Question 137
Suppose $a$, $b$, $c$ and $d$ are positive real numbers satisfying $ab + cd = 1$ and $P_i(x_i, y_i)$ ($i = 1, 2, 3, 4$) are four points on the unit circle which has the origin as its center. Prove that:
$$(ay_1 + by_2 + cy_3 + dy_4)^2 + (ax_4 + bx_3 + cx_2 + dx_1)^2 \le 2\left(\frac{a^2+b^2}{ab}+\frac{c^2+d^2}{cd}\right).$$
$$(ay_1 + by_2 + cy_3 + dy_4)^2 + (ax_4 + bx_3 + cx_2 + dx_1)^2 \le 2\left(\frac{a^2+b^2}{ab}+\frac{c^2+d^2}{cd}\right).$$