Problem

2025 AIME II Problem 5

Suppose \triangle ABC has angles \angle BAC = 84^\circ, \angle ABC=60^\circ, and \angle ACB = 36^\circ. Let D, E, and F be the midpoints of sides \overline{BC}, \overline{AC}, and \overline{AB}, respectively. The circumcircle of \triangle DEF intersects \overline{BD}, \overline{AE}, and \overline{AF} at points G, H, and J, respectively. The points G, D, E, H, J, and F divide the circumcircle of \triangle DEF into six minor arcs, as shown. Find \text{arc }DE+2\cdot \text{arc }HJ + 3\cdot \text{arc }FG, where the arcs are measured in degrees.

[asy] import olympiad; size(6cm); defaultpen(fontsize(10pt)); pair B = (0, 0), A = (Cos(60), Sin(60)), C = (Cos(60)+Sin(60)/Tan(36), 0), D = midpoint(B--C), E = midpoint(A--C), F = midpoint(A--B); guide circ = circumcircle(D, E, F); pair G = intersectionpoint(B--D, circ), J = intersectionpoints(A--F, circ)[0], H = intersectionpoints(A--E, circ)[0]; draw(B--A--C--cycle); draw(D--E--F--cycle); draw(circ); dot(A);dot(B);dot(C);dot(D);dot(E);dot(F);dot(G);dot(H);dot(J); label("$A$", A, (0, .8)); label("$B$", B, (-.8, -.8)); label("$C$", C, (.8, -.8)); label("$D$", D, (0, -.8)); label("$E$", E, (.8, .2)); label("$F$", F, (-.8, .2)); label("$G$", G, (0, .8)); label("$H$", H, (-.2, -1)); label("$J$", J, (.2, -.8)); [/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 II
Points: 3
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