2016 Philippine MO

Part of Philippine MO

Subcontests

(5)

1

Collinearity in cyclic pentagon

ABCDE

is inscribed in a circle. Its diagonals

AC

BD

F

. The bisectors of

\angle BAC

\angle CDB

G

AG

BD

H

DG

AC

I

EG

AD

J

FHGI

JA \cdot FC \cdot GH = JD \cdot FB \cdot GI,

G

F

E

1

2, 4, 5-Nim

A

(first player) and

B

, take alternate turns in playing a game using 2016 chips as follows: the player whose turn it is, must remove

s

chips from the remaining pile of chips, where

s \in \{ 2,4,5 \}

. No one can skip a turn. The player who at some point is unable to make a move (cannot remove chips from the pile) loses the game. Who among the two players can force a win on this game?

1

Inequality on 1/(i^2 + i)^(3/4)

n

be any positive integer. Prove that

\sum^n_{i=1} \frac{1}{(i^2+i)^{3/4}} > 2-\frac{2}{\sqrt{n+1}}

1

Infinite primes in arithmetic sequence

Prove that the arithmetic sequence

5, 11, 17, 23, 29, \ldots

contains infinitely many primes.

1

Operations on a quadratic

The operations below can be applied on any expression of the form

ax^2+bx+c

.

(\text{I})

c \neq 0

a

4a-\frac{3}{c}

c

\frac{c}{4}

.

(\text{II})

a \neq 0

a

-\frac{a}{2}

c

-2c+\frac{3}{a}

.

(\text{III}_t)

x

x-t

t

is an integer. (Different values of

t

can be used.)
Is it possible to transform

x^2-x-6

into each of the following by applying some sequence of the above operations?

(\text{a})

5x^2+5x-1

(\text{b})

x^2+6x+2