MathDB

f_1(f_2(y) - x)+ 2x = f_3(x + y) => f(x - f(x))= 0

Source: Dutch IMO TST 2018 day 1 p4

8/30/2019

Let

A

be a set of functions

f : R\to R

. For all

f_1, f_2 \in A

there exists a

f_3 \in A

such that

f_1(f_2(y) - x)+ 2x = f_3(x + y)

for all

x, y \in R

. Prove that for all

f \in A

, we have

f(x - f(x))= 0

for all

x \in R

.

functional equationfunctionalgebra

students in a classroom sit in a round table, possible to split into 3 groups

Source: Dutch IMO TST 2018 day 2 p4

8/30/2019

In the classroom of at least four students the following holds: no matter which four of them take seats around a round table, there is always someone who either knows both of his neighbours, or does not know either of his neighbours. Prove that it is possible to divide the students into two groups such that in one of them, all students know one another, and in the other, none of the students know each other.
(Note: if student A knows student B, then student B knows student A as well.)

combinatorics

FN is tangent to the circle through B, I,C, incenter and circumcircle related

Source: Dutch IMO TST3 2018 p4

8/5/2019

In a non-isosceles triangle

ABC

the centre of the incircle is denoted by

I

. The other intersection point of the angle bisector of

\angle BAC

and the circumcircle of

\vartriangle ABC

is

D

. The line through

I

perpendicular to

AD

intersects

BC

in

F

. The midpoint of the circle arc

BC

on which

A

lies, is denoted by

M

. The other intersection point of the line

MI

and the circle through

B, I

and

C

, is denoted by

N

. Prove that

FN

is tangent to the circle through

B, I

and

C

.

circumcircletangentgeometryincenterarc midpoint

4

Problems(3)