2017 Indonesia MO

Part of Indonesia MO

Subcontests

(6)

1

Gold detector

A field is made of

2017 \times 2017

unit squares. Luffy has

k

gold detectors, which he places on some of the unit squares, then he leaves the area. Sanji then chooses a

1500 \times 1500

area, then buries a gold coin on each unit square in this area and none other. When Luffy returns, a gold detector beeps if and only if there is a gold coin buried underneath the unit square it's on. It turns out that by an appropriate placement, Luffy will always be able to determine the

1500 \times 1500

area containing the gold coins by observing the detectors, no matter how Sanji places the gold coins. Determine the minimum value of

k

in which this is possible.

1

20^n + 17k

Find the number of positive integers

n

not greater than 2017 such that

n

20^n + 17k

for some positive integer

k

1

2017 is close enough to 0

P

has integral coefficients, and it has at least 9 different integral roots. Let

n

be an integer such that

|P(n)| < 2017

P(n) = 0

1

To the 100th... no, 200th power

Determine all pairs of distinct real numbers

(x, y)

such that both of the following are true:
[*]

x^{100} - y^{100} = 2^{99} (x-y)

x^{200} - y^{200} = 2^{199} (x-y)

1

Special numbers

A positive integer

d

is special if every integer can be represented as

a^2 + b^2 - dc^2

for some integers

a, b, c

.
[*]Find the smallest positive integer that is not special. [*]Prove 2017 is special.

1

Trios of handshakes

Five people are gathered in a meeting. Some pairs of people shakes hands. An ordered triple of people

(A,B,C)

is a trio if one of the following is true:
[*]A shakes hands with B, and B shakes hands with C, or [*]A doesn't shake hands with B, and B doesn't shake hands with C.
If we consider

(A,B,C)

(C,B,A)

as the same trio, find the minimum possible number of trios.