MathDB
Problems
Contests
National and Regional Contests
The Philippines Contests
Philippine MO
2023 Philippine MO
8
8
Part of
2023 Philippine MO
Problems
(1)
Geo FE with orthocenter
Source: Philippine MO 2023/8
3/19/2023
Let
S
\mathcal{S}
S
be the set of all points in the plane. Find all functions
f
:
S
→
R
f : \mathcal{S} \rightarrow \mathbb{R}
f
:
S
→
R
such that for all nondegenerate triangles
A
B
C
ABC
A
BC
with orthocenter
H
H
H
, if
f
(
A
)
≤
f
(
B
)
≤
f
(
C
)
f(A) \leq f(B) \leq f(C)
f
(
A
)
≤
f
(
B
)
≤
f
(
C
)
, then
f
(
A
)
+
f
(
C
)
=
f
(
B
)
+
f
(
H
)
.
f(A) + f(C) = f(B) + f(H).
f
(
A
)
+
f
(
C
)
=
f
(
B
)
+
f
(
H
)
.
geometry
functional equation
combinatorial geometry
PMO
function
nice