MathDB

1

Geo FE with orthocenter

\mathcal{S}

be the set of all points in the plane. Find all functions

f : \mathcal{S} \rightarrow \mathbb{R}

such that for all nondegenerate triangles

ABC

with orthocenter

H

, if

f(A) \leq f(B) \leq f(C)

, then

f(A) + f(C) = f(B) + f(H).

7

1

Divide factors into Fibonacci and triangular sets

A set of positive integers is said to be pilak if it can be partitioned into 2 disjoint subsets

F

and

T

, each with at least

2

elements, such that the elements of

F

are consecutive Fibonacci numbers, and the elements of

T

are consecutive triangular numbers. Find all positive integers

n

such that the set containing all the positive divisors of

n

except

n

itself is pilak.

6

1

Sliding FE

Find all functions

f : \mathbb{R} \rightarrow \mathbb{R}

such that

f(2f(x)) = f(x - f(y)) + f(x) + y

for all

x, y \in \mathbb{R}

.

5

1

Do operation on string finitely many times

Silverio is very happy for the 25th year of the PMO. In his jubilation, he ends up writing a finite sequence of As and Gs on a nearby blackboard. He then performs the following operation: if he finds at least one occurrence of the string "AG", he chooses one at random and replaces it with "GAAA". He performs this operation repeatedly until there is no more "AG" string on the blackboard. Show that for any initial sequence of As and Gs, Silverio will eventually be unable to continue doing the operation.

4

1

Knights on a 10x10 board

In chess, a knight placed on a chess board can move by jumping to an adjacent square in one direction (up, down, left, or right) then jumping to the next two squares in a perpendicular direction. We then say that a square in a chess board can be attacked by a knight if the knight can end up on that square after a move. Thus, depending on where a knight is placed, it can attack as many as eight squares, or maybe even less.
In a

10 \times 10

chess board, what is the maximum number of knights that can be placed such that each square on the board can be attacked by at most one knight?

3

1

Circumcircles and concurrence

In

\triangle ABC

,

AB > AC

. Point

P

is on line

BC

such that

AP

is tangent to its circumcircle. Let

M

be the midpoint of

AB

, and suppose the circumcircle of

\triangle PMA

meets line

AC

again at

N

. Point

Q

is the reflection of

P

with respect to the midpoint of segment

BC

. The line through

B

parallel to

QN

meets

PN

at

D

, and the line through

P

parallel to

DM

meets the circumcircle of

\triangle PMB

again at

E

. Show that the lines

PM

,

BE

, and

AC

are concurrent.

2

1

2^(p+1) - 4 / p a perfect square

Find all primes

p

such that

\dfrac{2^{p+1}-4}{p}

is a perfect square.

1

Diophantine muahahaha

Find all ordered pairs

(a, b)

of positive integers such that

a^2 + b^2 + 25 = 15ab

and

a^2 + ab + b^2

is prime.

2023 Philippine MO

Subcontests

Geo FE with orthocenter

Divide factors into Fibonacci and triangular sets

Sliding FE

Do operation on string finitely many times

Knights on a 10x10 board

Circumcircles and concurrence

2^(p+1) - 4 / p a perfect square

Diophantine muahahaha