Difference between revisions of "1989 AIME Problems/Problem 10"

Revision as of 23:15, 24 February 2007

Problem

Let $a_{}^{}$ , $b_{}^{}$ , $c_{}^{}$ be the three sides of a triangle, and let $\alpha_{}^{}$ , $\beta_{}^{}$ , $\gamma_{}^{}$ , be the angles opposite them. If $a^2+b^2=1989^{}_{}c^2$ , find

$\frac{\cot \gamma}{\cot \alpha+\cot \beta}$

Solution

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 == Problem ==
-A sample of 121 integers is given, each between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let <math>D^{}_{}</math> be the difference between the mode and the arithmetic mean of the sample. What is the largest possible value of <math>\lfloor D^{}_{}\rfloor</math>? (For real <math>x^{}_{}</math>, <math>\lfloor x^{}_{}\rfloor</math> is the greatest integer less than or equal to <math>x^{}_{}</math>.)
+Let <math>a_{}^{}</math>, <math>b_{}^{}</math>, <math>c_{}^{}</math> be the three sides of a triangle, and let <math>\alpha_{}^{}</math>, <math>\beta_{}^{}</math>, <math>\gamma_{}^{}</math>, be the angles opposite them. If <math>a^2+b^2=1989^{}_{}c^2</math>, find
+<center><math>\frac{\cot \gamma}{\cot \alpha+\cot \beta}</math></center>
 == Solution ==

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Difference between revisions of "1989 AIME Problems/Problem 10"

Revision as of 23:15, 24 February 2007

Problem

Solution

See also