Problem 57874. Easy Sequences 107: Minimized Circumcircle-Incircle Areas Ratio of Partial Pythagorean Triangles

Created by Ramon Villamangca

Solve Later

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Thus, the triples </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\{3,\\ 4,\\ 5\\}</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\{1,\\ \\sqrt{3}, \\ 2\\}</w:t></w:r></w:customXml><w:r><w:t> represent a PPT, while </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\{1,\\ 1,\\ \\sqrt{2} \\}</w:t></w:r></w:customXml><w:r><w:t>, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\{\\frac^{3}_5,\\ \\frac^{4}_5,\\ 1\\}</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\{{4.\\ 4,\\ 4\\}</w:t></w:r></w:customXml><w:r><w:t> do not.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Given the limit </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>P</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, find the </w:t></w:r><w:r><w:rPr><w:b/><w:u/></w:rPr><w:t>area of the PPT</w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t> with perimeter </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\le P</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t> , such that the ratio of the areas of the triangle's circumcircle to its incircle,</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\left(\\frac^{A_c}_{A_I}\\right)</w:t></w:r></w:customXml><w:r><w:t>, </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t> is as small as possible.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>                                                              </w:t></w:r><w:customXml w:element=\"image\"><w:customXmlPr><w:attr w:name=\"height\" w:val=\"232\"/><w:attr w:name=\"width\" w:val=\"256\"/><w:attr w:name=\"verticalAlign\" w:val=\"baseline\"/><w:attr w:name=\"altText\" w:val=\"\"/><w:attr w:name=\"relationshipId\" w:val=\"rId1\"/></w:customXmlPr></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Please present the answer rounded-off to the nearest 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E1YeveKbXRXW2SN7u/kGY7aLr9WPYVD4n8RPpnlafpirNqdz/q1PIiX++1Y2l6StjvnnkNxezHM1w/JY+g9q+uyLh+OIgsXjPgfwx2cvPyj+L6dzgxWL5HyU9/yIJbHUdek87xHcsIuqWMDbY1+p7n/Oa0bWytrKPZaQRxD/AGFxn6nvU9FfoEWoQVKmlGK2S0X3frueO227vVhRRRUgFZ99oen6hkz26iTtLH8rA/UdfxrQoq4TlB3i7Ctcxlv9c8OyRM9xLqWnRnnPMiL3B/vDH5ew69vpmq2esWYudPmEsZ69ip9CO1YNZE1vdaLqH9q6IP8Ar4tB92UeoHr/AJ+viZpkmHzOF6aUKq2asoy8paaPs++/c66GKlRfvaxO1ktrlLuSa0eMeaoDCQHgjoRirlUdH1a21rTY7yzbKNwy90YdQavV+ZYydfn9jXjaUPdelnppZ97JWR7cOVrmjswoooriNAooooAKKKKACiiigBsiCSNkflWBB+hql8L5SfBEVu5y1pcSwn67i3/s1X6y/hqdtprsX9zV5uPwX/Cv0ngWb/2in/hf3XX6nn4te/B+v6Ha0UUV+kHOFFFFAHCeKPl+Jnh88fNbTr/46TW1WP40Bh8c+Frg/dYzxfiVAH862K/KuOI2x1KXeC/9KkdWD2kvP9EV72VY4ApMgMjhF8rG4n2zx2qDTZZXe5SQyFY3Cr5pBYcZIJHBq3NBHcR7JkDrnOD60kFvFbIVgQIrHcQO5/yK+Yp4rDxwMqDjeb62Wmq1TvporWt136HS4ydRS6EtZ9zff2TpV3e6g3yQl3HPVc/Ko9zwPxrQriPFpaebTND83zFuLiS5nIP8AYsF/Uj/AIDW2S4JY/ERw8no2r9+VJt2fTRfNtGeIqeyg5/1cq6JBNO0usah815fHef+mafwqPwx+la9IBgYFLX6zKSb0Vlsl2S2XyPn/UKKKKkAooooAKKKKACiiigDGhceG/FUF4rMlhfP5c6qcBHPRj7H/Gu9trlbhWwrI6NtdGHKn/PeuQ1SyXUdMntWAy6naT2bsfzrU8M39xq/hO1ljl2XcP7uTeMhmTjDfUYP418txTg4V6VPFvSV1Bvtu4t6PzT9F8/QwNRxbh8zoaKYkqO7IGUumN6g/dyKfX5vKLi7M9m9wooopAFFFFABRRRQAVlfDjJbxG4+42ry4/z+IrVrN+GA3eH9Qn/576nNID6/dH9K/R+BY+/iJeUV+L/yODF/FBep2dFFFfpRzBRRRQBxXxKXybTRdQ/59dTiLH0U5z/IVqUePNP/ALS8D6nCBlkh85fXKHd/TFUNDvf7R0Gyu85aWFS3+9jn9c1+ecc0G6dDELpzRf4Nfqb4V2qSj3sy/RRRX5ieiFefYjl8c3XlyeYlraqiN67zvz/48a9Brzyy8oeMNU+z7vKeG3ePd12mMY/nX3XBqvVrvXSPy3S++zdvmeZmPwx9TZooor7g8kKKKKAGu6xxs7kKqgliewridG8S6pNrtpJqEg/s3U2lW2Ty1Gwq2BzjJ9Ovet3xc92PDc8OnwSzz3BEIESlioPUnHbGRn3rnNX8Kaza6FA0OpG9OnlXgtorQKwOR0I5OOvPXFZTbvp0Prslw+Clh2sVKKdVuK5rtrTdWTs+Zx1bWiep31FYOn6rd3XiYW8u6KFtOScwOmCjlsHORn2xW9Widz5rE4aeGkoT3aT+8KKKKZzBUPghvI1XXbHoqXCzKv8Avg5/kKmqt4U58aa4R08uEH67a8zOYqWVYhPok/mpx/zOjCu1eP8AXQ66SEbzNGi+eEKqx4/A+2aLeV5oFeSJon6Mjdj/AFHvUtQzW/mvHIrtG8bZBXuO4I7g1+TxnGa5Knyertvp6X+7p2fvNNO6JqKiiuY5pJI1yHjOGVhgj0P0PrUtZThKD5ZKzKTTV0FFFFSMKKKKAK2pXP2PSru56eTC8n5KTUnw6tPsfgHTFI+aRGlJ9dzEj9CKwvHNy8XheS3h5mvZEtox6ljyPyBru9Ps10/TLWzj5S3hSJfooA/pX61wTh3TwFSs/tyt8or/ADkzzMS71kuy/P8A4YsUUUV9uZBRRRQA2RFljaOQBkYFWB7g1514PD6c2p+H5iS+mXTCPPeJuVP8z+Nej1wXiqL+wvHGn64OLXUF+xXR7K/VGP5Y+imvFz7BPHZbVpRV5L3l6x/zV18wjLkqRn8vvNyiiivwg9gbIwSJmbkKCTXCagq2njWzlRPLhv7AKi/3WXnH4ACu5uHMVrLIq7iqFguOuB0rmvGtjLcaDBqVumLnTnE4Uf3f4h/I/hX1fDOJjh8ZFT0VRuF791pp/itqcOMg503bpqJRUVrcR3drFcQnKSKGFS1+kNNOzPECiiikAUUUUAU10yFdafUw0nnPAICuRt2g5z0zn8auUUUGk6k6lud3srfJBRRRQZhUXgVPOuda1Dqs10Ikb1CD/wCyqrrV/wD2dpM0y/6wjZEPVz0/x/Cuk8M6WdH8OWlo4xKqbpf99uT+WcfhXg8SYhUMscOtRpfKPvP8eX7zswUOatfsatFFFflZ7pFOkjRN9nZY5sfK7LkfQ+1LFIWVVl2rNtDOgbO3P9OtSVE9vG88cxX95HnawOOD2PqPatozi48k/k7a3tt6fluuqctO90S0VDbTtPG3mRNDIjbWVumfY9x71NUVISpycJboaaaugoopk00dvBJNMwSONSzsegAGSalJt2QzAuU/tn4jaRpyDdFpym+uPQH+D8c4/OvQ64r4cWr3FrqHiK6QrNqs5MYbqsS8KP5/kK7Wv6By3B/UMFSwvWK19Xq/xZ4ylztz7/l0Ciiiu8YUUUUAFZPifRE8Q+HLvTnwHlTMTH+Fxyp/P9M1rUU02ndCaTVmcH4V1Z9T0cR3YKX1mxt7pG+8HXjJ+v8APNbdYHiu2bwx4oi8SQIfsF5iDUVUfcb+GTH+f1reR1kRXRgysMhgcgj1r8W4nyn+zsa501+7qax8u8fk/wALHfhqrnHlluv6uRXbSrZTtbDdMI2MY9Wxx+tSsodCrgMrDBBHBqvqDlLGTbMkLsMK7ttAP1pDqmng4N9bA+8y/wCNeAqU5UYuEb6vZO/T8O3zNuZKTucP5DeFtbbTJzjT7pi9lKTwpPVCf8/rWxW1qOn2HiHSmt7jZPBJ910YHaRxlT6iuLE914buxp2vMWgJxbX2PlcejHsf8+9fqGT5tHM6fsqmleO6/mt1Xn/MvmutvExOHdF80fhf4G1RSKwZQykEEZBHelr2DlCiiigAooooAKQnAyeBSSSJFG0krqiKMlmOAKybeG78X3LW9kz2+ko22e6xgzf7Kf5/wKnKnSputWlywju3/WrfRdRxjKUuWKuybRbY+JvEK3rAnTNOf90SOJpfX6D/AA9a7yoLKyg0+yitLOMRwxLtVRU9flGdZpLMsT7RK0I6RXZefm93/wAA+gw9BUYW69QooorxToCiiigCG5heaMCKVopFO5WHIz6Edx7U5J43meEN+8jALKRjg9/cVJUcke7LoEEwUqjsuduf6cCt4zUo8k9unk3bfS7Xl03XVOWmndElc14peXVLux8M2LFZtRfM7L/yzhXlj+OP0I71r3OpLp2kSXupgQ+ShZ1DZzjjj68Y+tM8B6ROyz+JNWTF/qXMaH/ljD/Co+vB/L3r7LhPKHWxjxVVe5SfqnLpbo0t/u7nHiqt4qnHr+X9aHW2ttFZ2kVtbIEhhQIijsoGAKloor9YOUKKKKACiiigAooooAgvbODULGa0vIxLBMhR0PcGvPdIln8May3hjVnZo8ltNuX6Sx/3M+o/+t6V6TWR4l8O2viXSWtLn93Kp3wTr96F+xH9RXBmOX0cyw0sNW2eqfZ9H/n3Qk5Qkpx3X4+Ry3i6BLmxsIZQrJJqECsGGQQWxzV8aBo4GBpNiB/17J/hXOCa/wBRaPw7rNwlhrNlOkySvFvW5CHKsvIz05//AF41zpniEtn/AISGID0Gnr/8VX5VisLWwNOGDrV1TlFydvfs07WacYtNO39M64SjUk5qN727fqxfCChPDMKqAoWacAAYx++fitW6tLe+tnt7yFJoXGGRxkGqGkaRPpdpHAb9pQru74iVQ5Zix9cdfWr6wOLgym5lZe0RC7R/47n9a8fHThPG1a9KqtZOSfvJ73XRa/1c3ppqmoyj5dDj7nwrqmjO0nhyZbm1zn7DcNyvsrf44/GqX/CSQ28vk6vbXGnTf3Zozg/Qjr9a7yO28t3YzzPu7M3C/SmNptvJbSQTiSeOT7wlkZvyyePwr6fCcVSprkxi9pa2q0l53ezt5q77nFUwKlrT0/I5aHUbK4H7i7hk9lkBNT70xncv51oN4M8Ptj/iWxgjurMD+hqFfAmgLIzrayAt1xcOP616v+tGUv8A5+L/ALdi/wD285/qNfy+9/5GbNqdjbgma7hXHbeM/lWcfEa3Uog0a0nv5nO1SF2Jn/eNdba+FNDszmDTYuevmZfP/fRNaiW8MWPLhjTaMDaoGK56vFuCgv3NKUn/AHmkvuV/zRccvqP4pJehxdh4aOq3wPia/hlkQ5XTbeT5U/3sHJ/zzXXWs1msZgs9ipCuNka4Cj0AFWqK+QzHNquYy5q97LZJpRXeyt1+/u2ejRoKirRK6XsUkLyRiVlTr+6YE/QYyfwo+1g2/nLDOwzjb5ZDfkasUV5nNRvpF79X07bL7zW0u5Xe5dYUdLWeQt/Au0FfrkilkmmVkEdq8gYcnco2/Xn+VT0UKpBfYXXv/n0/4cfK+5C0k4uAqwKYu8hkxj8MUK1wbgh4ohD2cSEsfw24H51NRS9pG1uRbW6/fvv+HkFn3II/tRL+aIRx8m0k/nSKLsQP5ssAk/hYRnav1G7n8xViuXuprrxlqr6HokpisIjjUL5OmP8Anmh7k/546+vlWAxGaYn2NGKS0bdtIpev5dTGrONKN3q+nmR6ZZS+Otd3XJSTRNNcqZFj2i7kyDgZz8owO/8APj00DAwOBVXTdOtdJ06GxsIhFbwrtRR/M+pPXNWq/bsPQp4ajGhRVox2/V6aavX8jzYp3cnuwooorYoKKKKACiiigAooooAKKKKAMTxN4YtPEtiscxaC6hO62uo+Hib/AA9q5XTtdu9Nv10XxZGLa+6Q3P8Ayyuh2IPY/wCeDxXotUNZ0Sw1/T2s9UgWaJuQejIfVT2NebmWV4bNKPscQttmt4/8DuuvrqEXKEuaG/5mbRXMTx674IGL1ZNY0Rfu3MY/fW6/7Q7gev6jpW7p+pWeq2i3OnzpPE3dT0PoR1B9jX47m2RYzK5fvVeD2ktn/k/J/iejSrwq6bPsWqKKK8I3CiiigArPvJpF1S2SGNpCqOxRWAz0AzWhVOe2uPtoubZ48iPZskBwec9vwr08slShXbqW+GSV72baa1tr18vVGVVNx0JrW5W6gEiAryQVbqCOCKmqCztja2wjZt7klnb1JOTU9cuKVFYiaofBd29OhcL8q5twooormKCiiigApskiRRtJKyoijLMxwAPXNZus+ItP0OMfbJd0z/6u3jG6Rz7D+pqlZ+G9W8Xyrc+JhJp2lA7o9NRsSS+8h7fTr9OtfTZPw5i8zam/cpfzPr/hXV/h3ZzVcRGn7q1fb/Mr+ff+OLx9P0F2tdJjbbdajjBk9Uj/AMf8nvdI0iy0PTY7HTYRFBH+bHuSe5NWLW0t7G1jtrOFIIYxhI41wFFS1+v4HA4fAUFQw0bR/Fvu31f5dDz/AHpS5p6sKKKK7BhRRRQAUUUUAFFFFABRRRQAUUUUAFFFFAB161yGrfD60mumv/D1zJo1+eS0H+rc/wC0nT8vyNdfRQ7NOLV0909U/VEuKe55xJrWt+Hf3firSnkhX/mIWI3xkerD+H9PpWxputabq8e7TryKfjJUHDD6qeRXX1zmreAvD+ry+dJZ/ZbjOfPtG8ps+vHBPuRXyOP4Ry7FNyo3pS8tY/c9vk7eRtGvVh5rz3+8dRWLJ4M8RadzofiQ3CDpBqMe/wD8fGT+gqFpvG1l/wAffh62vVHVrS5C/o2TXyOI4LzGm/3LjNeTs/8Aya35m6xkPtJr8fyOgormz4suoOL3wvrURHXZbbx+fFMPjvT04msdSiPo9qQf515UuGs3jvh5fKz/ACNPrVH+Y6eiuZHjmyb/AFWm6rJ/uWmf604eJtTuf+Qd4U1aYHo00flA/ic04cM5xN2VBr1svzaD61R/mOkorn1j8daicQaXY6Uh/juZ/MI/75z+oqdPh/qOof8AIxeJbqdD96C0UQofbPcfhXsYbgnH1Na84wXrzP8ADT8TOWMj9mLf4fmLqfirR9Jyt1eoZRx5MXzvn0wOn44qrCPFfibH9n2o0Kwb/l5u1zMw9VTt+P4Gur0jwpomhAHTNOhikH/LVhuf/vo5NbFfY4DhbLcE1KS9pLvLb5R2++5zzq1am7svL/P/AIY57w/4L0zQJDcgNeag/Ml7cndIT7en4c+5roaKK+obuZqKirIKKKKQwooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooA//2Q==","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; "><div style="block-size: 427px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 213.5px; transform-origin: 407px 213.5px; vertical-align: baseline; "><div style="block-size: 50px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 25px; text-align: left; transform-origin: 384px 25px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">We define a Partial Pythagorean Triangle (PPT) as a right triangle wherein the hypotenuse and at least one leg are integers. Thus, the triples </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="62" height="19" style="width: 62px; height: 19px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> and </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="79.5" height="21" style="width: 79.5px; height: 21px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> represent a PPT, while </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="79.5" height="21" style="width: 79.5px; height: 21px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">, </span></span><span style="vertical-align:-10px"><img src="data:image/png;base64,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" width="67" height="29" style="width: 67px; height: 29px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> and </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="62" height="19" style="width: 62px; height: 19px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> do not.</span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""></span></span></div><div style="block-size: 61px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 30.5px; text-align: left; transform-origin: 384px 30.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; ">Given the limit </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">P</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; ">, find the </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; text-decoration-line: underline; ">area of the PPT</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; "> with perimeter </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="26.5" height="18" style="width: 26.5px; height: 18px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; "> , such that the ratio of the areas of the triangle's circumcircle to its incircle,</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> </span></span><span style="vertical-align:-15px"><img src="data:image/png;base64,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" width="36" height="40" style="width: 36px; height: 40px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">, </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; "> is as small as possible.</span></span></div><div style="block-size: 238px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; 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" data-image-state="image-loaded"></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">Please present the answer rounded-off to the nearest integer.</span></span></div></div></div>

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Last Solution submitted on Apr 26, 2023

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David Hill on 14 Jun 2023

Several tests are wrong (verified using symbolic numbers). Test 6 should be 970225, Test 8 should be 96049800, Test 9 should be 559819260, Test 10 should be 38034750625, and a suspect higher tests might also be incorrect. These results are consistent with what I expected by review of Lucas and Pell numbers.

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Problem 57874. Easy Sequences 107: Minimized Circumcircle-Incircle Areas Ratio of Partial Pythagorean Triangles

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Problem 57874. Easy Sequences 107: Minimized Circumcircle-Incircle Areas Ratio of Partial Pythagorean Triangles

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