2009 Philippine MO

Part of Philippine MO

Subcontests

(5)

1

2009 Philippines MO P5

AC

BD

intersect at point

P

PA = PD

PB = PC

E

be the foot of the perpendicular from

P

CD

. Prove that the line

PE

and the perpendicular bisectors of the segments

PA

PB

are concurrent.

1

2009 Philippines MO P4

k

be a positive real number such that

\frac{1}{k+a} + \frac{1}{k+b} + \frac{1}{k+c} \leq 1

for any positive positive real numbers

a

b

c

abc = 1

. Find the minimum value of

k

1

2009 Philippines MO P3

Each point of a circle is colored either red or blue.
(a) Prove that there always exists an isosceles triangle inscribed in this circle such that all its vertices are colored the same.
(b) Does there always exist an equilateral triangle inscribed in this circle such that all its vertices are colored the same?

1

2009 Philippines MO P2

(a) Find all pairs

(n,x)

of positive integers that satisfy the equation

2^n + 1 = x^2

.
(b) Find all pairs

(n,x)

of positive integers that satisfy the equation

2^n = x^2 + 1

1

2009 Philippines MO P1

{a_0, a_1, a_2, ...}

of real numbers satisfies the recursive relation

n(n+1)a_{n+1}+(n-2)a_{n-1} = n(n-1)a_n

for every positive integer

n

a_0 = a_1 = 1

. Calculate the sum

\frac{a_0}{a_1} + \frac{a_1}{a_2} + ... + \frac{a_{2008}}{a_{2009}}