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# help

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$$x$$ and $$y$$ are integers such that $$3x^2y^2 - 17xy + 20 = 0.$$ What is the maximum possible value of $$x+y?$$

Sep 2, 2019

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$$\text{Notice the expression is symmetrical in x and y}\\ \text{This suggests that any solution will be such that x=y}\\ \text{Letting x=y we have}\\ 3x^4 - 17x^2 + 20 = 0\\ \text{let u=x^2}\\ 3u^2 - 17u + 20 = 0\\ \text{Apply the quadratic formula}\\ u = \dfrac{17 \pm \sqrt{17^2 - (4)(3)(20)}}{6} = \left(\dfrac 5 3,~4\right)\\ \text{you want integer solutions so we can ignore the first one}\\ x^2 = 4\\ x=\pm 2$$

$$\text{So the max of x+y is 2+2=4}$$

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Sep 3, 2019