2017 APMO

Part of APMO

Subcontests

(5)

1

APMO 2017: Exquisite pairs

n

be a positive integer. A pair of

n

(a_1,\cdots{}, a_n)

(b_1,\cdots{}, b_n)

with integer entries is called an exquisite pair if

|a_1b_1+\cdots{}+a_nb_n|\le 1.

Determine the maximum number of distinct

n

-tuples with integer entries such that any two of them form an exquisite pair.
Pakawut Jiradilok and Warut Suksompong, Thailand

1

APMO 2017: Powerful rationals

Call a rational number

r

r

can be expressed in the form

\dfrac{p^k}{q}

for some relatively prime positive integers

p, q

and some integer

k >1

a, b, c

be positive rational numbers such that

abc = 1

. Suppose there exist positive integers

x, y, z

a^x + b^y + c^z

is an integer. Prove that

a, b, c

are all powerful.
Jeck Lim, Singapore

1

APMO 2017: Bijection between A(n) and B(n)

A(n)

denote the number of sequences

a_1\ge a_2\ge\cdots{}\ge a_k

of positive integers for which

a_1+\cdots{}+a_k = n

a_i +1

is a power of two

(i = 1,2,\cdots{},k)

B(n)

denote the number of sequences

b_1\ge b_2\ge \cdots{}\ge b_m

of positive integers for which

b_1+\cdots{}+b_m =n

and each inequality

b_j\ge 2b_{j+1}

(j=1,2,\cdots{}, m-1)

A(n) = B(n)

for every positive integer

n

.
Senior Problems Committee of the Australian Mathematical Olympiad Committee

1

APMO 2017: (ADZ) passes through M

ABC

be a triangle with

AB < AC

D

be the intersection point of the internal bisector of angle

BAC

and the circumcircle of

ABC

Z

be the intersection point of the perpendicular bisector of

AC

with the external bisector of angle

\angle{BAC}

. Prove that the midpoint of the segment

AB

lies on the circumcircle of triangle

ADZ

.
Olimpiada de Matemáticas, Nicaragua

1

APMO 2017: Arrangeable tuples

5

-tuple of integers arrangeable if its elements can be labeled

a, b, c, d, e

in some order so that

a-b+c-d+e=29

. Determine all

2017

-tuples of integers

n_1, n_2, . . . , n_{2017}

such that if we place them in a circle in clockwise order, then any

5

-tuple of numbers in consecutive positions on the circle is arrangeable.
Warut Suksompong, Thailand