Difference between revisions of "1983 AIME Problems/Problem 12"
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== Problem == | == Problem == | ||
+ | The length of diameter <math>AB</math> is a two digit integer. Reversing the digits gives the length of a perpendicular chord <math>CD</math>. The distance from their intersection point <math>H</math> to the center <math>O</math> is a positive rational number. Determine the length of <math>AB</math>. | ||
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+ | [img]http://www.artofproblemsolving.com/Forum/album_pic.php?pic_id=792&sid=cfd5dae222dd7b8944719b56de7b8bf7[/img] | ||
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+ | {{image}} | ||
== Solution == | == Solution == | ||
+ | Let <math>AB=10x+y</math> and <math>CD=10y+x</math>. It follows that <math>CO=\frac{AB}{2}=\frac{10x+y}{2}</math> and <math>CH=\frac{CD}{2}=\frac{10y+x}{2}</math>. Applying the [[Pythagorean Theorem]] on <math>CO</math> and <math>CH</math>, <math>OH=\sqrt{\left(\frac{10x+y}{2}\right)^2-\left(\frac{10y+x}{2}\right)^2}=\sqrt{\frac{9}{4}\cdot 11(x+y)(x-y)}=\frac{3}{2}\sqrt{11(x+y)(x-y)}</math>. | ||
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+ | Because <math>OH</math> is a positive rational number, the quantity <math>\sqrt{11(x+y)(x-y)}</math> cannot contain any square roots. Therefore, <math>x+y</math> must equal eleven and <math>x-y</math> must be a perfect square (since <math>x+y>x-y</math>). The only pair <math>(x,y)</math> that satisfies this condition is <math>(6,5)</math>, so our answer is <math>65</math>. | ||
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+ | ---- | ||
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+ | * [[1983 AIME Problems/Problem 11|Previous Problem]] | ||
+ | * [[1983 AIME Problems/Problem 13|Next Problem]] | ||
+ | * [[1983 AIME Problems|Back to Exam]] | ||
== See also == | == See also == | ||
− | * [[ | + | * [[AIME Problems and Solutions]] |
+ | * [[American Invitational Mathematics Examination]] | ||
+ | * [[Mathematics competition resources]] | ||
+ | |||
+ | [[Category:Intermediate Geometry Problems]] |
Revision as of 23:18, 23 July 2006
Problem
The length of diameter is a two digit integer. Reversing the digits gives the length of a perpendicular chord . The distance from their intersection point to the center is a positive rational number. Determine the length of .
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Solution
Let and . It follows that and . Applying the Pythagorean Theorem on and , .
Because is a positive rational number, the quantity cannot contain any square roots. Therefore, must equal eleven and must be a perfect square (since ). The only pair that satisfies this condition is , so our answer is .