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AMC (AIMO validation set) / https://artofproblemsolving.com/wiki/index.php/2022_AMC_12A_Problems/Problem_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≤i≤14 and are tangent to the circle, where …

Problem

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problem

A circle with integer radius

r

is centered at

(r, r)

. Distinct line segments of length

c_i

connect points

(0, a_i)

(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

Plain-text mathematical notation (without MathML)

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≤i≤14 and are tangent to the circle, where a_(i), b_(i), and c_(i) are all positive integers and c₁≤c₂≤⋯≤c₁₄. What is the ratio (c₁₄)/(c₁) for the least possible value of r?

Original LaTeX notation

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 $r$?

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Source and history

upstream_id / Original problem ID: https://artofproblemsolving.com/wiki/index.php/2022_AMC_12A_Problems/Problem_25
task_id: 2d701f58-1285-5fdc-8958-607838ed0eb8
task_key: default--train--https~3a~2f~2fartofproblemsolving~2ecom~2fwiki~2findex~2ephp~2f2022~5fAMC~5f12A~5fProblems~2fProblem~5f25
task_revision_id: 2
config / split: default / train

Official source

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