MathDB

A functional equation on positive integers with two conditions

Source: Dutch TST 2024, 2.2

6/28/2024

Find all functions

f:\mathbb{Z}_{>0} \to \mathbb{Z}_{>0}

such that for all positive integers

m,n

and

a

we have a)

f(f(m)f(n))=mn

and b)

f(2024a+1)=2024a+1

.

algebrafunctional equationalgebra proposed

A three-variable functional inequality on non-negative reals

Source: Dutch TST 2024, 1.2

6/28/2024

Find all functions

f:\mathbb{R}_{\ge 0} \to \mathbb{R}

with

2x^3zf(z)+yf(y) \ge 3yz^2f(x)

for all

x,y,z \in \mathbb{R}_{\ge 0}

.

inequalitiesfunctionfunctional equationalgebra proposedalgebraFunctional inequality

Compute side ratio in triangle if circle passes through incenter

Source: Dutch TST 2024, 3.2

6/28/2024

Let

ABC

be a triangle. A point

P

lies on the segment

BC

such that the circle with diameter

BP

passes through the incenter of

ABC

. Show that

\frac{BP}{PC}=\frac{c}{s-c}

where

c

is the length of segment

AB

and

2s

is the perimeter of

ABC

.

ratiogeometrygeometry proposedincenterSides of a triangle

2