2020 Korean MO winter camp

Part of Korea Winter Program Practice Test

Subcontests

(8)

1

Another FE

Find all functions

f:\mathbb{R}\rightarrow\mathbb{R}

2f(x^2+y^2)=(x+f(y))^2+f(x-f(y))^2

x,y\in\mathbb{R}

1

Integer sequence and divisibility

Find all strictly increasing sequences

\{a_n\}_{n=0}^\infty

of positive integers such that for all positive integers

k,m,n

\frac{a_{n+1} +a_{n+2} +\dots +a_{n+k}}{k+m}

is not an integer larger than

2020

1

Two tangent circles

\square ABCD

is a quadrilateral with

\angle A=2\angle C <90^\circ

I

is the incenter of

\triangle BAD

, and the line passing

I

and perpendicular to

AI

CB

CD

E,F

respectively. Denote

O

as the circumcenter of

\triangle CEF

. The line passing

E

and perpendicular to

OE

OF

Q

, and the line passing

F

and perpendicular to

OF

OE

P

. Prove that the circle with diameter

PQ

is tangent to the circumcircle of

\triangle BCD

1

Line DT tangent to incircle

I

is the incenter of a given triangle

\triangle ABC

. The angle bisectors of

ABC

meet the sides at

D,E,F

EF

(ABC)

L

T

F

LE

M

is the midpoint of

BC

. Prove that if

DT

is tangent to the incircle of

ABC

IL

\angle MLT

1

Polynomials and primes

Find all integer coefficient polynomials

Q

Q(n)\ge 1

\forall n\in \mathbb{Z}_+

Q(mn)

Q(m)Q(n)

have the same number of prime divisors

\forall m,n\in\mathbb{Z}_+

1

Minimizing sum of squares

X

2020

distinct real numbers. Prove that there exist

a,b\in \mathbb{R}

A\subset X

\sum_{x\in A}(x-a)^2 +\sum_{x\in X\backslash A}(x-b)^2\le \frac{1009}{1010}\sum_{x\in X}x^2

1

Subset sum problem

Call a positive integer challenging if it can be expressed as

2^a(2^b+1)

a,b

are positive integers. Prove that if

X

is a set of challenging numbers smaller than

2^n (n

is a given positive integer) and

|X|\ge \frac{4}{3}(n-1)

, there exist two disjoint subsets

A,B\subset X

|A|=|B|

\sum_{a\in A}a=\sum_{b \in B}b

1

A Challenging Graph Theory Problem

I've come across a challenging graph theory problem. Roughly translated, it goes something like this:
There are n lines drawn on a plane; no two lines are parallel to each other, and no three lines meet at a single point.
Those lines would partition the plane down into many 'area's. Suppose we select one point from each area. Also, should two areas share a common side, we connect the two points belonging to the respective areas with a line.
A graph consisted of points and lines will have been made. Find all possible 'n' that will make a hamiltonian circuit exist for the given graph