Problem

2013 AIME I Problem 9

A paper equilateral triangle ABC has side length 12. The paper triangle is folded so that vertex A touches a point on side \overline{BC} a distance 9 from point B. The length of the line segment along which the triangle is folded can be written as \frac{m\sqrt{p}}{n}, where m, n, and p are positive integers, m and n are relatively prime, and p is not divisible by the square of any prime. Find m+n+p.

[asy] import cse5; size(12cm); pen tpen = defaultpen + 1.337; real a = 39/5.0; real b = 39/7.0; pair B = MP("B", (0,0), dir(200)); pair A = MP("A", (9,0), dir(-80)); pair C = MP("C", (12,0), dir(-20)); pair K = (6,10.392); pair M = (a*B+(12-a)*K) / 12; pair N = (b*C+(12-b)*K) / 12; draw(B--M--N--C--cycle, tpen); draw(M--A--N--cycle); fill(M--A--N--cycle, mediumgrey); pair shift = (-20.13, 0); pair B1 = MP("B", B+shift, dir(200)); pair A1 = MP("A", K+shift, dir(90)); pair C1 = MP("C", C+shift, dir(-20)); draw(A1--B1--C1--cycle, tpen);[/asy]

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.


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Category: AIME I
Points: 4
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