82 lines
6.6 KiB
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82 lines
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HTML
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<h2>Three similar triangles</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 299</h3><span style="width:300px;color:#666;">Published on Saturday, 3rd July 2010, 01:00 am; Solved by 422</span></div>
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<p>Four points with integer coordinates are selected:<br />A(<var>a</var>, 0), B(<var>b</var>, 0), C(0, <var>c</var>) and D(0, <var>d</var>),
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with 0 <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> <var>a</var> <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> <var>b</var> and 0 <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> <var>c</var> <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> <var>d</var>.<br />
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Point P, also with integer coordinates, is chosen on the line AC so that the three triangles ABP, CDP and BDP are all <dfn title="Have equal angles">similar</dfn>.</p>
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<div align='center'><img src="project/images/p_299_ThreeSimTri.gif" /></div>
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<p>It is easy to prove that the three triangles can be similar, only if <var>a</var>=<var>c</var>.</p>
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<p>So, given that <var>a</var>=<var>c</var>, we are looking for triplets (<var>a</var>,<var>b</var>,<var>d</var>) such that at least one point P (with integer coordinates) exists on AC, making the three triangles ABP, CDP and BDP all similar.</p>
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<p>For example, if (<var>a</var>,<var>b</var>,<var>d</var>)=(2,3,4), it can be easily verified that point P(1,1) satisfies the above condition.
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Note that the triplets (2,3,4) and (2,4,3) are considered as distinct, although point P(1,1) is common for both.</p>
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<p>If <var>b</var>+<var>d</var> <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> 100, there are 92 distinct triplets (<var>a</var>,<var>b</var>,<var>d</var>) such that point P exists.<br />
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If <var>b</var>+<var>d</var> <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> 100 000, there are 320471 distinct triplets (<var>a</var>,<var>b</var>,<var>d</var>) such that point P exists.</p>
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<p>If <var>b</var>+<var>d</var> <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> 100 000 000, how many distinct triplets (<var>a</var>,<var>b</var>,<var>d</var>) are there such that point P exists?</p>
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