Problem 232

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Problem 232: (2022 AMC12A, 25)

A circle with integer radius $r$ is centered at $(r, r)$. Distinct line segments of length $c_i$ connect points $(0, a_i)$ to $(b_i, 0)$ for $1 \le i \le 14$ and are tangent to the circle, where $a_i$, $b_i$, and $c_i$ are all positive integers and $c_1 \le c_2 \le \cdots \le c_{14}$. What is the ratio $\frac{c_{14}}{c_1}$ for the least possible value of $r$? $\textbf{(A)} ~\frac{21}{5} \qquad$ $\textbf{(B)} ~\frac{85}{13} \qquad$ $\textbf{(C)} ~7 \qquad$ $\textbf{(D)} ~\frac{39}{5} \qquad$ $\textbf{(E)} ~17$

Solution

Exercise 232: Solution

A circle with radius $6$ is centered at $(6,6)$. Line segments of length $c_i$ connect the points $(0,a_i)$ and $(b_i,0)$ and are tangent to the circle, where $a_i$, $b_i$, and $c_i$ are all positive integers. How many such tangent segments have $c_i=5$, the minimum possible length?

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