China | Can you solve this ? | Math Olympiad

[@chaosredefined3834]

As an alternate approach for your first problem. x^2 - y^2 = 28 xy = 48 12x^2 - 12y^2 = 336 7xy = 336 12x^2 - 12y^2 = 7xy 12x^2 - 7xy - 12y^2 = 0 12x^2 - 16xy + 9xy - 12y^2 = 0 4x(3x - 4y) + 3y(3x - 4y) = 0 (4x + 3y)(3x - 4y) = 0 Case 1: 4x + 3y = 0. So 4x^2 + 3xy = 0, and therefore 4x^2 + 3(48) = 0. The minimum possible value for the LHS is 144, which is bigger than 0, so no real solutions. Case 2: 3x - 4y = 0. Multiplying through by x, we get 3x^2 - 4xy = 0. Since xy = 48, this means that 3x^2 - 4(48) = 0, or x^2 = 64, so x = +/- 8. y is then +/- 6. These do work with the original equations, so we have solutions.

[@guybayo2002]

I finished school two years ago, but the algorithm suggested this to me and I couldn't solve it in my head so I got curious

[@volvol1]

X= 8; y= 6