Problem

2022 AIME I Problem 11

Let ABCD be a parallelogram with \angle BAD < 90^\circ. A circle tangent to sides \overline{DA}, \overline{AB}, and \overline{BC} intersects diagonal \overline{AC} at points P and Q with AP < AQ, as shown. Suppose that AP=3, PQ=9, and QC=16. Then the area of ABCD can be expressed in the form m\sqrt{n}, where m and n are positive integers, and n is not divisible by the square of any prime. Find m+n.

[asy] defaultpen(linewidth(0.6)+fontsize(11)); size(8cm); pair A,B,C,D,P,Q; A=(0,0); label("$A$", A, SW); B=(6,15); label("$B$", B, NW); C=(30,15); label("$C$", C, NE); D=(24,0); label("$D$", D, SE); P=(5.2,2.6); label("$P$", (5.8,2.6), N); Q=(18.3,9.1); label("$Q$", (18.1,9.7), W); draw(A--B--C--D--cycle); draw(C--A); draw(Circle((10.95,7.45), 7.45)); dot(A^^B^^C^^D^^P^^Q); [/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: 5
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