Question

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 $1\le i\le14$ 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 $\dfrac{c_{14}}{c_1}$ for the least possible value of $r?$

$\dfrac{21}{5}$

$\dfrac{85}{13}$

$7$

$\dfrac{39}{5}$

$17$

Solution:

The circle centered $(r,r)$ with radius $r$ is tangent to both axes. A segment from $(0,a)$ to $(b,0)$ with $a^2+b^2=c^2$ is tangent to it when $r$ equals either the inradius $\tfrac{a+b-c}{2}$ or the semiperimeter $\tfrac{a+b+c}{2}$ of the right triangle with legs $a,b.$

The inradius case rearranges to $(a-2r)(b-2r)=2r^2,$ so the number of such segments grows with the number of divisors of $2r^2.$ The least $r$ that admits $14$ distinct segments is $r=6:$ the semiperimeter case gives the $3$ - $4$ - $5$ triangle (both orientations, $c=5$ ), and the inradius case gives twelve more, up to $13$ - $84$ - $85$ with $c=85.$

Then $c_1=5$ and $c_{14}=85,$ so $\dfrac{c_{14}}{c_1}=\dfrac{85}{5}=17.$

Thus, the correct answer is E.

2022 AMC 12A Problem 25

Solution:

Problem 25 in Other Years