{"id":3431,"date":"2023-02-03T21:45:00","date_gmt":"2023-02-04T01:45:00","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=3431"},"modified":"2024-10-25T10:28:08","modified_gmt":"2024-10-25T14:28:08","slug":"aime-2022-ii-problem-15","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=3431","title":{"rendered":"AIME 2022 II &#8211; Problem 15"},"content":{"rendered":"\n<p>Two externally tangent circles $\\omega_1$ and $\\omega_2$ have centers $O_1$ and $O_2$, respectively. A third circle $\\Omega$ passing through $O_1$ and $O_2$ intersects $\\omega_1$ at $B$ and $C$ and $\\omega_2$ at $A$ and $D$, as shown. Suppose that $AB = 2$, $O_1O_2 = 15$, $CD = 16$, and $ABO_1CDO_2$ is a convex hexagon. Find the area of this hexagon. <a onclick=\"toggle_visibility('aime-2022-ii-15');\">\ud83d\udd11<\/a><\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" src=\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2022\/12\/74a9883958ea503b78cb2c07bc71aefe4d830ce9.png\" alt=\"\" class=\"wp-image-3433\" width=\"475\" height=\"342\" srcset=\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2022\/12\/74a9883958ea503b78cb2c07bc71aefe4d830ce9.png 950w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2022\/12\/74a9883958ea503b78cb2c07bc71aefe4d830ce9-300x216.png 300w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2022\/12\/74a9883958ea503b78cb2c07bc71aefe4d830ce9-768x553.png 768w\" sizes=\"(max-width: 475px) 100vw, 475px\" \/><\/figure>\n\n\n\n<div id=\"aime-2022-ii-15\" style=\"display:none\">\n\n\n\n<p><strong>Solution: <\/strong>Let the $a$ and $b$ the radius of circle $\\omega_1$ and $\\omega_2$ respectively. We have $a+b=15$.<\/p>\n\n\n\n<p>Based on <a href=\"https:\/\/en.wikipedia.org\/wiki\/Brahmagupta%27s_formula\">Brahmagupta&#8217;s Formula,<\/a> the area of cyclic quadrilateral $ABO_1O_2$ is $$S_1=\\sqrt{(s-a)(s-2)(s-b)(s-15)}\\tag{1}$$ where $$s=\\dfrac{a+2+b+15}{2}=16$$ Simplifying (1), we have $$S_1=\\sqrt{14(16+ab)}\\tag{2}$$<\/p>\n\n\n\n<p>Similarly, the area of cyclic quadrilateral $DCO_1O_2$ is $$S_2=2\\sqrt{14(184+ab)}\\tag{3}$$<\/p>\n\n\n\n<p>Therefore, the area of the hexagon is $$S=S_1+S_2=\\sqrt{14(16+ab)}+2\\sqrt{14(184+ab)}\\tag{4}$$<\/p>\n\n\n\n<p>Assume $$16+ab=14x^2\\tag{5}$$ $$184+ab=14y^2\\tag{6}$$ Therefore $$S=14x+28y\\tag{7}$$<\/p>\n\n\n\n<p>Eliminating $ab$ in (5) and (6), we have $$y^2-x^2=12\\tag{8}$$<\/p>\n\n\n\n<p>The integer solution for (8) is $x=2$ and $y=4$. Plugging $x$ and $y$ in (7), we have $$S=14x+28y=14\\times 2+28\\times 4=\\boxed{140}$$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Two externally tangent circles $\\omega_1$ and $\\omega_2$ have centers $O_1$ and $O_2$, respectively. A third circle $\\Omega$ passing through $O_1$ and $O_2$ intersects $\\omega_1$ at $B$ and $C$ and $\\omega_2$ at $A$ and $D$, as shown. Suppose that $AB = &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=3431\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[13,9,15],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/3431"}],"collection":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=3431"}],"version-history":[{"count":15,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/3431\/revisions"}],"predecessor-version":[{"id":4595,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/3431\/revisions\/4595"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3431"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3431"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3431"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}