Given $a,b,c>0$ such that $$a+b+c+\frac{1}{abc}=\frac{19}{2}$$ What is the greatest value for $a$?

Let $(a_n)$ and $(b_n)$ be sequences of real numbers, such that $a_1 = b_1 = 1$, $a_{n+1} = a_n + \sqrt{a_n}$, $b_{n+1} = b_n + \sqrt[3]{b_n}$ for all positive integers $n$. Prove that there is a positive integer $n$ for which the inequality $a_n \leq b_k < a_{n+1}$ holds for exactly 2021 values of $k$.

Find all functions $f : \mathbb{R^{+}}\to \mathbb{R^{+}}$ such that $$f(x)^2=f(xy)+f(x+f(y))-1$$for all $x, y\in \mathbb{R^{+}}$

Given acute triangle $ABC$ with circumcircle $\Gamma$ and altitudes $AD, BE, CF$, line $AD$ cuts $\Gamma$ again at $P$ and $PF, PE$ meet $\Gamma$ again at $R, Q$. Let $O_1, O_2$ be the circumcenters of $\triangle BFR$ and $\triangle CEQ$ respectively. Prove that $O_{1}O_{2}$ bisects $\overline{EF}$.

Let $a$ be a positive integer. Prove that for any pair $(x,y)$ of integer solutions of equation $$x(y^2-2x^2)+x+y+a=0$$we have: $$|x| \leqslant a+\sqrt{2a^2+2}$$