MathDB

1

Equal distance to triangle vertices

ABC

with incenter

I

, the line

AI

intersects the circumcircle again at

D

, and let

J

be a point such that

D

is the midpoint of

IJ

. Consider points

E

and

F

on line

BC

such that

IE

and

JF

are perpendicular to

AI

. Consider points

G

on

AE

and

H

on

AF

such that

IG

and

JH

are perpendicular to

AE

and

AF

, respectively. Prove that

BG=CH

.

5

1

Moving one or 20 tokens at a time

We have a row of 203 cells. Initially the leftmost cell contains 203 tokens, and the rest are empty. On each move we can do one of the following: 1)Take one token, and move it to an adjacent cell (left or right). 2)Take exactly 20 tokens from the same cell, and move them all to an adjacent cell (all left or all right). After 2023 moves each cell contains one token. Prove that there exists a token that moved left at least nine times.

4

1

Roots of polynomial composition

Let

x_1\leq x_2\leq x_3\leq x_4

be real numbers. Prove that there exist polynomials of degree two

P(x)

and

Q(x)

with real coefficients such that

x_1

,

x_2

,

x_3

and

x_4

are the roots of

P(Q(x))

if and only if

x_1+x_4=x_2+x_3

.

3

1

Divisibility with sum of squares

Find all quadruples

(a,b,c,d)

of positive integers satisfying that

a^2+b^2=c^2+d^2

and such that

ac+bd

divides

a^2+b^2

.

2

1

Equal lengths on altitude

Let

ABC

be an acute scalene triangle with incenter

I

and orthocenter

H

. Let

M

be the midpoint of

AB

. On the line

AH

we consider points

D

and

E

, such that the line

MD

is parallel to

CI

and

ME

is perpendicular to

CI

. Prove that

AE=DH

.

1

Lamp switches in cube pieces

A

3\times 3\times 3

cube is made of 27 unit cube pieces. Each piece contains a lamp, which can be on or off. Every time a piece is pressed (the center piece cannot be pressed), the state of that piece and the pieces that share a face with it changes. Initially all lamps are off. Determine which of the following states are achievable: (1) All lamps are on. (2) All lamps are on except the central one. (3) Only the central lamp is on.

2023 Spain Mathematical Olympiad

Subcontests

Equal distance to triangle vertices

Moving one or 20 tokens at a time

Roots of polynomial composition

Divisibility with sum of squares

Equal lengths on altitude

Lamp switches in cube pieces