MathDB

1

2019 Iberoamerican Mathematical Olympiad P5

(n+1)^2

vertices determined by an

n \times n

board. A move consists of moving the token from the vertex on which it is placed to an adjacent vertex which is at most

\sqrt2

away, as long as it stays on the board. A path is a sequence of moves such that the token was in each one of the

(n+1)^2

vertices exactly once. What is the maximum number of diagonal moves (those of length

\sqrt2

) that a path can have in total?

6

1

2019 Iberoamerican Mathematical Olympiad P6

Let

a_1, a_2, \dots, a_{2019}

be positive integers and

P

a polynomial with integer coefficients such that, for every positive integer

n

,

P(n) \text{ divides } a_1^n+a_2^n+\dots+a_{2019}^n.

Prove that

P

is a constant polynomial.

4

1

2019 Iberoamerican Mathematical Olympiad P4

Let

ABCD

be a trapezoid with

AB\parallel CD

and inscribed in a circumference

\Gamma

. Let

P

and

Q

be two points on segment

AB

(

A

,

P

,

Q

,

B

appear in that order and are distinct) such that

AP=QB

. Let

E

and

F

be the second intersection points of lines

CP

and

CQ

with

\Gamma

, respectively. Lines

AB

and

EF

intersect at

G

. Prove that line

DG

is tangent to

\Gamma

.

1

2019 Iberoamerican Mathematical Olympiad, P1

For each positive integer

n

, let

s(n)

be the sum of the squares of the digits of

n

. For example,

s(15)=1^2+5^2=26

. Determine all integers

n\geq 1

such that

s(n)=n

.

2

1

2019 Iberoamerican Mathematical Olympiad, P2

Determine all polynomials

P(x)

with degree

n\geq 1

and integer coefficients so that for every real number

x

the following condition is satisfied

P(x)=(x-P(0))(x-P(1))(x-P(2))\cdots (x-P(n-1))

3

1

2019 Iberoamerican Mathematical Olympiad, P3

Let

\Gamma

be the circumcircle of triangle

ABC

. The line parallel to

AC

passing through

B

meets

\Gamma

at

D

(

D\neq B

), and the line parallel to

AB

passing through

C

intersects

\Gamma

to

E

(

E\neq C

). Lines

AB

and

CD

meet at

P

, and lines

AC

and

BE

meet at

Q

. Let

M

be the midpoint of

DE

. Line

AM

meets

\Gamma

at

Y

(

Y\neq A

) and line

PQ

at

J

. Line

PQ

intersects the circumcircle of triangle

BCJ

at

Z

(

Z\neq J

). If lines

BQ

and

CP

meet each other at

X

, show that

X

lies on the line

YZ

.

2019 IberoAmerican

Subcontests

2019 Iberoamerican Mathematical Olympiad P5

2019 Iberoamerican Mathematical Olympiad P6

2019 Iberoamerican Mathematical Olympiad P4

2019 Iberoamerican Mathematical Olympiad, P1

2019 Iberoamerican Mathematical Olympiad, P2

2019 Iberoamerican Mathematical Olympiad, P3