2011 Polish MO Finals

Part of Polish MO Finals

Subcontests

(3)

2

Number of Real Solutions of The System of The Equations

n\geq 3

be an odd integer. Determine how many real solutions there are to the set of

n

\left\{\begin{array}{cc}x_1(x_1+1)=x_2(x_2-1)\\x_2(x_2+1)=x_3(x_3-1)\\ \vdots \\ x_n(x_n+1) = x_1(x_1-1)\end{array}\right.

Sum of polynomials is not equal to x^2+7

Prove that it is impossible for polynomials

f_1(x),f_2(x),f_3(x),f_4(x)\in \mathbb{Q}[x]

f_1^2(x)+f_2^2(x)+f_3^2(x)+f_4^2(x) = x^2+7.

2

Sum of the numbers is divisible by k

Find all integers

n\geq 1

such that there exists a permutation

(a_1,a_2,...,a_n)

(1,2,...,n)

a_1+a_2+...+a_k

is divisible by

k

k=1,2,...,n

All functions with f(x)f(y)=g(x)g(y)+g(x)+g(y)

Determine all pairs of functions

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

such that for any

x,y\in \mathbb{R}

f(x)f(y)=g(x)g(y)+g(x)+g(y).

2

Circumcenters coincide

The incircle of triangle

ABC

BC,CA,AB

D,E,F

respectively. Consider the triangle formed by the line joining the midpoints of

AE,AF

, the line joining the midpoints of

BF,BD

, and the line joining the midpoints of

CD,CE

. Prove that the circumcenter of this triangle coincides with the circumcenter of triangle

ABC

Prove that PQ=2HP

In a tetrahedron

ABCD

, the four altitudes are concurrent at

H

DH

intersects the plane

ABC

P

and the circumsphere of

ABCD

Q\neq D

PQ=2HP