https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Slicedclementines&feedformat=atom AoPS Wiki - User contributions [en] 2021-09-24T15:57:54Z User contributions MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=2016_IMO_Problems/Problem_3&diff=85315 2016 IMO Problems/Problem 3 2017-04-20T03:25:09Z <p>Slicedclementines: /* Problem */</p> <hr /> <div>==Problem==<br /> Let &lt;math&gt;P = A_1A_2 \cdots A_k&lt;/math&gt; be a convex polygon in the plane. The vertices &lt;math&gt;A_1,A_2,\dots, A_k&lt;/math&gt; have integral coordinates and lie on a circle. Let &lt;math&gt;S&lt;/math&gt; be the area of &lt;math&gt;P&lt;/math&gt;. And odd positive integer &lt;math&gt;n&lt;/math&gt; is given such that the squares of the side lengths of &lt;math&gt;P&lt;/math&gt; are integers divisible by &lt;math&gt;n&lt;/math&gt;. Prove that &lt;math&gt;2S&lt;/math&gt; is an integer divisible by &lt;math&gt;n&lt;/math&gt;.</div> Slicedclementines https://artofproblemsolving.com/wiki/index.php?title=2016_IMO_Problems/Problem_3&diff=85314 2016 IMO Problems/Problem 3 2017-04-20T03:24:50Z <p>Slicedclementines: added problem</p> <hr /> <div>==Problem==<br /> Let &lt;math&gt;P = A_1A_2 \cdots A_k&lt;/math&gt; be a convex polygon in the plane. The vertices &lt;math&gt;A_1,A_2,\dots, A_k&lt;/math&gt; have integral coordinates and lie on a circle. Let &lt;math&gt;S&lt;/math&gt; be the area of &lt;math&gt;P&lt;/math&gt;. And odd positive integer &lt;math&gt;n&lt;/math&gt; is given such that the squares of the side lenghts of &lt;math&gt;P&lt;/math&gt; are integers divisible by &lt;math&gt;n&lt;/math&gt;. Prove that &lt;math&gt;2S&lt;/math&gt; is an integer divisible by &lt;math&gt;n&lt;/math&gt;.</div> Slicedclementines