BG-MO 2020

On the sides of $\triangle{ABC}$ points $P,Q \in{AB}$ ( $P$ is between $A$ and $Q$ ) and $R\in{BC}$ are chosen. The points $M$ and $N$ are defined as the intersection point of $AR$ with the segments $CP$ and $CQ$ , respectively. If $BC=BQ$ , $CP=AP$ , $CR=CN$ and $\angle{BPC}=\angle{CRA}$ , prove that $MP+NQ=BR$ .

Let $b_1$ , $\dots$ , $b_n$ be nonnegative integers with sum $2$ and $a_0$ , $a_1$ , $\dots$ , $a_n$ be real numbers such that $a_0=a_n=0$ and $|a_i-a_{i-1}|\leq b_i$ for each $i=1$ , $\dots$ , $n$ . Prove that $$\sum_{i=1}^n(a_i+a_{i-1})b_i\leq 2$$ Click to expandI believe that the original problem was for nonnegative real numbers and it was a typo on the version of the exam paper we had but I'm not sure the inequality would hold

Let $a_1\in\mathbb{Z}$ , $a_2=a_1^2-a_1-1$ , $\dots$ , $a_{n+1}=a_n^2-a_n-1$ . Prove that $a_{n+1}$ and $2n+1$ are coprime.

Are there positive integers $m>4$ and $n$ , such that a) ${m \choose 3}=n^2;$ b) ${m \choose 4}=n^2+9?$

There are $n$ points in the plane, some of which are connected by segments. Some of the segments are colored in white, while the others are colored black in such a way that there exist a completely white as well as a completely black closed broken line of segments, each of them passing through every one of the $n$ points exactly once. It is known that the segments $AB$ and $BC$ are white. Prove that it is possible to recolor the segments in red and blue in such a way that $AB$ and $BC$ are recolored as red, not all of which segments are recolored redmeaning that recoloring every white as red and every black as blue is not acceptable, and that there exist a completely red as well as a completely blue closed broken line of segments, each of them passing through every one of the $n$ points exactly once.

Let $f(x)$ be a nonconstant real polynomial. The sequence $\{a_i\}_{i=1}^{\infty}$ of real numbers is strictly increasing and unbounded, as $$a_{i+1}<a_i+2020.$$ The integers $\lfloor{|f(a_1)|}\rfloor$ , $\lfloor{|f(a_2)|}\rfloor$ , $\lfloor{|f(a_3)|}\rfloor$ , $\dots$ are written consecutively in such a way that their digits form an infinite sequence of digits $\{s_k\}_{k=1}^{\infty}$ (here $s_k\in\{0, 1, \dots, 9\}$ ). $$ If $n\in\mathbb{N}$ , prove that among the numbers $\overline{s_{n(k-1)+1}s_{n(k-1)+2}\cdots s_{nk}}$ , where $k\in\mathbb{N}$ , all $n$ -digit numbers appear.