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Difference between revisions of "2022 AMC 12A Problems/Problem 25"
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<math>\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 </math>
 
<math>\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 </math>
  
−
==Solution==
+
==Solution 1==
 
Suppose that with a pair <math>(a_i,b_i)</math> the circle is an excircle. Then notice that the hypotenuse must be <math>(r-x)+(r-y)</math>, so it must be the case that <cmath>a_i^2+b_i^2=(2r-a_i-b_i)^2.</cmath> Similarly, if with a pair <math>(a_i,b_i)</math> the circle is an excircle, the hypotenuse must be <math>(x-r)+(y-r)</math>, leading to the same equation.  
 
Suppose that with a pair <math>(a_i,b_i)</math> the circle is an excircle. Then notice that the hypotenuse must be <math>(r-x)+(r-y)</math>, so it must be the case that <cmath>a_i^2+b_i^2=(2r-a_i-b_i)^2.</cmath> Similarly, if with a pair <math>(a_i,b_i)</math> the circle is an excircle, the hypotenuse must be <math>(x-r)+(y-r)</math>, leading to the same equation.  
  
Line 14: Line 14:
  
 
~ bluelinfish
 
~ bluelinfish
 +
 +
==Solution 2==
 +
 +
Case 1: The tangent and the origin are on the opposite sides of the circle.
 +
 +
In this case, <math>a, b > 2r</math>.
 +
 +
We can easily prove that
 +
<cmath>
 +
a + b - 2 r = c . \hspace{1cm} (1)
 +
</cmath>
 +
 +
Recall that <math>c = \sqrt{a^2 + b^2}</math>.
 +
 +
Taking square of (1) and reorganizing all terms, (1) is converted as
 +
<cmath>
 +
\left( a - 2 r \right) \left( b - 2 r \right) = 2 r^2 .
 +
</cmath>
 +
 +
 +
Case 2: The tangent and the origin are on the opposite sides of the circle.
 +
 +
In this case, <math>0 < a, b < r</math>.
 +
 +
We can easily prove that
 +
<cmath>
 +
2 r - a - b  = c . \hspace{1cm} (2)
 +
</cmath>
 +
 +
Recall that <math>c = \sqrt{a^2 + b^2}</math>.
 +
 +
Taking square of (2) and reorganizing all terms, (2) is converted as
 +
<cmath>
 +
\left( a - 2 r \right) \left( b - 2 r \right) = 2 r^2 .
 +
</cmath>
 +
 +
Putting both cases together, for given <math>r</math>, we look for solutions of <math>a</math> and <math>b</math> satisfying
 +
<cmath>
 +
\left( a - 2 r \right) \left( b - 2 r \right) = 2 r^2 ,
 +
</cmath>
 +
with either <math>a, b > 2r</math> or <math>0 < a, b < r</math>.
 +
 +
Now, we need to find the smallest <math>r</math>, such that the number of feasible solutions of <math>(a, b)</math> is at least 14.
 +
 +
For equation
 +
<cmath>
 +
uv = 2 r^2 ,
 +
</cmath>
 +
we observe that the R.H.S. is a not a perfect square. Thus, the number of positive <math>(u, v)</math> is equal to the number of positive divisors of <math>2 r^2</math>.
 +
 +
Second, for each feasible positive solution <math>(u, v)</math>, its opposite <math>(-u, -v)</math> is also a solution. However, <math>(u,v)</math> corresponds to a feasible solution if <math>(a, b)</math> with <math>a = u + 2r</math> and <math>b = v + 2r</math>, but <math>(-u, -v)</math> may not lead to a feasible solution if <math>(a, b)</math> with <math>a = 2 r - u</math> and <math>b = 2 r - v</math>.
 +
 +
Recall that we are looking for <math>r</math> that leads to at least 14 solutions.
 +
Therefore, the above observations imply that we must have <math>r</math>, such that <math>2 r^2</math> has least 7 positive divisors.
 +
 +
Following this guidance, we find the smallest <math>r</math> is 6. This leads to the following solutions:
 +
 +
<math>\left( a_1, b_1, c_1 \right) = \left( 3, 4, 5 \right)</math>, <math>\left( a_2, b_2, c_2 \right) = \left( 4, 3, 5 \right)</math>.
 +
 +
<math>\left( a_3, b_3, c_3 \right) = \left( 20, 21, 29 \right)</math>, <math>\left( a_4, b_4, c_4 \right) = \left( 21, 20, 29 \right)</math>.
 +
 +
<math>\left( a_5, b_5, c_5 \right) = \left( 18, 24, 30 \right)</math>, <math>\left( a_6, b_6, c_6 \right) = \left( 24, 18, 30 \right)</math>.
 +
 +
<math>\left( a_7, b_7, c_7 \right) = \left( 16, 30, 34 \right)</math>, <math>\left( a_8, b_8, c_8 \right) = \left( 30, 16, 34 \right)</math>.
 +
 +
<math>\left( a_9, b_9, c_9 \right) = \left( 15, 36, 39 \right)</math>, <math>\left( a_{10}, b_{10}, c_{10} \right) = \left( 36, 15, 39 \right)</math>.
 +
 +
<math>\left( a_{11}, b_{11}, c_{11} \right) = \left( 14, 48, 50 \right)</math>, <math>\left( a_{12}, b_{12}, c_{12} \right) = \left( 48, 14, 50 \right)</math>.
 +
 +
<math>\left( a_{13}, b_{13}, c_{13} \right) = \left( 13, 84, 85 \right)</math>, <math>\left( a_{14}, b_{14}, c_{14} \right) = \left( 84, 13, 85 \right)</math>.
 +
 +
 +
Therefore, <math>\frac{c_{14}}{c_1} = \boxed{\textbf{(E) 17}}</math>.
 +
 +
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
 +
 +
==Video Solution by Steven Chen==
 +
 +
https://youtu.be/6RfGCTNQ2Jw
  
 
==See also==
 
==See also==
 
{{AMC12 box|year=2022|ab=A|num-b=24|after=Last Problem}}
 
{{AMC12 box|year=2022|ab=A|num-b=24|after=Last Problem}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 21:05, 11 November 2022

Problem

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 1

Suppose that with a pair $(a_i,b_i)$ the circle is an excircle. Then notice that the hypotenuse must be $(r-x)+(r-y)$, so it must be the case that \[a_i^2+b_i^2=(2r-a_i-b_i)^2.\] Similarly, if with a pair $(a_i,b_i)$ the circle is an excircle, the hypotenuse must be $(x-r)+(y-r)$, leading to the same equation.

Notice that this equation can be simplified through SFFT to \[(a_i-2r)(b_i-2r)=2r^2.\] Thus, we want the smallest $r$ such that this equation has at least $14$ distinct pairs $(a_i,b_i)$ for which this holds. The obvious choice to check is $r=6$. In this case, since $2r^2=2^3\cdot 3^2$ has $12$ positive factors, we get $12$ pairs, and we get another two if the factors are $-8,-9$ or vice versa. One can check that for smaller values of $r$, we do not even get close to $14$ possible pairs.

When $r=6$, the smallest possible $c$-value is clearly when the factors are negative. When this occurs, $a_i=4, b_i=3$ (or vice versa), so the mimimal $c$ is $5$. The largest possible $c$-value occurs when the largest of $a_i$ and $b_i$ are maximized. This occurs when the factors are $72$ and $1$, leading to $a_i=84, b_i=13$ (or vice-versa), leading to a maximal $c$ of $85$.

Hence the answer is $\frac{85}5=\boxed{17}$.

~ bluelinfish

Solution 2

Case 1: The tangent and the origin are on the opposite sides of the circle.

In this case, $a, b > 2r$.

We can easily prove that \[a + b - 2 r = c . \hspace{1cm} (1)\]

Recall that $c = \sqrt{a^2 + b^2}$.

Taking square of (1) and reorganizing all terms, (1) is converted as \[\left( a - 2 r \right) \left( b - 2 r \right) = 2 r^2 .\]


Case 2: The tangent and the origin are on the opposite sides of the circle.

In this case, $0 < a, b < r$.

We can easily prove that \[2 r - a - b = c . \hspace{1cm} (2)\]

Recall that $c = \sqrt{a^2 + b^2}$.

Taking square of (2) and reorganizing all terms, (2) is converted as \[\left( a - 2 r \right) \left( b - 2 r \right) = 2 r^2 .\]

Putting both cases together, for given $r$, we look for solutions of $a$ and $b$ satisfying \[\left( a - 2 r \right) \left( b - 2 r \right) = 2 r^2 ,\] with either $a, b > 2r$ or $0 < a, b < r$.

Now, we need to find the smallest $r$, such that the number of feasible solutions of $(a, b)$ is at least 14.

For equation \[uv = 2 r^2 ,\] we observe that the R.H.S. is a not a perfect square. Thus, the number of positive $(u, v)$ is equal to the number of positive divisors of $2 r^2$.

Second, for each feasible positive solution $(u, v)$, its opposite $(-u, -v)$ is also a solution. However, $(u,v)$ corresponds to a feasible solution if $(a, b)$ with $a = u + 2r$ and $b = v + 2r$, but $(-u, -v)$ may not lead to a feasible solution if $(a, b)$ with $a = 2 r - u$ and $b = 2 r - v$.

Recall that we are looking for $r$ that leads to at least 14 solutions. Therefore, the above observations imply that we must have $r$, such that $2 r^2$ has least 7 positive divisors.

Following this guidance, we find the smallest $r$ is 6. This leads to the following solutions:

$\left( a_1, b_1, c_1 \right) = \left( 3, 4, 5 \right)$, $\left( a_2, b_2, c_2 \right) = \left( 4, 3, 5 \right)$.

$\left( a_3, b_3, c_3 \right) = \left( 20, 21, 29 \right)$, $\left( a_4, b_4, c_4 \right) = \left( 21, 20, 29 \right)$.

$\left( a_5, b_5, c_5 \right) = \left( 18, 24, 30 \right)$, $\left( a_6, b_6, c_6 \right) = \left( 24, 18, 30 \right)$.

$\left( a_7, b_7, c_7 \right) = \left( 16, 30, 34 \right)$, $\left( a_8, b_8, c_8 \right) = \left( 30, 16, 34 \right)$.

$\left( a_9, b_9, c_9 \right) = \left( 15, 36, 39 \right)$, $\left( a_{10}, b_{10}, c_{10} \right) = \left( 36, 15, 39 \right)$.

$\left( a_{11}, b_{11}, c_{11} \right) = \left( 14, 48, 50 \right)$, $\left( a_{12}, b_{12}, c_{12} \right) = \left( 48, 14, 50 \right)$.

$\left( a_{13}, b_{13}, c_{13} \right) = \left( 13, 84, 85 \right)$, $\left( a_{14}, b_{14}, c_{14} \right) = \left( 84, 13, 85 \right)$.


Therefore, $\frac{c_{14}}{c_1} = \boxed{\textbf{(E) 17}}$.

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Video Solution by Steven Chen

https://youtu.be/6RfGCTNQ2Jw

See also

2022 AMC 12A (Problems • Answer Key • Resources)
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Problem 24 		Followed by
Last Problem
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