CN-SMO 2020

Given a polynomial $f(x)=x^{2020}+\sum_{i=0}^{2019} c_ix^i$ , where $c_i \in \{ -1,0,1 \}$ . Denote $N$ the number of positive integer roots of $f(x)=0$ (counting multiplicity). If $f(x)=0$ has no negative integer roots, find the maximum of $N$ .

Consider the set $I=\{ 1,2, \cdots, 2020 \}$ . Let $W= \{w(a,b)=(a+b)+ab | a,b \in I \} \cap I$ , $Y=\{y(a,b)=(a+b) \cdot ab | a,b \in I \} \cap I$ be its two subsets. Further, let $X= W \cap Y$ . (1) Find the sum of maximal and minimal elements in $X$ .(2) An element $n=y(a,b) (a \le b)$ in $Y$ is called excellent, if its representation is not unique (for instance, $20=y(1,5)=y(2,3)$ ). Find the number of excellent elements in $Y$ .

In a quadrilateral $ABCD$ , $\angle ABC=\angle ADC <90^{\circ}$ . The circle with diameter $AC$ intersects $BC$ and $CD$ again at $E,F$ , respectively. $M$ is the midpoint of $BD$ , and $AN \perp BD$ at $N$ . Prove that $M,N,E,F$ is concyclic.

Using a nozzle to paint each square in a $1 \times n$ stripe, when the nozzle is aiming at the $i$ -th square, the square is painted black, and simultaneously, its left and right neighboring square (if exists) each has an independent probability of $\tfrac{1}{2}$ to be painted black.

In the optimal strategy (i.e. achieving least possible number of painting), the expectation of number of painting to paint all the squares black, is $T(n)$ . Find the explicit formula of $T(n)$ .