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1991 AHSME Problems/Problem 10

Problem

Point $P$ is $9$ units from the center of a circle of radius $15$. How many different chords of the circle contain $P$ and have integer lengths?

(A) 11 (B) 12 (C) 13 (D) 14 (E) 29

Solution $\fbox{B}$ Let the chord be $AB$, and let the diameter passing through $P$ be $CD$. Then we have $PC = 15-9 = 6$ and $PD = 15+9 = 24$. By power of a point, we now get $6 \times 24 = AP \times PB \implies AP \times PB = 144$. Now AM-GM gives $AB = AP + PB \geq 2 \sqrt{AP \times PB} = 2 \times 12 = 24$. Clearly we also know $AC \leq 30$ as the diameter is the longest chord of a circle. Hence there are 7 possible values of $AC$, and each gives two chords as we can reflect the chord, except that for the $24$ and the $30$ we can't do this as it gives the same chord, so the answer is $7 \times 2 - 2 = 12.$

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