2017 Philippine MO

Part of Philippine MO

Subcontests

(4)

1

Varying point along an arc

\mathcal{C}_1

\mathcal{C}_2

with centers at

C_1

C_2

respectively, intersect at two points

A

B

P

Q

are varying points on

\mathcal{C}_1

\mathcal{C}_2

, respectively, such that

P

Q

B

are collinear and

B

is always between

P

Q

PC_1

QC_2

R

I

be the incenter of

\Delta PQR

S

be the circumcenter of

\Delta PIQ

P

Q

S

traces the arc of a circle whose center is concyclic with

A

C_1

C_2

1

Coloring problem with the color pink

Each of the numbers in the set

A = \{1,2, \cdots, 2017\}

is colored either red or white. Prove that for

n \geq 18

, there exists a coloring of the numbers in

A

such that any of its n-term arithmetic sequences contains both colors.

1

Some inequality with min function

Find all positive real numbers

(a,b,c) \leq 1

\huge \min \Bigg\{ \sqrt{\frac{ab+1}{abc}}\, \sqrt{\frac{bc+1}{abc}}, \sqrt{\frac{ac+1}{abc}} \Bigg \} = \sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}

1

Inequality involving integer functions

n \in \mathbb{N}

\sigma (n)

denote the sum of the divisors of

n

\phi (n)

denote the number of integers

n \geq m

\gcd(m,n) = 1

. Show that for all

n \in \mathbb{N}

,

\large \frac{1}{\sigma (n)} + \frac{1}{\phi (n)} \geq \frac{2}{n}

and determine when equality holds.