{"id":3458,"date":"2022-12-31T00:52:07","date_gmt":"2022-12-31T04:52:07","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=3458"},"modified":"2024-10-25T12:14:33","modified_gmt":"2024-10-25T16:14:33","slug":"aime-2017-i-problem-15","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=3458","title":{"rendered":"AIME 2017 I &#8211; Problem 15"},"content":{"rendered":"\n<p>The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths&nbsp;$2\\sqrt{3}$, $5$, and $\\sqrt{37}$ as shown, is $\\dfrac{m\\sqrt{p}}{n}$, where $m$, $n$, and $p$ are positive integers, and $m$, $n$, and $p$ are relative prime, and $p$ is not divisible by the square of any prime. Find&nbsp;$m+n+p$.<\/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\/000.png\" alt=\"\" class=\"wp-image-3464\" width=\"239\" height=\"169\" srcset=\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2022\/12\/000.png 477w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2022\/12\/000-300x212.png 300w\" sizes=\"(max-width: 239px) 100vw, 239px\" \/><\/figure>\n\n\n\n<p><strong>Solution: <\/strong>Let the coordinates of the triangle vertices as $O=(0,0)$, $X=(5,0)$, $Y=(0,2\\sqrt{3})$. And the coordinates of the equilateral triangle on the $x$-axis and $y$-axis as $A=(a,0)$ and $B=(0,b)$. Then the coordinate of the third vertex $C$ of the equilateral triangle can be calculated by rotating line $AB$ $60^\\circ$ clockwise around $A$, which is $$(a+(-a\\cdot\\cos(60^\\circ)+b\\cdot\\sin(60^\\circ)), b\\cdot\\cos(60^\\circ)-(-a\\cdot\\sin(60^\\circ)))$$<\/p>\n\n\n\n<p>Simplifying the above, we have $$C=(\\dfrac{a+b\\sqrt{3}}{2},\\dfrac{a\\sqrt{3}+b}{2})$$<\/p>\n\n\n\n<p>As $C$ is on the line of $XY$, which is $$\\dfrac{x}{5}+\\dfrac{y}{2\\sqrt{3}}=1$$ We have $$\\dfrac{\\dfrac{a+b\\sqrt{3}}{2}}{5}+\\dfrac{\\dfrac{a\\sqrt{3}+b}{2}}{2\\sqrt{3}}=1$$<\/p>\n\n\n\n<p>Simplifying the above, we have $$a=\\dfrac{60-11\\sqrt{3}b}{21}$$<\/p>\n\n\n\n<p>The area of the equilateral $\\triangle{ABC}$ is $$\\dfrac{\\sqrt{3}}{4}\\overline{AB}^2=\\dfrac{\\sqrt{3}}{4}(a^2+b^2)$$<\/p>\n\n\n\n<p>$$a^2+b^2=(\\dfrac{60-11\\sqrt{3}b}{21})^2+b^2=\\dfrac{3600-1320\\sqrt{3}b+363b^2}{441}+b^2$$ $$=\\dfrac{3600-1320\\sqrt{3}b+804b^2}{441}$$<\/p>\n\n\n\n<p>Since the minimum value of $f(x)=Ax^2+Bx+C$ is $C-\\dfrac{B^2}{4A}$, when $A&gt;0$, the minimum value of $a^2+b^2$ is $$\\dfrac{3600-\\dfrac{{1320\\sqrt{3}}^2}{4\\cdot 804}}{441}=\\dfrac{300}{67}$$<\/p>\n\n\n\n<p>Therefore the minimum area of the $\\triangle{ABC}$ is $\\dfrac{\\sqrt{3}}{4}\\cdot\\dfrac{300}{67}=\\dfrac{75\\sqrt{3}}{67}$. Therefore the answer is $75+67+3=\\boxed{145}$.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths&nbsp;$2\\sqrt{3}$, $5$, and $\\sqrt{37}$ as shown, is $\\dfrac{m\\sqrt{p}}{n}$, where $m$, $n$, and $p$ are positive integers, and $m$, $n$, &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=3458\">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\/3458"}],"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=3458"}],"version-history":[{"count":22,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/3458\/revisions"}],"predecessor-version":[{"id":4611,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/3458\/revisions\/4611"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3458"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3458"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3458"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}