2024 Turkey EGMO TST

Part of Turkey EGMO TST

Subcontests

(6)

1

Geometry with Miquel configuration

ABC

be a triangle and its circumcircle be

\omega

I

be the incentre of the

ABC

BI

AC

E

\omega

M

for the second time. The line

CI

AB

F

\omega

N

for the second time. Let the circumcircles of

BFI

CEI

meet again at point

K

. Prove that the lines

BN

CM

AK

are concurrent.

1

Euler totient and functional equation

Find all functions

f:\mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}

such that the conditions
a) a-b \mid f(a)-f(b) for all

a\neq b

a,b \in \mathbb{Z}^{+}

b) f(\varphi(a))=\varphi(f(a)) for all

a \in \mathbb{Z}^{+}

\varphi

is the Euler's totient function.
holds

1

Unlucky Graph with 13 colors

Initially, all edges of the

K_{2024}

are painted with

13

different colors. If there exist

k

colors such that the subgraph constructed by the edges which are colored with these

k

colors is connected no matter how the initial coloring was, find the minimum value of

k

1

Induction screaming sequence

(a_n)_{n=1}^{\infty}

be a strictly increasing sequence such that inequality

a_n(a_n-2a_{n-1})+a_{n-1}(a_{n-1}-2a_{n-2})\geq 0

n \geq 3

. Prove that for all

n\geq2

a_n \geq a_{n-1}+a_{n-2}+\dots+a_1

1

Combinatorics with prime and chess board

p

be a given prime number. For positive integers

n,k\geq2

S_1, S_2,\dots, S_n

be unit square sets constructed by choosing exactly one unit square from each of the columns from

p\times k

chess board. If

|S_i \cap S_j|=1

1\leq i < j \leq n

and for any duo of unit squares which are located at different columns there exists

S_i

such that both of these unit squares are in

S_i

find all duos of

(n,k)

p

.
Note: Here we denote the number of rows by

p

and the number of columns by

k

1

Tangency at everywhere

\omega_1

\omega_2

be two different circles that intersect at two different points,

X

Y

l_1

l_2

be common tangent lines of these circles such that

l_1

\omega_1

A

\omega_2

C

l_2

\omega_1

B

\omega_2

D

Z

be the reflection of

Y

l_1

BC

\omega_1

K

for the second time. Let

AD

\omega_2

L

for the second time. Prove that the line tangent to

\omega_1

and passes through

K

and the line tangent to

\omega_2

and passes through

L

meet on the line

XZ