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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Revision as of 22:26, 24 March 2023
Problem
Find all triples of positive integers that satisfy the equation
Solution 1
We claim that the only solutions are and its permutations.
Factoring the above squares and canceling the terms gives you:
Jumping on the coefficients in front of the ,
,
terms, we factor into:
Realizing that the only factors of 2023 that could be expressed as are
,
, and
, we simply find that the only solutions are
by inspection.
-FFAmplify