2022 Indonesia MO

Part of Indonesia MO

Subcontests

(8)

1

GET WRECKED!

ABC

D

E

AB

AC

DE

BC

. There exists point

P

in the interior of

BDEC

\angle BPD = \angle CPE = 90^{\circ}

Prove that the line

AP

passes through the circumcenter of triangles

EPD

BPC

1

Get bored of inequalities yet?

Determine the smallest positive real

K

such that the inequality

K + \frac{a + b + c}{3} \ge (K + 1) \sqrt{\frac{a^2 + b^2 + c^2}{3}}

holds for any real numbers

0 \le a,b,c \le 1

.
Proposed by Fajar Yuliawan, Indonesia

1

How is it not prime relatives???

N\ge2

be a positive integer. Given a sequence of natural numbers

a_1,a_2,a_3,\dots,a_{N+1}

such that for every integer

1\le i\le j\le N+1

a_ia_{i+1}a_{i+2}\dots a_j \not\equiv1\mod{N}

Prove that there exist a positive integer

k\le N+1

\gcd(a_k, N) \neq 1

1

Most Number of Sequences

A

be the sequence of zeroes and ones (binary sequence). The sequence can be modified by the following operation: we may pick a block or a contiguous subsequence where there are an unequal number of zeroes and ones, and then flip their order within the block (so block

a_1, a_2, \ldots, a_r

a_r, a_{r-1}, \ldots, a_1

).
As an example, let

A

be the sequence

1,1,0,0,1

. We can pick block

1,0,0

and flip it, so the sequence

1,\boxed{1,0,0},1

1,\boxed{0,0,1},1

. However, we cannot pick block

1,1,0,0

and flip their order since they contain the same number of

1

0

s.
Two sequences

A

B

are called related if

A

can be transformed into

B

using a finite number the operation mentioned above.
Determine the largest natural number

n

for which there exists

n

different sequences

A_1, A_2, \ldots, A_n

where each sequence consists of 2022 digits, and for every index

i \neq j

A_i

is not related to

A_j

1

A particular case of an "unsolved" problem

Given a regular

26

-gon. Prove that for any

9

vertices of that regular

26

-gon, then there exists three vertices that forms an isosceles triangle.

1

Rectangle geometry with concurrency

ABCD

be a rectangle. Points

E

F

are on diagonal

AC

F

A

E

E

C

F

. The circumcircle of triangle

BEF

AB

BC

G

H

respectively, and the circumcircle of triangle

DEF

AD

CD

I

J

respectively. Prove that the lines

GJ, IH

AC

concur at a point.

1

Standard Polynomial NT

P(x)

be a polynomial with integer coefficient such that

P(1) = 10

P(-1) = 22

. (a) Give an example of

P(x)

P(x) = 0

has an integer root. (b) Suppose that

P(0) = 4

P(x) = 0

does not have an integer root.

1

Boring Wrapped FE

Determine all functions

f: \mathbb{R} \to \mathbb{R}

such that for any

x,y \in \mathbb{R}

f(f(f(x)) + f(y)) = f(y) - f(x)