Problem

2025 AIME I Problem 2

On \triangle ABC points A, D, E, and B lie in that order on side \overline{AB} with AD = 4, DE = 16, and EB = 8. Points A, F, G, and C lie in that order on side \overline{AC} with AF = 13, FG = 52, and GC = 26. Let M be the reflection of D through F, and let N be the reflection of G through E. Quadrilateral DEGF has area 288. Find the area of heptagon AFNBCEM.


[asy] unitsize(14); pair A = (0, 9), B = (-6, 0), C = (12, 0), D = (5A + 2B)/7, E = (2A + 5B)/7, F = (5A + 2C)/7, G = (2A + 5C)/7, M = 2F - D, N = 2E - G; filldraw(A--F--N--B--C--E--M--cycle, lightgray); draw(A--B--C--cycle); draw(D--M); draw(N--G); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); dot(G); dot(M); dot(N); label("$A$", A, dir(90)); label("$B$", B, dir(225)); label("$C$", C, dir(315)); label("$D$", D, dir(135)); label("$E$", E, dir(135)); label("$F$", F, dir(45)); label("$G$", G, dir(45)); label("$M$", M, dir(45)); label("$N$", N, dir(135)); [/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: 3
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