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Difference between revisions of "2022 AMC 12A Problems/Problem 25"

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==Problem==
 
==Problem==
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A circle with integer radius <math>r</math> is centered at <math>(r, r)</math>. Distinct line segments of length <math>c_i</math> connect points <math>(0,a_i)</math> to <math>(b_i, 0)</math> for <math>1 \leq i \leq 14</math> and are tangent to the circle, where <math>a_i</math>, <math>b_i</math>, and <math>c_i</math> are all positive integers and
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A circle with integer radius <math>r</math> is centered at <math>(r, r)</math>. Distinct line segments of length <math>c_i</math> connect points <math>(0, a_i)</math> to <math>(b_i, 0)</math> for <math>1 \le i \le 14</math> and are tangent to the circle, where <math>a_i</math>, <math>b_i</math>, and <math>c_i</math> are all positive integers and <math>c_1 \le c_2 \le \cdots \le c_{14}</math>. What is the ratio <math>\frac{c_{14}}{c_1}</math> for the least possible value of <math>r</math>?
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<math>c_1 \leq c_2 \leq \cdots \leq c_{14}</math>. What is the ratio <math>\frac{c_{14}}{c_1}</math> for the least possible value of <math>r</math>?
 
  
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==Solution==
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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>
 +
 
 +
==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 incircle, the hypotenuse must be <math>(x-r)+(y-r)</math>, leading to the same equation.
 +
 
 +
Notice that this equation can be simplified through SFFT to <cmath>(a_i-2r)(b_i-2r)=2r^2.</cmath> Thus, we want the smallest <math>r</math> such that this equation has at least <math>14</math> distinct pairs <math>(a_i,b_i)</math> for which this holds. The obvious choice to check is <math>r=6</math>. In this case, since <math>2r^2=2^3\cdot 3^2</math> has <math>12</math> positive factors, we get <math>12</math> pairs, and we get another two if the factors are <math>-8,-9</math> or vice versa. One can check that for smaller values of <math>r</math>, we do not even get close to <math>14</math> possible pairs.
 +
 
 +
When <math>r=6</math>, the smallest possible <math>c</math>-value is clearly when the factors are negative. When this occurs, <math>a_i=4, b_i=3</math> (or vice versa), so the mimimal <math>c</math> is <math>5</math>. The largest possible <math>c</math>-value occurs when the largest of <math>a_i</math> and <math>b_i</math> are maximized. This occurs when the factors are <math>72</math> and <math>1</math>, leading to <math>a_i=84, b_i=13</math> (or vice-versa), leading to a maximal <math>c</math> of <math>85</math>.
 +
 
 +
Hence the answer is <math>\frac{85}5=\boxed{17}</math>.
 +
 
 +
~ bluelinfish
 +
 
 +
==Solution 2==
  
 
Case 1: The tangent and the origin are on the opposite sides of the circle.
 
Case 1: The tangent and the origin are on the opposite sides of the circle.
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Case 2: The tangent and the origin are on the opposite sides of the circle.
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Case 2: The tangent and the origin are on the same sides of the circle.
  
 
In this case, <math>0 < a, b < r</math>.
 
In this case, <math>0 < a, b < r</math>.
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Following this guidance, we find the smallest <math>r</math> is 6. This leads to the following solutions:
 
Following this guidance, we find the smallest <math>r</math> is 6. This leads to the following solutions:
  
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\begin{enumerate}
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<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>.
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\item <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>.
+
 
−
\item <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>.
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<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>.
−
\item <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>.
+
 
−
\item <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>.
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<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>.
−
\item <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>.
+
 
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\item <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>.
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<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>.
−
\item <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>.
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−
\end{enumerate}
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<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>.
 
Therefore, <math>\frac{c_{14}}{c_1} = \boxed{\textbf{(E) 17}}</math>.
 +
 +
==Solution 3==
 +
 +
As <math>c_i</math> is the length of the segment <math>(0,a_i)</math> and <math>(b_i,0)</math>， <math>a_i^2+b_i^2=c_i^2</math>. The equation for the line that passes through <math>(0,a_i)</math> and <math>(b_i,0)</math> is <math>a_ix+b_iy-a_ib_i=0</math>.
 +
 +
By the point-line distance formula from point <math>(r,r)</math> to line <math>a_ix+b_iy-a_ib_i=0</math>
 +
 +
<cmath>r = \frac{ |a_ir+b_ir-a_ib_i| }{ \sqrt{a_i^2+b_i^2} }, \quad c_i = \frac{ |a_ir+b_ir-a_ib_i| }{r} = |a_i+b_i- \frac{a_ib_i}{r}|</cmath>
 +
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<cmath>\because c_i \quad \text{is an integer}, \quad \therefore r|a_ib_i</cmath>
 +
 +
To find <math>c_1</math> and <math>c_{14}</math> for the smallest <math>r</math>, we will list Pythagorean triples <math>(a_i,b_i,c_i)</math> with relatively prime elements from the smallest. Notice that we only need to find <math>7</math> triples with <math>a_i<b_i</math>, the <math>7</math> other triples will simply be <math>(b_i,a_i,c_i)</math>. <math>a_i</math> will not equal <math>b_i</math> because then <math>c_i = a_i \cdot \sqrt{2}</math>, meaning that <math>a_i</math>, <math>b_i</math>, and <math>c_i</math> cannot all be integers.
 +
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The <math>7</math> smallest triples are: <math>(3,4,5)</math>, <math>(5,12,13)</math>, <math>(7,24,25)</math>, <math>(8,15,17)</math>, <math>(11,60,61)</math>, <math>(12,35,37)</math>, <math>(13,84,85)</math>
 +
 +
Therefore, <math>\frac{c_{14}}{c_1} = \frac{85}{5} = \boxed{\textbf{(E) 17}}</math>.
 +
 +
~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]
 +
 +
==Video Solution by Math-X (Let's understand the question first)==
 +
https://youtu.be/7yAh4MtJ8a8?si=2dTtBDEZLIjN3YWS&t=9728 ~Math-X
  
 
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
 
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
  
−
==Video Solution==
+
==Video Solution by MOP 2024==
 +
https://youtu.be/5jO__fUbgs8
 +
 
 +
~r00tsOfUnity
 +
 
 +
==Video Solution by Steven Chen==
  
 
https://youtu.be/6RfGCTNQ2Jw
 
https://youtu.be/6RfGCTNQ2Jw
  
−
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
+
==Video Solution (5 Minutes)==
 +
 
 +
https://youtu.be/79CLeiJrPvs
 +
 
 +
~MathProblemSolvingSkills.com
 +
 
 +
==See also==
 +
{{AMC12 box|year=2022|ab=A|num-b=24|after=Last Problem}}
 +
 
 +
[[Category:Intermediate Geometry Problems]]
 +
[[Category:Intermediate Number Theory Problems]]
 +
{{MAA Notice}}

Latest revision as of 11:09, 4 January 2024

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

Solution 3

As $c_i$ is the length of the segment $(0,a_i)$ and $(b_i,0)$， $a_i^2+b_i^2=c_i^2$. The equation for the line that passes through $(0,a_i)$ and $(b_i,0)$ is $a_ix+b_iy-a_ib_i=0$.

By the point-line distance formula from point $(r,r)$ to line $a_ix+b_iy-a_ib_i=0$

\[r = \frac{ |a_ir+b_ir-a_ib_i| }{ \sqrt{a_i^2+b_i^2} }, \quad c_i = \frac{ |a_ir+b_ir-a_ib_i| }{r} = |a_i+b_i- \frac{a_ib_i}{r}|\]

\[\because c_i \quad \text{is an integer}, \quad \therefore r|a_ib_i\]

To find $c_1$ and $c_{14}$ for the smallest $r$, we will list Pythagorean triples $(a_i,b_i,c_i)$ with relatively prime elements from the smallest. Notice that we only need to find $7$ triples with $a_i<b_i$, the $7$ other triples will simply be $(b_i,a_i,c_i)$. $a_i$ will not equal $b_i$ because then $c_i = a_i \cdot \sqrt{2}$, meaning that $a_i$, $b_i$, and $c_i$ cannot all be integers.

The $7$ smallest triples are: $(3,4,5)$, $(5,12,13)$, $(7,24,25)$, $(8,15,17)$, $(11,60,61)$, $(12,35,37)$, $(13,84,85)$

Therefore, $\frac{c_{14}}{c_1} = \frac{85}{5} = \boxed{\textbf{(E) 17}}$.

~isabelchen

Video Solution by Math-X (Let's understand the question first)

https://youtu.be/7yAh4MtJ8a8?si=2dTtBDEZLIjN3YWS&t=9728 ~Math-X

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

Video Solution by MOP 2024

https://youtu.be/5jO__fUbgs8

~r00tsOfUnity

Video Solution by Steven Chen

https://youtu.be/6RfGCTNQ2Jw

Video Solution (5 Minutes)

https://youtu.be/79CLeiJrPvs

~MathProblemSolvingSkills.com

See also

2022 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 24 		Followed by
Last Problem
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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