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Difference between revisions of "1976 USAMO Problems/Problem 3"

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* Case 1: <math>3|a</math>
 
* Case 1: <math>3|a</math>
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Thus the RHS is a multiple of 3, and <math>b</math> and <math>c</math> are also multiples of 3. Let <math>a=3a_1</math>, <math>b=3b_1</math>, and <math>c=3c_1</math>. Thus <math>a_1^2+b_1^2+c_1^2=9a_1^2b_1^2</math>. Thus the new variables are all multiples of 3, and we continue like this infinitely, and thus there are no solutions with <math>3|a</math>.
+
Thus the RHS is a multiple of 3, and <math>b</math> and <math>c</math> are also multiples of 3. Let <math>a=3a_1</math>, <math>b=3b_1</math>, and <math>c=3c_1</math>. Thus <math>a_1^2+b_1^2+c_1^2=9a_1^2b_1^2</math>. Thus the new variables are all multiples of 3, and we continue like this indefinitely, and thus there are no solutions with <math>3|a</math>.
  
 
*Case 2: 3 is not a divisor of <math>a</math>.
 
*Case 2: 3 is not a divisor of <math>a</math>.

Revision as of 17:07, 4 October 2008

Problem

Determine all integral solutions of $a^2+b^2+c^2=a^2b^2$.

Solution

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

We have the trivial solution, $(a,b,c)=(0,0,0)$. Now WLOG, let the variables be positive.


  • Case 1: $3|a$

Thus the RHS is a multiple of 3, and $b$ and $c$ are also multiples of 3. Let $a=3a_1$, $b=3b_1$, and $c=3c_1$. Thus $a_1^2+b_1^2+c_1^2=9a_1^2b_1^2$. Thus the new variables are all multiples of 3, and we continue like this indefinitely, and thus there are no solutions with $3|a$.

  • Case 2: 3 is not a divisor of $a$.

Thus $a^2\equiv b^2\equiv 1\bmod{3}$, but for $a^2+b^2+c^2$ to be a quadratic residue, $c^2\equiv 1\bmod{3}$, and we have that a multiple of 3 equals something that isn't a multiple of 3, which is a contradiction.


Thus after both cases, the only solution is the trivial solution stated above.

See also

1976 USAMO (Problems • Resources)
Preceded by
Problem 2 		Followed by
Problem 4
1 • 2 • 3 • 4 • 5
All USAMO Problems and Solutions
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