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Difference between revisions of "2021 Fall AMC 12B Problems/Problem 16"

m (fixing some typos)
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<math>(\textbf{A})\: 259\qquad(\textbf{B}) \: 438\qquad(\textbf{C}) \: 516\qquad(\textbf{D}) \: 625\qquad(\textbf{E}) \: 687</math>
 
<math>(\textbf{A})\: 259\qquad(\textbf{B}) \: 438\qquad(\textbf{C}) \: 516\qquad(\textbf{D}) \: 625\qquad(\textbf{E}) \: 687</math>
  
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==Solution==
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==Solution 1==
 
Let <math>\gcd(a,b)=x</math>, <math>\gcd(b,c)=y</math>, <math>\gcd(c,a)=z</math>. WLOG, let <math>x \le y \le z</math>. We can split this off into cases:  
 
Let <math>\gcd(a,b)=x</math>, <math>\gcd(b,c)=y</math>, <math>\gcd(c,a)=z</math>. WLOG, let <math>x \le y \le z</math>. We can split this off into cases:  
  
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-ConcaveTriangle
 
-ConcaveTriangle
 +
 +
== Solution 2 ==
 +
Because <math>a + b + c</math> is odd, <math>a</math>, <math>b</math>, <math>c</math> are either one odd and two evens or three odds.
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<math>\textbf{Case 1}</math>: <math>a</math>, <math>b</math>, <math>c</math> have one odd and two evens.
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 +
Without loss of generality, we assume <math>a</math> is odd and <math>b</math> and <math>c</math> are even.
 +
 +
Hence, <math>{\rm gcd} \left( a , b \right)</math> and <math>{\rm gcd} \left( a , c \right)</math> are odd, and <math>{\rm gcd} \left( b , c \right)</math> is even.
 +
Hence, <math>{\rm gcd} \left( a , b \right) + {\rm gcd} \left( b , c \right) + {\rm gcd} \left( c , a \right)</math> is even. This violates the condition given in the problem.
 +
 +
Therefore, there is no solution in this case.
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 +
<math>\textbf{Case 2}</math>: <math>a</math>, <math>b</math>, <math>c</math> are all odd.
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In this case, <math>{\rm gcd} \left( a , b \right)</math>, <math>{\rm gcd} \left( a , c \right)</math>, <math>{\rm gcd} \left( b , c \right)</math> are all odd.
 +
 +
Without loss of generality, we assume
 +
<cmath>
 +
\[
 +
{\rm gcd} \left( a , b \right) \leq {\rm gcd} \left( b , c \right) \leq {\rm gcd} \left( c , a \right) .
 +
\]
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</cmath>
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<math>\textbf{Case 2.1}</math>: <math>{\rm gcd} \left( a , b \right) = 1</math>, <math>{\rm gcd} \left( b , c \right) = 1</math>, <math>{\rm gcd} \left( c , a \right) = 7</math>.
 +
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The only solution for <math>\left( a, b, c \right)</math> is (7, 9, 7).
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 +
Hence, <math>a^2 + b^2 + c^2 = 179</math>.
 +
 +
<math>\textbf{Case 2.2}</math>: <math>{\rm gcd} \left( a , b \right) = 1</math>, <math>{\rm gcd} \left( b , c \right) = 3</math>, <math>{\rm gcd} \left( c , a \right) = 5</math>.
 +
 +
The only solution for <math>\left( a, b, c \right)</math> is (5, 3, 15).
 +
 +
Hence, <math>a^2 + b^2 + c^2 = 259</math>.
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<math>\textbf{Case 2.3}</math>: <math>{\rm gcd} \left( a , b \right) = 3</math>, <math>{\rm gcd} \left( b , c \right) = 3</math>, <math>{\rm gcd} \left( c , a \right) = 3</math>.
 +
 +
There is no solution in this case.
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 +
Therefore, putting all cases together, the answer is <math>179 + 259 = 438</math>.
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 +
Therefore, the answer is <math>\boxed{\textbf{(B) }438}</math>.
 +
 +
~Steven Chen (www.professorchenedu.com)

Revision as of 00:15, 26 November 2021

Problem

Suppose $a$, $b$, $c$ are positive integers such that \[a+b+c=23\] and \[\gcd(a,b)+\gcd(b,c)+\gcd(c,a)=9.\] What is the sum of all possible distinct values of $a^2+b^2+c^2$?

$(\textbf{A})\: 259\qquad(\textbf{B}) \: 438\qquad(\textbf{C}) \: 516\qquad(\textbf{D}) \: 625\qquad(\textbf{E}) \: 687$

Solution 1

Let $\gcd(a,b)=x$, $\gcd(b,c)=y$, $\gcd(c,a)=z$. WLOG, let $x \le y \le z$. We can split this off into cases:

$x=1,y=1,z=7$: let $a=7A, c=7C,$ we can try all possibilities of $A$ and $C$ to find that $a=7, b=9, c=7$ is the only solution.

$x=1,y=2,z=6$: No solutions. By $y$ and $z$, we know that $a$, $b$, and $c$ have to all be divisible by $2$. Therefore, $x$ cannot be equal to $1$.

$x=1,y=3,z=5$: C has to be both a multiple of $3$ and $5$. Therefore, $c$ has to be a multiple of $15$. The only solution for this is $a=5, b=3, c=15$.

$x=1,y=4,z=4$: No solutions. By $y$ and $z$, we know that $a$, $b$, and $c$ have to all be divisible by $4$. Therefore, $x$ cannot be equal to $1$.

$x=2,y=2,z=5$: No solutions. By $x$ and $y$, we know that $a$, $b$, and $c$ have to all be divisible by $2$. Therefore, $z$ cannot be equal to $5$.

$x=2,y=3,z=4$: No solutions. By $x$ and $z$, we know that $a$, $b$, and $c$ have to all be divisible by $2$. Therefore, $y$ cannot be equal to $3$.

$x=3,y=3,z=3$: No solutions. As $a$, $b$, and $c$ have to all be divisible by $3$, $a+b+c$ has to be divisible by $3$. This contradicts the sum $a+b+c=23$.

Putting these solutions together, we have $(7^2+9^2+7^2)+(5^2+3^2+15^2)=179+259=\boxed{\textbf{(B) }438}$

-ConcaveTriangle

Solution 2

Because $a + b + c$ is odd, $a$, $b$, $c$ are either one odd and two evens or three odds.

$\textbf{Case 1}$: $a$, $b$, $c$ have one odd and two evens.

Without loss of generality, we assume $a$ is odd and $b$ and $c$ are even.

Hence, ${\rm gcd} \left( a , b \right)$ and ${\rm gcd} \left( a , c \right)$ are odd, and ${\rm gcd} \left( b , c \right)$ is even. Hence, ${\rm gcd} \left( a , b \right) + {\rm gcd} \left( b , c \right) + {\rm gcd} \left( c , a \right)$ is even. This violates the condition given in the problem.

Therefore, there is no solution in this case.

$\textbf{Case 2}$: $a$, $b$, $c$ are all odd.

In this case, ${\rm gcd} \left( a , b \right)$, ${\rm gcd} \left( a , c \right)$, ${\rm gcd} \left( b , c \right)$ are all odd.

Without loss of generality, we assume \[ {\rm gcd} \left( a , b \right) \leq {\rm gcd} \left( b , c \right) \leq {\rm gcd} \left( c , a \right) . \]

$\textbf{Case 2.1}$: ${\rm gcd} \left( a , b \right) = 1$, ${\rm gcd} \left( b , c \right) = 1$, ${\rm gcd} \left( c , a \right) = 7$.

The only solution for $\left( a, b, c \right)$ is (7, 9, 7).

Hence, $a^2 + b^2 + c^2 = 179$.

$\textbf{Case 2.2}$: ${\rm gcd} \left( a , b \right) = 1$, ${\rm gcd} \left( b , c \right) = 3$, ${\rm gcd} \left( c , a \right) = 5$.

The only solution for $\left( a, b, c \right)$ is (5, 3, 15).

Hence, $a^2 + b^2 + c^2 = 259$.

$\textbf{Case 2.3}$: ${\rm gcd} \left( a , b \right) = 3$, ${\rm gcd} \left( b , c \right) = 3$, ${\rm gcd} \left( c , a \right) = 3$.

There is no solution in this case.

Therefore, putting all cases together, the answer is $179 + 259 = 438$.

Therefore, the answer is $\boxed{\textbf{(B) }438}$.

~Steven Chen (www.professorchenedu.com)

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