2013 Philippine MO

Part of Philippine MO

Subcontests

(5)

1

PMO Finals #5

r

s

be positive real numbers such that

(r+s-rs)(r+s+rs)=rs

. Find the minimum value of

r+s-rs

r+s+rs

1

PMO Finals #4

a

p

q

be positive integers with

p \le q

. Prove that if one of the numbers

a^p

a^q

is divisible by

p

, then the other number must also be divisible by

p

1

PMO Finals #3

3. Let n be a positive integer. The numbers 1, 2, 3,....., 2n are randomly assigned to 2n distinct points on a circle. To each chord joining two of these points, a value is assigned equal to the absolute value of the difference between the assigned numbers at its endpoints. Show that one can choose n pairwise non-intersecting chords such that the sum of the values assigned to them is

n^2

1

PMO Finals #2

2. Let P be a point in the interior of triangle ABC . Extend AP, BP, and CP to meet BC, AC, and AB at D, E, and F, respectively. If triangle APF, triangle BPD and triangle CPE have equal areas, prove that P is the centroid of triangle ABC .

1

PMO Finals #1

1. Determine, with proof, the least positive integer

n

for which there exist

n

distinct positive integers,

\left(1-\frac{1}{x_1}\right)\left(1-\frac{1}{x_2}\right)......\left(1-\frac{1}{x_n}\right)=\frac{15}{2013}