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AIME 2024 / AIME 2024 train cedfddc6-965b-5cbd-bddd-0a83e0fb41c3

Problem

Answer published by the source. Consult the official source to check your work against its answer.

Problem

Let ABCABC be a triangle inscribed in circle ω\omega. Let the tangents to ω\omega at BB and CC intersect at point DD, and let AD¯\overline{AD} intersect ω\omega at PP. If AB=5AB=5, BC=9BC=9, and AC=10AC=10, APAP can be written as the form mn\frac{m}{n}, where mm and nn are relatively prime integers. Find m+nm + n. [asy] import olympiad; unitsize(15); pair A, B, C, D, E, F, P, O; C = origin; A = (10,0); B = (7.8, 4.4899); draw(A--B--C--cycle); draw(A..B..C..cycle, red+dotted); O = circumcenter(A, B, C); E = rotate(90,B) * (O); F = rotate(90,C) * (O); D = IP(B..E + (B-E)*4, C..F + (C-F)*-3); draw(B--D--C--D--A); P = IP(D..A, A..B..C); dot(A); dot(B); dot(C); dot(D); dot(P); label("AA", A, dir(335)); label("BB", B, dir(65)); label("CC", C, dir(200)); label("DD", D, dir(135)); label("PP", P, dir(235)); [/asy]
Plain-text mathematical notation (without MathML)
Let ABC be a triangle inscribed in circle ω. Let the tangents to ω at B and C intersect at point D, and let (AD)¯ intersect ω at P. If AB=5, BC=9, and AC=10, AP can be written as the form (m)/(n), where m and n are relatively prime integers. Find m+n.

[asy] import olympiad;  unitsize(15);  pair A, B, C, D, E, F, P, O;  C = origin; A = (10,0); B = (7.8, 4.4899); draw(A--B--C--cycle); draw(A..B..C..cycle, red+dotted);  O = circumcenter(A, B, C);  E = rotate(90,B) * (O); F = rotate(90,C) * (O);  D = IP(B..E + (B-E)*4, C..F + (C-F)*-3);  draw(B--D--C--D--A);  P = IP(D..A, A..B..C);  dot(A); dot(B); dot(C); dot(D); dot(P); label("A", A, dir(335)); label("B", B, dir(65)); label("C", C, dir(200)); label("D", D, dir(135)); label("P", P, dir(235)); [/asy]
Original LaTeX notation
Let $ABC$ be a triangle inscribed in circle $\omega$. Let the tangents to $\omega$ at $B$ and $C$ intersect at point $D$, and let $\overline{AD}$ intersect $\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be written as the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m + n$.

[asy] import olympiad;  unitsize(15);  pair A, B, C, D, E, F, P, O;  C = origin; A = (10,0); B = (7.8, 4.4899); draw(A--B--C--cycle); draw(A..B..C..cycle, red+dotted);  O = circumcenter(A, B, C);  E = rotate(90,B) * (O); F = rotate(90,C) * (O);  D = IP(B..E + (B-E)*4, C..F + (C-F)*-3);  draw(B--D--C--D--A);  P = IP(D..A, A..B..C);  dot(A); dot(B); dot(C); dot(D); dot(P); label("$A$", A, dir(335)); label("$B$", B, dir(65)); label("$C$", C, dir(200)); label("$D$", D, dir(135)); label("$P$", P, dir(235)); [/asy]

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Source and history

task_id
cedfddc6-965b-5cbd-bddd-0a83e0fb41c3
task_key
default--train--cedfddc6-965b-5cbd-bddd-0a83e0fb41c3
task_revision_id
3
config / split
default / train

Official source

initial import