2018 AIME I Problems/Problem 4

Problem 4

In $\triangle ABC, AB = AC = 10$ and $BC = 12$ . Point $D$ lies strictly between $A$ and $B$ on $\overline{AB}$ and point $E$ lies strictly between $A$ and $C$ on $\overline{AC}$ ) so that $AD = DE = EC$ . Then $AD$ can be expressed in the form $\dfrac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p+q$ .

Solution 1

$[asy] import cse5; unitsize(10mm); pathpen=black; dotfactor=3; pair B = (0,0), A = (6,8), C = (12,0); D(A--B); D(C--B); D(A--C); dot(dotted); label("$A$",A,SW); label("$B$",B,S); label("$C$",C,SE); [/asy]$

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2018 AIME I Problems/Problem 4

Problem 4

Solution 1