BG-JBMO-TST 2022

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.)