MathDB

1

Geometry with orthocenter II

ABC

be an acute triangle with orthocenter

H

, circumcenter

O

, and circumcircle

\Omega

. Points

E

and

F

are the feet of the altitudes from

B

to

AC

, and from

C

to

AB

, respectively. Let line

AH

intersect

\Omega

again at

D

. The circumcircle of

DEF

intersects

\Omega

again at

X

, and

AX

intersects

BC

at

I

. The circumcircle of

IEF

intersects

BC

again at

G

. If

M

is the midpoint of

BC

, prove that lines

MX

and

OG

intersect on

\Omega

.

P3

1

Geometry with orthocenter I

Given triangle

ABC

with orthocenter

H

, the lines through

B

and

C

perpendicular to

AB

and

AC

, respectively, intersect line

AH

at

X

and

Y

, respectively. The circle with diameter

XY

intersects lines

BX

and

CY

a second time at

K

and

L

, respectively. Prove that points

H, K

and

L

are collinear.

P2

1

Double factorial divisibility

Let

0!!=1!!=1

and

n!!=n\cdot (n-2)!!

for all integers

n\geq 2

. Find all positive integers

n

such that

\dfrac{(2^n+1)!!-1}{2^{n+1}}

is an integer.

P1

1

A two-variable functional equation

Let

f:\mathbb{Z}^2\rightarrow\mathbb{Z}

be a function satisfying

f(x+1,y)+f(x,y+1)+1=f(x,y)+f(x+1,y+1)

for all integers

x

and

y

. Can it happen that

|f(x,y)|\leq 2024

for all

x,y\in\mathbb{Z}

?

P5

1

oops! all PALINDROME

Find the largest positive integer

k

so that any binary string of length

2024

contains a palindromic substring of length at least

k

.

P4

1

Weird function on subsets

Let

n

be a positive integer. Suppose for any

\mathcal{S} \subseteq \{1, 2, \cdots, n\}

,

f(\mathcal{S})

is the set containing all positive integers at most

n

that have an odd number of factors in

\mathcal{S}

. How many subsets of

\{1, 2, \cdots, n\}

can be turned into

\{1\}

after finitely many (possibly zero) applications of

f

?

P8

1

$\phi(\phi (n)) |n$

Find all positive integers

n

for wich

\phi(\phi (n))

divides

n

.

P6

1

$a_{n+1}=\frac{1}{2\lfloor a_n \rfloor -a_n+1}$

The sequence

\{a_n\}_{n\ge 1}

of real numbers is defined as follows: a_1=1, \text{and} a_{n+1}=\frac{1}{2\lfloor a_n \rfloor -a_n+1} \text{for all} n\ge 1 Find

a_{2024}

.

2024 Philippine Math Olympiad

Subcontests

Geometry with orthocenter II

Geometry with orthocenter I

Double factorial divisibility

A two-variable functional equation

oops! all PALINDROME

Weird function on subsets

$\phi(\phi (n)) |n$

$a_{n+1}=\frac{1}{2\lfloor a_n \rfloor -a_n+1}$