2019 Nepal TST

Part of Nepal TST

Subcontests

(3)

1

NEPAL TST 2019

Find all functions

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

such that for any real

x, y

f(xf(y)) + f(xy) = 2f(x)y

Proposed by Arseniy Nikolaev

1

NEPAL TST 2019

H

be orthocenter of an acute

\Delta ABC

M

is a midpoint of

AC

MH

AB, BC

A_1, C_1

A_2

C_2

are projections of

A_1, C_1

BH

respectively. Prove that lines

CA_2, AC_2

meet at circumscribed circle of

\Delta ABC

.
Proposed by Anton Trygub

1

NEPAL TST 2019

Prove that there exist infinitely many pairs of different positive integers

(m, n)

m!n!

is a square of an integer.
Proposed by Anton Trygub