2015 APMO

Part of APMO

Subcontests

(5)

1

Sequence of integers with divisibility condition

Determine all sequences

a_0 , a_1 , a_2 , \ldots

of positive integers with

a_0 \ge 2015

such that for all integers

n\ge 1

a_{n+2}

is divisible by

a_n

|s_{n+1} - (n + 1)a_n | = 1

s_{n+1} = a_{n+1} - a_n + a_{n-1} - \cdots + (-1)^{n+1} a_0

.
Proposed by Pakawut Jiradilok and Warut Suksompong, Thailand

1

Blue and red lines

n

be a positive integer. Consider

2n

distinct lines on the plane, no two of which are parallel. Of the

2n

n

are colored blue, the other

n

are colored red. Let

\mathcal{B}

be the set of all points on the plane that lie on at least one blue line, and

\mathcal{R}

the set of all points on the plane that lie on at least one red line. Prove that there exists a circle that intersects

\mathcal{B}

2n - 1

points, and also intersects

\mathcal{R}

2n - 1

points.
Proposed by Pakawut Jiradilok and Warut Suksompong, Thailand

1

Existence of function on a set

S = \{2, 3, 4, \ldots\}

denote the set of integers that are greater than or equal to

2

. Does there exist a function

f : S \to S

f (a)f (b) = f (a^2 b^2 )\text{ for all }a, b \in S\text{ with }a \ne b?

Proposed by Angelo Di Pasquale, Australia

1

APMO 2015 P3

A sequence of real numbers

a_0, a_1, . . .

is said to be good if the following three conditions hold. (i) The value of

a_0

is a positive integer. (ii) For each non-negative integer

i

a_{i+1} = 2a_i + 1

a_{i+1} =\frac{a_i}{a_i + 2}

(iii) There exists a positive integer

k

a_k = 2014

.
Find the smallest positive integer

n

such that there exists a good sequence

a_0, a_1, . . .

of real numbers with the property that

a_n = 2014

.
Proposed by Wang Wei Hua, Hong Kong

1

APMO 2015 P1

ABC

be a triangle, and let

D

be a point on side

BC

. A line through

D

intersects side

AB

X

AC

Y

. The circumcircle of triangle

BXD

intersects the circumcircle

\omega

ABC

Z

distinct from point

B

ZD

ZY

\omega

V

W

respectively. Prove that

AB = V W

Proposed by Warut Suksompong, Thailand