2017 Ukraine Team Selection Test

Part of Ukraine Team Selection Test

Subcontests

(3)

1

P9 Sequence

There're two positive inegers

a_1<a_2

. For every positive integer

n \geq 3

a_n

be the smallest integer that bigger than

a_{n-1}

and such that there's unique pair

1\leq i< j\leq n-1

such that this number equals to

a_i+a_j

. Given that there're finitely many even numbers in this sequence. Prove that sequence

\{a_{n+1}-a_n \}

is periodic starting from some element.

1

P4 Polynomial

Whether exist set

A

that contain 2016 real numbers (some of them may be equal) not all of which equal 0 such that next statement holds. For arbitrary 1008-element subset of

A

there is a monic polynomial of degree 1008 such that elements of this subset are roots of the polynomial and other 1008 elements of

A

are coefficients of this polynomial's degrees from 0 to 1007.

1

P3 Stripes

Andriyko has rectangle desk and a lot of stripes that lie parallel to sides of the desk. For every pair of stripes we can say that first of them is under second one. In desired configuration for every four stripes such that two of them are parallel to one side of the desk and two others are parallel to other side, one of them is under two other stripes that lie perpendicular to it. Prove that Andriyko can put stripes one by one such way that every next stripe lie upper than previous and get desired configuration.
Proposed by Denys Smirnov