AMO-Bench 2025 Problem 15

Let $a, b, c, d$ be real numbers such that $$4a^4 + \frac{c^4}{16} + b^4 + b^3d^2 + 27 + \frac{d^2}{4} + a^2c^2 = 3c^2 + 24a^2 + 8b^2 + 4d^2.$$ Find the minimum value of $$\frac{4a^4 + \frac{c^4}{16} + a^2c^2 + 12a^2 + \frac{3}{2}c^2 + b^4 + \frac{d^4}{4} + 10 + b^2d^2 + 2b^2 + d^2 }{\sqrt{2a^2b^2 + a^2d^2 + 2a^2 + \frac{b^2c^2}{4} + \frac{c^2d^2}{8} + \frac{c^2}{4} + 3b^2 + 3 + \frac{3}{2}d^2}}.$$