[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-272565-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-272565-en":130},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","2020-chapter-competition-solutions-mathcounts","2020 Chapter Competition Solutions - MATHCOUNTS","","Comprehensive answer key for the 2020 MATHCOUNTS Chapter Competition, covering the Sprint Round, Target Round, and Team Round. Each problem is paired with a creative, concise solution path that demonstrates clear mathematical reasoning across a wide range of topics, from number patterns and geometry to ratios, areas, and average-rate comparisons. The collection highlights multiple strategies leading to correct results and encourages exploring alternate approaches. Solutions are attributed to Howard Ludwig, who shared work with the MATHCOUNTS community.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/exam/","Exam",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/2020-chapter-competition-solutions-mathcounts/272565/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/2020-chapter-competition-solutions-mathcounts/272565.png","ImageObject",300,407,{"name":42,"@type":43},"Elsa","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-22","2026-09-15",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",10,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"Which rounds are covered in the 2020 Chapter Competition solutions?","Question",{"text":63,"@type":64},"The solutions cover the Sprint Round, Target Round, and Team Round of the 2020 MATHCOUNTS Chapter Competition.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"What is the style of the solutions in this document?",{"text":68,"@type":64},"Each solution provides a creative and concise way to solve the competition problems, while acknowledging that other correct approaches may exist.",{"name":70,"@type":61,"acceptedAnswer":71},"Who contributed the solutions included here?",{"text":72,"@type":64},"Howard Ludwig is thanked as the solutions author for voluntarily sharing his solutions with the MATHCOUNTS community.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},272565,1789466125,{"code":4,"msg":82,"data":83},"success",[84,88,92,95,100,105,110,115,120,123,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":93,"slug":94},70,"exam",{"id":96,"doc_module":4,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},5,"Comic",60,"comic",{"id":101,"doc_module":4,"doc_module_name":25,"category_name":102,"show_sort_weight":103,"slug":104},6,"Technology",50,"technology",{"id":106,"doc_module":4,"doc_module_name":25,"category_name":107,"show_sort_weight":108,"slug":109},7,"Healthcare",40,"healthcare",{"id":111,"doc_module":4,"doc_module_name":25,"category_name":112,"show_sort_weight":113,"slug":114},8,"Research & Report",30,"research-report",{"id":116,"doc_module":4,"doc_module_name":25,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":25,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":55,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":55,"slug":125},"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":96,"slug":129},19,"General","general",{"code":4,"msg":82,"data":131},{"doc_id":79,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":33,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":139,"language":140,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":12,"update_tm":80,"read_time":144},137455077381,"https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d","2020 Chapter Competition Solutions  \nAre you wondering how we could have possibly thought that a Mathlete® would be able to answer a particular Sprint Round problem without a calculator?  \nAre you wondering how we could have possibly thought that a Mathlete would be able to answer a particular Target Round problem in less 3 minutes?  \nAre you wondering how we could have possibly thought that a particular Team Round problem would be solved by a team of only four Mathletes?  \nThe following pages provide solutions to the Sprint, Target and Team Rounds of the 2020 MATHCOUNTS® Chapter Competition. These solutions provide creative and concise ways of solving the problems from the competition.  \nThere are certainly numerous other solutions that also lead to the correct answer, some even more creative and more concise!  \nWe encourage you to find a variety of approaches to solving these fun and challenging MATHCOUNTS problems.  \nSpecial thanks to solutions author  \nHoward Ludwig  \nfor graciously and voluntarily sharing his solutions with the MATHCOUNTS community.  \n2020 Chapter Competition Sprint Round  \n1. Each hour has 60 minutes. Therefore, 4.5 hours has 4.5 × 60 minutes = 45 × 6 minutes = 270 minutes.  \n2. The ratio of apples to oranges is 5 to 3. Therefore, 53 = 􀯔9, where a represents the number of apples. So, 3􀜽 = 5 × 9 = 45. Thus, 􀜽 = 453 = 􀫚􀫞 apples.  \n3. Substituting for x andy yields 12􀝔􀝕 = 12 × 12 × 6 = 6 × 6 = 􀫜􀫟 .  \n4. The minimum Category 4 speed is 130 mi/h. The maximum Category 1 speed is 95 mi/h.  \nAbsolute difference is |130 − 95| mi/h = 35 mi/h.  \n5. Area 􀜣 = 􀝏 2 = 144 cm2. Perimeter 􀝌 = 4􀝏 = 4√􀜣 = 4√144  = 4 × 12 cm = 􀫝􀫡 cm.  \n6. Since 3 miles = 1 league, 1 mile = 13 league = 13 × 24 furlongs = 8 furlongs.  \n7. The sum of the measures of the 4 interior angles of a quadrilateral is 360 degrees. Therefore, 360 = 119 + 89 + 49 + 􀝉∠D = 257 + 􀝉∠D, so 􀝉∠D = 360 − 257 = 􀫚􀫙􀫜 degrees.  \n8. In any nonzero geometric sequence, the ratio of the nth term to the (n – 1)th term, for n > 1, is constant, and is called the common ratio. Therefore, 􀝐5 /􀝐4 = 􀝐2 /􀝐1 = 4/2 = 2, so 􀝐5 = 2􀝐4 = 2 × 16 = 32.  \n9. The first polygon, a triangle, has 3 sides. The other polygon has 2 × 3 – 2 = 6 – 2 = 4 sides.  \n10. Solve using cross cancelation to get 50 cm × 1$4m.00eter = 50 cm × 1~~ ~~m100eerm × $4me.~~ ~~0t0er = 12 × $4 .00 = $􀫛 . 􀫙􀫙 􀝋􀝎 $􀫛 .  \n11. The average of 70 over 3 rounds is the sum of the scores for the 3 rounds divided by 3. Therefore, the sum of the scores over 3 rounds is 3 × 70 = 210. The two known scores are 68 and 72, which sum to 140. Therefore, the third round score must have been 210 – 140 = 70.  \n12. The diagonals of the inner shaded square are congruent to the sides of the outer square—call that length 􀝏 . Then the area of the inner shaded square is 􀝏2 /2 while the area of the outer square is 􀝏 2, which is 2 times as big as the inner square. Because the inner square has area 36 cm², the area of the outer square is  \n2 × 36 cm² = 72 cm².  \nAlternatively, if we draw the diagonals of the shaded square, we create 8 congruent 45 45 45 right triangles. Since the shaded square, which is made up of four of these right triangles, has area 36 cm2, it follows that each triangle has area 36/4 = 9 cm2. Therefore, the outer square, composed of 8 of these right triangles, has area 9 × 8 = 72 cm2.  \n13. Because Rafa averages moving 12.7 meters per point and Sascha averages 11.8 meters per point, Rafa averages 12.7 – 11.8 = 0.9 meters per point more than Sascha. Given 320 points for the match at that rate, Rafa will run a total of 320 × 0.9 = 288 meters more than Sascha.  \n14. Let 􀝈 be the length and 􀝓 be the width of the rectangle. Then 􀝈 + 􀝓 = 16 u and 􀝈 = 3􀝓 . Thus, we have  \n16 units = 3􀝓 + 􀝓 = 4􀝓, 􀝓 = 4 units, 􀝈 = 16 units − 4 units = 12 units, and 􀝈􀝓 = 12 units × 4 units = 48 units2.  \n15. The expression √5 × 6 × 10 × 12 can be rewritten as √5 × 10 × 6 × 12. Simplifying, we get 5√2 × 6√2 = 30 × 2 = 􀫟􀫙 .  \n16. The smallest, unshaded sq","cbCairPtRhEgJu0c","https://ap.wps.com/l/cbCairPtRhEgJu0c","pdf",415246,11,"English","# 2020 MATHCOUNTS Chapter Competition Solutions\n## Sprint Round\n## Target Round\n## Team Round","[{\"question\":\"Which rounds are covered in the 2020 Chapter Competition solutions?\",\"answer\":\"The solutions cover the Sprint Round, Target Round, and Team Round of the 2020 MATHCOUNTS Chapter Competition.\"},{\"question\":\"What is the style of the solutions in this document?\",\"answer\":\"Each solution provides a creative and concise way to solve the competition problems, while acknowledging that other correct approaches may exist.\"},{\"question\":\"Who contributed the solutions included here?\",\"answer\":\"Howard Ludwig is thanked as the solutions author for voluntarily sharing his solutions with the MATHCOUNTS community.\"}]","2020 Chapter Competition Solutions - MATHCOUNTS | PDF",28]