2022 Philippine MO

Part of Philippine MO

Subcontests

(8)

1

GCD-related set partition

S = \{1, 2, \dots, 2022\}

is to be partitioned into

n

disjoint subsets

S_1, S_2, \dots, S_n

such that for each

i \in \{1, 2, \dots, n\}

, exactly one of the following statements is true:
(a) For all

x, y \in S_i

x \neq y, \gcd(x, y) > 1.

x, y \in S_i

x \neq y, \gcd(x, y) = 1.

Find the smallest value of

n

for which this is possible.

1

Fourth power inequality

a, b,

c

be positive real numbers such that

ab + bc + ca = 3

\dfrac{bc}{1 + a^4} + \dfrac{ca}{1 + b^4} + \dfrac{ab}{1 + c^4} \geq \dfrac{3}{2}.

1

Circumcircle tangency

\triangle ABC

D

be the point on side

BC

AB+BD=DC+CA.

AD

intersects the circumcircle of

\triangle ABC

X \neq A

. Prove that one of the common tangents of the circumcircles of

\triangle BDX

\triangle CDX

BC

1

Rattling reciprocal sum

Find all positive integers

n

for which there exists a set of exactly

n

distinct positive integers, none of which exceed

n^2

, whose reciprocals add up to

1

1

Center on line IG

\triangle ABC

I

G

P_A

is the foot of the perpendicular from

C

to the exterior angle bisector of

B

Q_A

is the foot of the perpendicular from

B

to the exterior angle bisector of

C

P_B

P_C

Q_B

Q_C

similarly. Show that

P_A, P_B, P_C, Q_A, Q_B,

Q_C

lie on a circle whose center is on line

IG

1

Visible points on a circle

Call a lattice point visible if the line segment connecting the point and the origin does not pass through another lattice point. Given a positive integer

k

S_k

the set of all visible lattice points

(x, y)

x^2 + y^2 = k^2

D

denote the set of all positive divisors of

2021 \cdot 2025

. Compute the sum

\sum_{d \in D} |S_d|

Here, a lattice point is a point

(x, y)

on the plane where both

x

y

are integers, and

|A|

denotes the number of elements of the set

A

1

The PMO Magician

The PMO Magician has a special party game. There are

n

chairs, labelled

1

n

n

sheets of paper, labelled

1

n

.
[*] On each chair, she attaches exactly one sheet whose number does not match the number on the chair. [*] She then asks

n

party guests to sit on the chairs so that each chair has exactly one occupant. [*] Whenever she claps her hands, each guest looks at the number on the sheet attached to their current chair, and moves to the chair labelled with that number.
Show that if

1 < m \leq n

m

is not a prime power, it is always possible for the PMO Magician to choose which sheet to attach to each chair so that everyone returns to their original seats after exactly

m

1

What's the difference?

Find all functions

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

f(a-b)f(c-d) + f(a-d)f(b-c) \leq (a-c)f(b-d)

for all real numbers

a, b, c,

d