter of 26, and one side of length 4. What is the greatest possible length of one side of this quadrilateral?  \nProblem 5  \nHow many digits are in the base-ten representation of 85 ⋅ 510 ⋅ 155 ?  \nProblem 6  \nAn integer is assigned to each vertex of a cube. The value of an edge is defined tobe the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of the values of the four edges surrounding it. The value of the cube is defined as the sum of the values of its six faces. Suppose the sum of the integers assigned to the vertices is 21. What is the value of the cube?  \nProblem 7  \nJanet rolls a standard 6-sided die 4 times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal 3?  \nProblem 8  \nBarb the baker has developed a new temperature scale for her bakery called the Breadus scale, which is a linear function of the Fahrenheit scale. Bread rises at 110 degrees Fahrenheit, which is 0 degrees on the Breadus scale. Bread is baked at 350 degrees Fahrenheit, which is 100 degrees on the Breadus scale. Bread is done when its internal temperature is 200 degrees Fahrenheit. What is this in degrees on the Breadus scale?  \nProblem 9  \nA digital display shows the current date as an 8-digit integer, consisting of a 4-digit year, followed by a 2-digit month, followed by a 2-digit date within the month. For how many dates in 2023 will each digit appear an even number of times in the digital display for that date?  \nProblem 10  \nIf Mareen scores an 11 on her next test, her mean score will go up by 1. If she gets three 11's in a row, her mean score will increase by 2. What is her current mean test score?  \nProblem 11  \nA square with area 3 has a square with area 2 inscribed in it. This creates 4 smaller congruent right triangles. What is the ratio of the smaller leg to the larger leg in the shaded right triangle?  \nProblem 12  \nHow many three-digit positive integers 􀜰 satisfy the following properties?  \n• The number N is divisible by 7.  \n• The number formed by reversing the digits of 􀜰 is divisible by 5.  \nProblem 13  \nAbdul and Chiang are standing 48 feet apart in a field. Bharat is standing in the same field as far from Abdul as possible so that the angle formed by his lines of sight to Abdul and Chiang measures 60° . What is the square of the distance (in feet) between Abdul and Bharat?  \nProblem 14  \nA number is chosen at random from among the first 100 positive integers, and a positive integer divisor of that number is then chosen at random. What is the probability that the chosen divisor is divisible by 11?  \nProblem 15  \nAn even number of circles are nested, starting with a radius of 1 and increasing by 1 each time, all sharing a common point. The region between every other circle is shaded, starting with the region inside the circle of radius 2 but outside the circle of radius 1. An example showing 8 circles is displayed below. What is the least number of circles needed to make the total shaded area at least 2023􀟨?  \n(A) 46 (B) 48 (C) 56 (D) 60 (E) 64  \nProblem 16  \nIn a tennis tournament, each person plays every other person once. In this tournament, there are twice as many right-handed players than left-handed players, but lefthanded players won 40% more","cbCainhbHgsyjRgS","https://ap.wps.com/l/cbCainhbHgsyjRgS","pdf",363549,11,"English","# Problem 1 - Problem 12\n## Problem 13 - Problem 24","[{\"question\":\"What topics are covered across the 2023 AMC 10A Problems set?\",\"answer\":\"The set spans number theory, geometry, probability, and algebraic/polynomial reasoning, with many questions requiring divisibility, counting, and optimization steps.\"},{\"question\":\"Does the document include problems with probability and geometry together?\",\"answer\":\"Yes. For example, it contains die-based probability problems and geometry problems involving rotations, polygons, and tangency of circles.\"},{\"question\":\"How many problems are included in the document text provided?\",\"answer\":\"The provided content lists Problems 1 through 24, indicating 24 questions in total.\"}]","2023 AMC 10A Problems | PDF",1790707476,28]