Are there positive integers $a$, $b$, $c$ and $d$ such that: a) $a^{2021} + b^{2023} = 11(c^{2022} + d^{2024})$; b) $a^{2022} + b^{2022} = 11(c^{2022} + d^{2022})$?

Let $a$, $b$ and $c$ be positive real numbers with $abc = 1$. Determine the minimum possible value of $$ \left(\frac{a}{b} + \frac{b}{c} + \frac{c}{a}\right) \cdot \left(\frac{ab}{a+b} + \frac{bc}{b+c} + \frac{ca}{c+a}\right) $$as well as all triples $(a,b,c)$ which attain the minimum.

For a positive integer $n$ let $t_n$ be the number of unordered triples of non-empty and pairwise disjoint subsets of a given set with $n$ elements. For example, $t_3 = 1$. Find a closed form formula for $t_n$ and determine the last digit of $t_{2022}$. (I also give here that $t_4 = 10$, for a reader to check his/her understanding of the problem statement.)

Let $ABC (AC < BC)$ be an acute triangle with circumcircle $k$ and midpoint $P$ of $AB$. The altitudes $AM$ and $BN$ ($M\in BC$, $N\in AC$) intersect at $H$. The point $E$ on $k$ is such that the segments $CE$ and $AB$ are perpendicular. The line $EP$ intersects $k$ again at point $K$ and the point $Q$ on $k$ is such that $KQ$ and $AB$ are parallel. The circumcircle of $AHB$ intersects the segment $CP$ at an interior point $R$. Prove that the points $C$, $M$, $R$, $H$, $N$ and $Q$ are concyclic.

Determine all triples $(a,b,c)$ of real numbers such that $$ (2a+1)^2 - 4b = (2b+1)^2 - 4c = (2c+1)^2 - 4a = 5. $$

Let $ABC$ ($AB < AC$) be a triangle with circumcircle $k$. The tangent to $k$ at $A$ intersects the line $BC$ at $D$ and the point $E\neq A$ on $k$ is such that $DE$ is tangent to $k$. The point $X$ on line $BE$ is such that $B$ is between $E$ and $X$ and $DX = DA$ and the point $Y$ on the line $CX$ is such that $Y$ is between $C$ and $X$ and $DY = DA$. Prove that the lines $BC$ and $YE$ are perpendicular.

The integers $a$, $b$, $c$ and $d$ are such that $a$ and $b$ are relatively prime, $d\leq 2022$ and $a+b+c+d = ac + bd = 0$. Determine the largest possible value of $d$,

There are $n\leq 99$ people around a circular table. At every moment everyone can either be truthful (always says the truth) or a liar (always lies). Initially some of people (possibly none) are truthful and the rest are liars. At every minute everyone answers at the same time the question "Is your left neighbour truthful or a liar?" and then becomes the same type of person as his answer. Determine the largest $n$ for which, no matter who are the truthful people in the beginning, at some point everyone will become truthful forever.