Bulgarian Spring Tournament

Question

The reals x, y satisfy x(x - 6) y(4 - y) + 7. Find the minimal and maximal values of the expression x + 2y.

Accepted Answer

Let a = x + 2y. Then x = a - 2y and aligned (a - 2y)(a - 2y - 6) & y(4 - y) + 7 a^2 - 2ay - 6a - 2ay + 4y^2 + 12y & 4y - y^2 + 7 5y^2 - 2(2a - 4)y + (a^2 - 6a - 7) & 0 D &= (2a - 4)^2 - 5(a^2 - 6a - 7) 0 4a^2 - 16a + 16 - 5a^2 + 30a + 35 & 0 a^2 - 14a - 51 & 0 (a - 17)(a + 3) & 0. aligned Finally, a [-3; 17].