1. Given positive numbers $x>y$. Let $z=\frac{x+y}{2}$ and $t=\sqrt{x y}$. It turns out that $(x-y)=3(z-t)$. Find

$x / y$.
Answer: 25 (in the second variant 9). Note that $z-t=(\sqrt{} x-\sqrt{} y)^{2} / 2$. Denoting $a=\sqrt{} x, b=\sqrt{} y$, we get that $a^{2}-b^{2}=3(a-b)^{2} / 2$, which transforms into $\kappa(a-b)(2 a+2 b-3 a+3 b)=(a-b)(5 b-a)=0$. By the condition $a \neq b$, therefore $a=5 b$, i.e. $x / y=25$.