MathDB

2

Graph of polynomial with integer coefficients

On the graph of a polynomial with integer coefficients, two points are chosen with integer coordinates. Prove that if the distance between them is an integer, then the segment that connects them is parallel to the horizontal axis.

Three colours for each face of a polyhedron

All faces of a polyhedron are triangles. Each of the vertices of this polyhedron is assigned independently one of three colors : green, white or black. We say that a face is Extremadura if its three vertices are of different colors, one green, one white and one black. Is it true that regardless of how the vertices's color, the number of Extremadura faces of this polyhedron is always even?

3

2

Game with subtract 1 or divide by 2.

On the board is written an integer

N \geq 2

. Two players

A

and

B

play in turn, starting with

A

. Each player in turn replaces the existing number by the result of performing one of two operations: subtract 1 and divide by 2, provided that a positive integer is obtained. The player who reaches the number 1 wins. Determine the smallest even number

N

requires you to play at least

2015

times to win (

B

shifts are not counted).

Simple but not easy. 2015 Spain Math Olympiad.

Let

ABC

be a triangle.

M

, and

N

points on

BC

, such that

BM=CN

, with

M

in the interior of

BN

. Let

P

and

Q

be points in

AN

and

AM

respectively such that

\angle PMC= \angle MAB

, and

\angle QNB= \angle NAC

. Prove that

\angle QBC= \angle PCB

.

2

symmetrical with respect to the circumcenter

In triangle

ABC

, let

A'

is the symmetrical of

A

with respect to the circumcenter

O

of

ABC

. Prove that: a) The sum of the squares of the tangents segments drawn from

A

and

A'

to the incircle of

ABC

equals

4R^2-4Rr-2r^2

where

R

and

r

are the radii of the circumscribed and inscribed circles of

ABC

respectively. b) The circle with center

A'

and radius

A'I

intersects the circumcircle of

ABC

in a point

L

such that

AL=\sqrt{ AB.AC}

where

I

is the centre of the inscribed circle of

ABC

.

sum of perfect squares

Let

p

and

n

be a natural numbers such that

p

is a prime and

1+np

is a perfect square. Prove that the number

n+1

is sum of

p

perfect squares.

2015 Spain Mathematical Olympiad

Subcontests

Graph of polynomial with integer coefficients

Three colours for each face of a polyhedron

Game with subtract 1 or divide by 2.

Simple but not easy. 2015 Spain Math Olympiad.

symmetrical with respect to the circumcenter

sum of perfect squares