2022 Peru MO (ONEM)

Part of Peru MO (ONEM)

Subcontests

(4)

1

peruvian area chasing in final round

D

be the midpoint of the side

BC

ABC

G

be the point of the segment

AD

AG = 2GD

E

F

be points on the sides

AB

AC

, respectively, such that

G

lies on the segment

EF

M

N

be points of the segments

AE

AF

, respectively, such that

ME = EB

NF = FC

. a) Prove that the area of the quadrilateral

BMNC

is equal to four times the area of the triangle

DEF

. b) Prove that the quadrilaterals

MNFE

AMDN

have the same area.

1

sum of 8 numbers, after 8 operations remains the same

The following figure is made up of

12

8

circles. As you can see, at the beginning all the circles are empty. In each operation an empty circle is chosen, it is painted red and inside it the number of red neighboring circles that the chosen circle has is written (in the first operation the chosen circle is painted red and the number

0

is written). After

8

operations all the circles are painted red and each one has a number written on it. Prove that, no matter how the operations are done, the sum of all the numbers at the end is the same. https://cdn.artofproblemsolving.com/attachments/3/a/8cd74a0fdc7bb9bc5d1bc764e80ffb58159c0c.png

1

n=(odd prime)^a, R(n) = 11 ... 1

For each positive integer n, the number

R(n) = 11 ... 1

is defined, which is made up of exactly

n

digits equal to

1

R(5) = 11111

n > 4

be an integer for which, by writing all the positive divisors of

R(n)

, it is true that each written digit belongs to the set

\{0, 1\}

n

is a power of an odd prime number.
Clarification: A power of an odd prime number is a number of the form

p^a

p

is an odd prime number and

a

is a positive integer.

1

f(xy) + y + f(x + f(y)) = (y + 1)f(x)

R

be the set of real numbers and

f : R \to R

be a function that satisfies:

f(xy) + y + f(x + f(y)) = (y + 1)f(x),

for all real numbers

x, y

. a) Determine the value of

f(0)

. b) Prove that

f(x) = 2-x

for every real number

x