2013 AMC 12A Problems/Problem 19

Revision as of 20:10, 7 February 2013 by Epicwisdom (talk | contribs) (moved 2013 AMC 12A Problems/Problems 19 to 2013 AMC 12A Problems/Problem 19: Wrong title ("Problems 19"))

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.

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