MathDB

1

n points on plane partitioned into m sets whose convex hulls have common point

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.

3

1

(-1)^dP (-x), powerful + non-increasing polynomials with real coefficients

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.

2

1

SK // BC wanted, CM x CL = AM x BL, circumcircle related

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

.

1

a_n = n if strictly increasing, a_n<=n+2020m , n^3a_n-1 divisible by a_ {n+1}

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

.

2020 Silk Road

Subcontests

n points on plane partitioned into m sets whose convex hulls have common point

(-1)^dP (-x), powerful + non-increasing polynomials with real coefficients

SK // BC wanted, CM x CL = AM x BL, circumcircle related

a_n = n if strictly increasing, a_n<=n+2020m , n^3a_n-1 divisible by a_ {n+1}

2020 Silk Road

Subcontests

n points on plane partitioned into m sets whose convex hulls have common point

(-1)^dP (-x), powerful + non-increasing polynomials with real coefficients

SK // BC wanted, CM x CL = AM x BL, circumcircle related

a_n = n if strictly increasing, a_n&lt;=n+2020m , n^3a_n-1 divisible by a_ {n+1}

a_n = n if strictly increasing, a_n<=n+2020m , n^3a_n-1 divisible by a_ {n+1}