2016 APMO

Part of APMO

Subcontests

(5)

1

APMO 2016: Functional equation

Find all functions

f: \mathbb{R}^+ \to \mathbb{R}^+

(z + 1)f(x + y) = f(xf(z) + y) + f(yf(z) + x),

for all positive real numbers

x, y, z

.
Fajar Yuliawan, Indonesia

1

APMO 2016: one-way flights between cities

The country Dreamland consists of

2016

cities. The airline Starways wants to establish some one-way flights between pairs of cities in such a way that each city has exactly one flight out of it. Find the smallest positive integer

k

such that no matter how Starways establishes its flights, the cities can always be partitioned into

k

groups so that from any city it is not possible to reach another city in the same group by using at most

28

flights.
Warut Suksompong, Thailand

1

APMO 2016: Sum of power of 2

A positive integer is called fancy if it can be expressed in the form

2^{a_1}+2^{a_2}+ \cdots+ 2^{a_{100}},

a_1,a_2, \cdots, a_{100}

are non-negative integers that are not necessarily distinct. Find the smallest positive integer

n

such that no multiple of

n

is a fancy number.
Senior Problems Committee of the Australian Mathematical Olympiad Committee

1

APMO 2016: Line is tangent to circle

AB

AC

be two distinct rays not lying on the same line, and let

\omega

be a circle with center

O

that is tangent to ray

AC

E

AB

F

R

be a point on segment

EF

. The line through

O

EF

intersects line

AB

P

N

be the intersection of lines

PR

AC

M

be the intersection of line

AB

and the line through

R

AC

. Prove that line

MN

\omega

.
Warut Suksompong, Thailand

1

APMO 2016: Great triangle

We say that a triangle

ABC

is great if the following holds: for any point

D

BC

P

Q

are the feet of the perpendiculars from

D

AB

AC

, respectively, then the reflection of

D

PQ

lies on the circumcircle of the triangle

ABC

. Prove that triangle

ABC

is great if and only if

\angle A = 90^{\circ}

AB = AC

.
Senior Problems Committee of the Australian Mathematical Olympiad Committee