Difference between revisions of "2005 JBMO Problems/Problem 1"

(Created page with "==Problem 1== Find all positive integers <math>x,y</math> satisfying the equation <cmath> 9(x^2+y^2+1) + 2(3xy+2) = 2005 . </cmath> == Solution == We can re-write the equa...")
 
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Since <math>498 - 2y^2 \ge 0</math>. this implies that <math>y \le 15</math>
 
Since <math>498 - 2y^2 \ge 0</math>. this implies that <math>y \le 15</math>
  
Trying all values of <math>y</math> from <math>0</math> to <math>15</math>, we find that <math>y = 7, 11</math> result in perfect squares.
+
Also, taking <math>mod 3</math> on both sides we see that <math>y</math> cannot be a multiple of <math>3</math>. Also, note that <math>249 - y^2</math> has to be even since <math>(498 - 2y^2) = 2(249 - y^2)</math> is a perfect square.
 +
So, <math>y^2</math> cannot be even, implying that <math>y</math> is odd.
 +
 
 +
So we have only <math>\{1, 5, 7, 11, 13\}</math> to consider for <math>y</math>.
 +
 
 +
Trying above 5 values for <math>y</math> we find that <math>y = 7, 11</math> result in perfect squares.
  
 
Thus, we have <math>2</math> cases to check:
 
Thus, we have <math>2</math> cases to check:
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<math>Case 1: y = 7</math>
 
<math>Case 1: y = 7</math>
  
<math> (3x + y)^2 = 4(498 - 2y^2)</math>
+
<math> (3x + 7)^2 = 4(498 - 2(7^2))</math>
<math> => (3x + 7)^2 = 4(498 - 2(7^2))</math>
 
 
<math> =>  (3x + 7)^2 = 4(400)</math>
 
<math> =>  (3x + 7)^2 = 4(400)</math>
 
<math> => x = 11</math>
 
<math> => x = 11</math>
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<math>Case 2: y = 11</math>
 
<math>Case 2: y = 11</math>
  
<math> (3x + y)^2 = 4(498 - 2y^2)</math>
+
<math> (3x + 11)^2 = 4(498 - 2(11^2))</math>
<math> => (3x + 11)^2 = 4(498 - 2(11^2))</math>
 
 
<math> => (3x + 11)^2 = 4(256)</math>
 
<math> => (3x + 11)^2 = 4(256)</math>
 
<math> => x = 7</math>
 
<math> => x = 7</math>

Revision as of 01:18, 25 December 2018

Problem 1

Find all positive integers $x,y$ satisfying the equation \[9(x^2+y^2+1) + 2(3xy+2) = 2005 .\]


Solution

We can re-write the equation as:

$3x^2 + y^2 + 2(3x)(y) + 8y^2 + 9 + 4 = 2005$

or $(3x + y)^2 = 4(498 - 2y^2)$

The above equation tells us that $(498 - 2y^2)$ is a perfect square. Since $498 - 2y^2 \ge 0$. this implies that $y \le 15$

Also, taking $mod 3$ on both sides we see that $y$ cannot be a multiple of $3$. Also, note that $249 - y^2$ has to be even since $(498 - 2y^2) = 2(249 - y^2)$ is a perfect square. So, $y^2$ cannot be even, implying that $y$ is odd.

So we have only $\{1, 5, 7, 11, 13\}$ to consider for $y$.

Trying above 5 values for $y$ we find that $y = 7, 11$ result in perfect squares.

Thus, we have $2$ cases to check:

$Case 1: y = 7$

$(3x + 7)^2 = 4(498 - 2(7^2))$ $=>  (3x + 7)^2 = 4(400)$ $=> x = 11$

$Case 2: y = 11$

$(3x + 11)^2 = 4(498 - 2(11^2))$ $=> (3x + 11)^2 = 4(256)$ $=> x = 7$

Thus all solutions are $(7, 11)$ and $(11, 7)$.


$Kris17$

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