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71 lines
3.0 KiB
71 lines
3.0 KiB
<!DOCTYPE html>
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<html lang="en">
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<head>
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<title>USAMTS 2/5/35 Solution</title>
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<meta charset="UTF-8">
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<meta name="viewport" content="width=device-width, initial-scale=1">
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<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/@picocss/pico@1/css/pico.min.css">
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<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16.9/dist/katex.min.css" integrity="sha384-n8MVd4RsNIU0tAv4ct0nTaAbDJwPJzDEaqSD1odI+WdtXRGWt2kTvGFasHpSy3SV" crossorigin="anonymous">
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<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.9/dist/katex.min.js" integrity="sha384-XjKyOOlGwcjNTAIQHIpgOno0Hl1YQqzUOEleOLALmuqehneUG+vnGctmUb0ZY0l8" crossorigin="anonymous"></script>
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<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.9/dist/contrib/auto-render.min.js" integrity="sha384-+VBxd3r6XgURycqtZ117nYw44OOcIax56Z4dCRWbxyPt0Koah1uHoK0o4+/RRE05" crossorigin="anonymous"
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onload="renderMathInElement(document.body);"></script>
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</head>
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<body>
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<main class="container">
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<h1> USAMTS 5/2/35 </h1>
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<strong><h3>Problem</h3></strong>
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<p>
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Let \(m\) and \(n\) be positive integers. Let \(S\) be the set of all points
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\((x, y)\) with integer coordinates such that \(1 \leq x, y \leq m + n − 1\)
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and \(m + 1 \leq x + y \leq 2m + n − 1\). Let L be the set of the \(3m + 3n −
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3\) lines parallel to one of \(x = 0\), \(y = 0\), or \(x + y = 0\) and passing
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through at least one point in \(S\). For which pairs \((m, n)\) does there
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exist a subset \(T\) of \(S\) such that every line in \(L\) intersects an odd
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number of elements of \(T\)?
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</p>
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<strong><h3>Proof</h3></strong>
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<p>
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The answer is just all \((m,n)\) such that \( (m+n-1)(m-n) \) is divisible
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by 4. The sum of all the x and y coordinates should clearly be even, but
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the sum of all \(x+y \mod 2\) is \( \frac{(m+n-1)(3m+n)}{2} \). For a
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construction to even be viable, this quantity clearly needs to be even.
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</p>
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<strong><h3>Constructor</h3></strong>
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<form method="POST" action="{{url_for ('index')}}">
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<table role="grid">
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<tbody>
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<tr>
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<td>
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<label for="M">Value for M:</label>
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</td>
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<td>
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<input type="number" name="M" min="1" max="40" value="{{M}}" required />
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</td>
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</tr>
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<tr>
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<td>
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<label for="N">Value for N:</label>
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</td>
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<td>
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<input type="number" name="N" min="1" max="40" value="{{N}}" required />
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</td>
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</tr>
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</tbody>
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</table>
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<input type="submit" value="Construct">
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</form>
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{% if invalid %}
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<p> Invalid Input! </p>
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{% endif %}
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{% if nosol %}
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<p> No Valid Solution is Possible! </p>
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{% endif %}
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{% if image %}
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<img src="{{ image }}" alt="Solution Image" />
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{% endif %}
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{% if works == False %}
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<p> Claimed Construction Fails!!!! </p>
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{% endif %}
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</main>
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</body>
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</html>
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