Difference between revisions of "1983 AIME Problems/Problem 12"
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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>. | 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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== See also == | == See also == | ||
| + | {{AIME box|year=1983|num-b=11|num-a=13}} | ||
* [[AIME Problems and Solutions]] | * [[AIME Problems and Solutions]] | ||
* [[American Invitational Mathematics Examination]] | * [[American Invitational Mathematics Examination]] | ||
Revision as of 14:12, 6 May 2007
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 .
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 
.
See also
| 1983 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 11 |
Followed by Problem 13 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
