Difference between revisions of "2013 AMC 12A Problems/Problem 19"
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<math>x(x+y) = 2013</math> | <math>x(x+y) = 2013</math> | ||
− | The prime factors of 2013 are 3, 11, and 61. Obviously <math>x < x+y</math>. In addition, by the Triangle Inequality, <math>BC < AB + AC</math>, so <math>x+y < 183</math>. Therefore, <math>x</math> must equal 33, and <math>x+y</math> must equal 61. <math>\boxed{D}</math> | + | The prime factors of 2013 are 3, 11, and 61. Obviously, <math>x < x+y</math>. In addition, by the Triangle Inequality, <math>BC < AB + AC</math>, so <math>x+y < 183</math>. Therefore, <math>x</math> must equal 33, and <math>x+y</math> must equal 61. <math>\boxed{D}</math> |
-Solution by '''fowlmaster''' | -Solution by '''fowlmaster''' |
Revision as of 19:08, 8 February 2013
Let CX=x, BX=y. Let the circle intersect AC at D and the diameter including AD intersect the circle again at E. Use power of a point on point C to the circle centered at A.
So CX*CB=CD*CE x(x+y)=(97-86)(97+86) x(x+y)=3*11*61.
Obviously x+y>x so we have three solution pairs for (x,x+y)=(1,2013),(3,671),(11,183),(33,61). By the Triangle Inequality, only x+y=61 yields a possible length of BX+CX=BC.
Therefore, the answer is D) 61.
Solution 2
Let represent
, and let
represent
. Since the circle goes through
and
,
=
= 86.
Then by Stewart's Theorem,
(Since cannot be equal to 0, dividing both sides of the equation by
is allowed.)
The prime factors of 2013 are 3, 11, and 61. Obviously, . In addition, by the Triangle Inequality,
, so
. Therefore,
must equal 33, and
must equal 61.
-Solution by fowlmaster