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<!DOCTYPE html><html lang="en"> <head> <title>USAMTS 2/5/35 Solution</title> <meta charset="UTF-8"> <meta name="viewport" content="width=device-width, initial-scale=1"> <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/@picocss/pico@1/css/pico.min.css"> <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16.9/dist/katex.min.css" integrity="sha384-n8MVd4RsNIU0tAv4ct0nTaAbDJwPJzDEaqSD1odI+WdtXRGWt2kTvGFasHpSy3SV" crossorigin="anonymous"> <script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.9/dist/katex.min.js" integrity="sha384-XjKyOOlGwcjNTAIQHIpgOno0Hl1YQqzUOEleOLALmuqehneUG+vnGctmUb0ZY0l8" crossorigin="anonymous"></script> <script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.9/dist/contrib/auto-render.min.js" integrity="sha384-+VBxd3r6XgURycqtZ117nYw44OOcIax56Z4dCRWbxyPt0Koah1uHoK0o4+/RRE05" crossorigin="anonymous" onload="renderMathInElement(document.body);"></script> </head> <body> <main class="container"> <strong><h3>Problem</h3></strong> <p> Let \(m\) and \(n\) be positive integers. Let \(S\) be the set of all points \((x, y)\) with integer coordinates such that \(1 \leq x, y \leq m + n − 1\) and \(m + 1 \leq x + y \leq 2m + n − 1\). Let L be the set of the \(3m + 3n − 3\) lines parallel to one of \(x = 0\), \(y = 0\), or \(x + y = 0\) and passing through at least one point in \(S\). For which pairs \((m, n)\) does there exist a subset \(T\) of \(S\) such that every line in \(L\) intersects an odd number of elements of \(T\)? </p> <strong><h3>Proof</h3></strong> <p> The answer is just all \((m,n)\) such that \( (m+n-1)(m-n) \) is divisible by 4. The sum of all the x and y coordinates should clearly be even, but the sum of all \(x+y \mod 2\) is \( \frac{(m+n-1)(3m+n)}{2} \). For a construction to even be viable, this quantity clearly needs to be even. </p> <strong><h3>Constructor</h3></strong> {% if invalid %} <p> Invalid Input! </p> {% endif %} {% if nosol %} <p> No Valid Solution is Possible! </p> {% endif %} {% if image %} <img src="{{ image }}" alt="Solution Image" /> {% endif %} {% if works == False %} <p> Claimed Construction Fails!!!! </p> {% endif %} <form method="POST" action="{{url_for ('index')}}"> <table role="grid"> <tbody> <tr> <td> <label for="M">Value for M:</label> </td> <td> <input type="number" name="M" min="1" max="40" value="{{M}}" required /> </td> </tr> <tr> <td> <label for="N">Value for N:</label> </td> <td> <input type="number" name="N" min="1" max="40" value="{{N}}" required /> </td> </tr> </tbody> </table> <input type="submit" value="Construct"> </form> </main> </body></html>
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