SILK-ROAD 2020

Given a strictly increasing infinite sequence of natural numbers $a_1,$ $a_2,$ $a_3,$ $\ldots$ . It is known that $a_n \leq n + 2020$ and the number $n ^ 3 a_n - 1$ is divisible by $a_ {n + 1}$ for all natural numbers $n$ . Prove that $a_n = n$ for all natural numbers $n$ .

The triangle $ABC$ is inscribed in the circle $\omega$ . Points $K, L, M$ are marked on the sides $AB, BC, CA$ , respectively, and $CM \cdot CL = AM \cdot BL$ . Ray $LK$ intersects line $AC$ at point $P$ . The common chord of the circle $\omega$ and the circumscribed circle of the triangle $KMP$ meets the segment $AM$ at the point $S$ . Prove that $SK \parallel BC$ .

A polynomial $Q (x) = k_n x ^ n + k_ {n-1} x ^ {n-1} + \ldots + k_1 x + k_0$ with real coefficients is called powerful if the equality $| k_0 | = | k_1 | + | k_2 | + \ldots + | k_ {n-1} | + | k_n |$ , and non-increasing , if $k_0 \geq k_1 \geq \ldots \geq k_ {n-1} \geq k_n$ . Let for the polynomial $P (x) = a_d x ^ d + a_ {d-1} x ^ {d-1} + \ldots + a_1 x + a_0$ with nonzero real coefficients, where $a_d> 0$ , the polynomial $P (x) (x-1) ^ t (x + 1) ^ s$ is powerful for some non-negative integers $s$ and $t$ ( $s + t> 0$ ). Prove that at least one of the polynomials $P (x)$ and $(- 1) ^ d P (-x)$ is nonincreasing.

Prove that for any natural number $m$ there exists a natural number $n$ such that any $n$ distinct points on the plane can be partitioned into $m$ non-empty sets whose convex hulls have a common point. The convex hull of a finite set $X$ of points on the plane is the set of points lying inside or on the boundary of at least one convex polygon with vertices in $X$ , including degenerate ones, that is, the segment and the point are considered convex polygons. No three vertices of a convex polygon are collinear. The polygon contains its border.