Difference between revisions of "2018 AIME I Problems/Problem 7"
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+ | We can consider two cases: when the three vertices are on one base, and when the vertices are on two bases. | ||
+ | Case 1: vertices are on one base. Then we can call one of the vertices <math>A</math> for distinction. Either the triangle can have sides <math>1, 1, \sqrt{3}</math> with 6 cases or <math>2, 2, 2</math> with 2 cases. This can be repeated on the other base for <math>16</math> cases. | ||
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+ | Case 2: The vertices span two bases. WLOG call the only vertex on one of the bases <math>X</math>. Call the closest vertex on the other base <math>B</math>, and label clockwise <math>C, D, E, F, G</math>. We will multiply the following scenarios by <math>12</math>, because the top vertex can have <math>6</math> positions and the top vertex can be on the other base. We can have <math>XCG, XDF</math>, but we are not done! Don't forget that the problem statement implies that the longest diagonal in a base is <math>2</math> and the height is <math>2</math>, so <math>XBE</math> is also correct! Those are the only three cases, so there are <math>12*3=36</math> cases for this case. | ||
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+ | In total there's <math>\boxed{052}</math> cases. |
Revision as of 17:48, 7 March 2018
We can consider two cases: when the three vertices are on one base, and when the vertices are on two bases.
Case 1: vertices are on one base. Then we can call one of the vertices for distinction. Either the triangle can have sides
with 6 cases or
with 2 cases. This can be repeated on the other base for
cases.
Case 2: The vertices span two bases. WLOG call the only vertex on one of the bases . Call the closest vertex on the other base
, and label clockwise
. We will multiply the following scenarios by
, because the top vertex can have
positions and the top vertex can be on the other base. We can have
, but we are not done! Don't forget that the problem statement implies that the longest diagonal in a base is
and the height is
, so
is also correct! Those are the only three cases, so there are
cases for this case.
In total there's cases.