Given $x,\,y,\,z$ are real such that $2x+y+z+14=2\sqrt{2x}+4\sqrt{y+1}+6\sqrt{z-1}$. Evaluate $\dfrac{x-y}{z}$.

Given $x,\,y,\,z$ are real such that $2x+y+z+14=2\sqrt{2x}+4\sqrt{y+1}+6\sqrt{z-1}$.

Evaluate $\dfrac{x-y}{z}$.

My solution:

This IMO Problem should look easy to you if you're being careful and look at the problem with your heart and mind, not merely with the eyes.

For me, I will try to group

Mock Olympiad Math Problem: Solve $\large 3^{\sin^4 x-\cos^2 x}-3^{\cos^4 x-\sin^2 x}=\cos 2x$.

Solve $\large 3^{\sin^4 x-\cos^2 x}-3^{\cos^4 x-\sin^2 x}=\cos 2x$.

$\large 3^{\sin^4 x-\cos^2 x}-3^{\cos^4 x-\sin^2 x}=\cos 2x$

$\large 3^{\sin^4 x-\cos^2 x}\left(1-\dfrac{3^{\cos^4 x-\sin^2 x}}{3^{\sin^4 x-\cos^2 x}}\right)=\cos 2x$