2024 Nepal TST

Part of Nepal TST

Subcontests

(4)

1

Angle geometry

ABC

be an acute triangle so that

2BC = AB + AC,

I{}.

AI{}

BC{}

A'.{}

The perpendicular bisector of

AA'{}

BI{}

CI{}

B'{}

C'{}

respectively. Let

AB'{}

(ABC)

X{}

XI{}

AC'{}

X'{}.

2\angle XX'A'=\angle ABC.{}

(Proposed by Kang Taeyoung, South Korea)

1

A strange divisibility condition

Prove that there are infinitely many integers

k\geqslant 2024

for which there exists a set

\{a_1,\ldots,a_k\}

with the following properties: [*]

a_1{}

is a positive integer and

a_{i+1}=a_i+1

1\leqslant i<k,

2(a_1\cdots a_{k-2}-1)^2

is divisible by

2(a_1+\cdots+a_k)+a_1-a_1^2.

(Proposed by Prajit Adhikari, Nepal)

1

Function can't not satisfy this!

f: \mathbb{N} \to \mathbb{N}

be an arbitrary function. Prove that there exist two positive integers

x

y

f(x+y) \le f(2x+f(y))

.
(Proposed by David Anghel, Romania)

1

Exponents modulo 4

a, b

be positive integers. Prove that if

a^b + b^a \equiv 3 \pmod{4}

a+1

b+1

can't be written as the sum of two integer squares.
(Proposed by Orestis Lignos, Greece)