[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-344182-105":59,"doc-detail-344182-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","79th-william-lowell-putnam-mathematical-competition-problems","79th William Lowell Putnam Mathematical Competition - Problems","","The document presents the full set of problems from the 79th William Lowell Putnam Mathematical Competition held on December 1, 2018. It lists questions A1 through A6 and B1 through B6, covering diverse fields such as number theory, linear algebra, real analysis, combinatorics, algebraic structure in groups, geometry with rational distance constraints, polynomial roots, divisibility properties, sequences and periodicity, and multivariable calculus. Each problem specifies precise conditions and asks for proofs or determinations of values.",{"@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/79th-william-lowell-putnam-mathematical-competition-problems/344182/",{"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/79th-william-lowell-putnam-mathematical-competition-problems/344182.png","ImageObject",300,407,{"name":92,"@type":93},"Quinn","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-04","2026-09-22",true,{"@type":102,"interactionType":103,"userInteractionCount":24},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What competition and date does the document correspond to?","Question",{"text":112,"@type":113},"It corresponds to the 79th William Lowell Putnam Mathematical Competition, dated Saturday, December 1, 2018.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How are the problems organized in the document?",{"text":117,"@type":113},"The problems are organized into Section A (A1 through A6) and Section B (B1 through B6).",{"name":119,"@type":110,"acceptedAnswer":120},"What types of tasks do the problems request?",{"text":121,"@type":113},"The tasks include finding ordered pairs, calculating determinants, proving inequalities or existence statements, and demonstrating properties such as commutativity in groups and periodicity in sequences.","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},344182,1792182939,{"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":24,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":8,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":81,"preview_page_count":4},962075114765,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","The 79th William Lowell Putnam Mathematical Competition Saturday, December 1, 2018  \nA1 Find all ordered pairs (a; b) of positive integers for which  \n1  1  3  a + b = 2018 :  \nA2 Let S1 ; S2 ; : : : ; S2n􀀀1 be the nonempty subsets off1; 2 ; : : : ; ng in some order, and let M be the (2n 􀀀 1) 􀀂(2n 􀀀 1) matrix whose (i; j) entry is  \nmi j = (01 Shie\\ Srwijse= /0; Calculate the determinant of M.  \nA3 Determine the greatest possible value of å1i1 cos (3xi) for real numbers x 1 ; x 2 ; : : : ; x 10 satisfying å1i1 cos (xi) = 0.  \nA4 Let m and n be positive integers with gcd(m; n) = 1, and let  \nak = 􀀖 m~~n~~k 􀀗􀀀 􀀖 m~~ ~~(k~~n~~􀀀~~ ~~1)􀀗  \nfor k = 1 ; 2 ; : : : ; n. Suppose that g and h are elements ina group G and that  \ngha1 gha2 􀀁􀀁􀀁ghan = e ;  \nwhere e is the identity element. Show that gh = hg. (As usual, bxc denotes the greatest integer less than or equal to x.)  \nA5 Let f : R ! R be an infinitely differentiable function satisfying f (0) = 0, f (1) = 1, and f (x) 􀀕 0 for all x 2 R. Show that there exist a positive integer n and a real number x such that f (n)(x) \u003C 0.  \nA6 Suppose that A; B ; C ; and D are distinct points, no three of which lie on a line, in the Euclidean plane. Show that if the squares of the lengths of the line segments AB, AC, AD, BC, BD, and CD are rational numbers, then the quotient  \narea (4ABC)  \narea (4ABD)  \nis a rational number.  \nB1 Let P be the set of vectors defined by  \nP = 􀀚 􀀒ab􀀓 􀀌 0 􀀔 a 􀀔 2; 0 􀀔 b 􀀔 100; and a; b 2 Z􀀛 :  \nFind all v 2 P such that the set P n fvg obtained by omitting vector v from P can be partitioned into two sets of equal size and equal sum.  \nB2 Let n be a positive integer, and let fn(z) = n +(n 􀀀1)z +(n 􀀀 2)z2 + 􀀁􀀁􀀁 + zn􀀀1 . Prove that fn has no roots in the closed unit disk fz 2 C : jz j 􀀔 1g.  \nB3 Find all positive integers n \u003C 10100 for which simultaneously n divides 2n , n 􀀀 1 divides 2n 􀀀 1, and n 􀀀 2 divides 2n 􀀀 2.  \nB4 Given a real number a, we define a sequence by x 0 = 1, x 1 = x 2 = a, and xn+1 = 2xnxn􀀀1 􀀀xn􀀀2 for n 􀀕 2. Prove that if xn = 0 for some n, then the sequence is periodic.  \nB5 Let f = ( f 1 ; f 2) be a function from R2 to R2 with continuous partial derivatives ~~¶¶f ix ~~j that are positive everywhere. Suppose that  \n¶~~¶~~f~~ ~~1~~x ~~1 ¶~~¶~~f~~ ~~2~~x ~~2 􀀀 1~~4~~ 􀀒 ¶~~¶~~f~~ ~~1~~x ~~2 + ¶~~¶~~f~~ ~~2~~x ~~1 􀀓 2 > 0 everywhere. Prove that f is one-to-one.  \nB6 Let S be the set of sequences of length 2018 whose terms are in the set f1; 2 ; 3 ; 4 ; 5 ; 6 ; 10g and sum to 3860 . Prove that the cardinality of S is at most  \n23860 􀀁 􀀒 2018~~2048~~ 􀀓 2018 :","cbCaipzqXcsyhgic","https://ap.wps.com/l/cbCaipzqXcsyhgic","pdf",84855,"English","# Section A\n## A1-A6\n# Section B\n## B1-B6","[{\"question\":\"What competition and date does the document correspond to?\",\"answer\":\"It corresponds to the 79th William Lowell Putnam Mathematical Competition, dated Saturday, December 1, 2018.\"},{\"question\":\"How are the problems organized in the document?\",\"answer\":\"The problems are organized into Section A (A1 through A6) and Section B (B1 through B6).\"},{\"question\":\"What types of tasks do the problems request?\",\"answer\":\"The tasks include finding ordered pairs, calculating determinants, proving inequalities or existence statements, and demonstrating properties such as commutativity in groups and periodicity in sequences.\"}]","79th William Lowell Putnam Mathematical Competition - Problems | PDF",1790053179]