2021 Iberoamerican

Part of IberoAmerican

Subcontests

(6)

1

Cyclic equality implies equal sum of squares

a,b,c,x,y,z

be real numbers such that

a^2+x^2=b^2+y^2=c^2+z^2=(a+b)^2+(x+y)^2=(b+c)^2+(y+z)^2=(c+a)^2+(z+x)^2

a^2+b^2+c^2=x^2+y^2+z^2

1

Grand finale of 2021 Iberoamerican MO

n

-sided regular polygon,

n \geq 4

V

r

vertices of the polygon. Show that if

r(r-3) \geq n

, then there exist at least two congruent triangles whose vertices belong to

V

1

Constructing two sets from conditions on their intersection, union and product

For a finite set

C

of integer numbers, we define

S(C)

as the sum of the elements of

C

. Find two non-empty sets

A

B

whose intersection is empty, whose union is the set

\{1,2,\ldots, 2021\}

and such that the product

S(A)S(B)

is a perfect square.

1

Sequence with condition on million consecutive terms

a_1,a_2,a_3, \ldots

be a sequence of positive integers and let

b_1,b_2,b_3,\ldots

be the sequence of real numbers given by b_n = \dfrac{a_1a_2\cdots a_n}{a_1+a_2+\cdots + a_n},\ \mbox{for}\ n\geq 1 Show that, if there exists at least one term among every million consecutive terms of the sequence

b_1,b_2,b_3,\ldots

that is an integer, then there exists some

k

b_k > 2021^{2021}

1

Midpoints and circumcircles lead to another midpoint

Consider an acute-angled triangle

ABC

AC>AB

\Gamma

be its circumcircle. Let

E

F

be the midpoints of the sides

AC

AB

, respectively. The circumcircle of the triangle

CEF

\Gamma

X

C

X\neq C

BX

and the tangent to

\Gamma

A

Y

P

be the point on segment

AB

YP = YA

P\neq A

Q

be the point where

AB

and the parallel to

BC

Y

meet each other. Show that

F

is the midpoint of

PQ

1

A prime number and its same-colored multiples

P = \{p_1,p_2,\ldots, p_{10}\}

10

different prime numbers and let

A

be the set of all the integers greater than

1

so that their prime decomposition only contains primes of

P

. The elements of

A

are colored in such a way that:
[*] each element of

P

has a different color, [*] if

m,n \in A

mn

is the same color of

m

n

, [*] for any pair of different colors

\mathcal{R}

\mathcal{S}

j,k,m,n\in A

(not necessarily distinct from one another), with

j,k

\mathcal{R}

m,n

\mathcal{S}

j

is a divisor of

m

n

is a divisor of

k

, simultaneously.
Prove that there exists a prime of

P

so that all its multiples in

A

are the same color.