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Difference between revisions of "2023 USAJMO Problems/Problem 1"

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Realizing that the only factors of 2023 that could be expressed as <math>(2x^2 - 1)</math> are <math>1</math>, <math>7</math>, and <math>17</math>, we simply find that the only solutions are <math>(2,3,3)</math> by inspection.
 
Realizing that the only factors of 2023 that could be expressed as <math>(2x^2 - 1)</math> are <math>1</math>, <math>7</math>, and <math>17</math>, we simply find that the only solutions are <math>(2,3,3)</math> by inspection.
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-FFAmplify

Revision as of 21:26, 24 March 2023

Problem

Find all triples of positive integers $(x,y,z)$ that satisfy the equation

\begin{align*} 2(x+y+z+2xyz)^2=(2xy+2yz+2zx+1)^2+2023 \end{align*}


Solution 1

We claim that the only solutions are $(2,3,3)$ and its permutations.

Factoring the above squares and canceling the terms gives you:

$8(xyz)^2 + 2(x^2 +y^2 + z^2) = 4((xy)^2 + (yz)^2 + (zx)^2) + 2024$

Jumping on the coefficients in front of the $x^2$, $y^2$, $z^2$ terms, we factor into:

$(2x^2 - 1)(2y^2 - 1)(2z^2 - 1) = 2023$

Realizing that the only factors of 2023 that could be expressed as $(2x^2 - 1)$ are $1$, $7$, and $17$, we simply find that the only solutions are $(2,3,3)$ by inspection.

-FFAmplify

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