1

Part of 2002 Tuymaada Olympiad

Problems(2)

One hexagon and 18 segments.

Source: Tuymaada 2002, day 1, problem 1. - Author : A. Golovanov.

Each of the points

G

H

lying from different sides of the plane of hexagon

ABCDEF

is connected with all vertices of the hexagon. Is it possible to mark 18 segments thus formed by the numbers

1, 2, 3, \ldots, 18

and arrange some real numbers at points

A, B, C, D, E, F, G, H

so that each segment is marked with the difference of the numbers at its ends?
Proposed by A. Golovanov

modular arithmeticcombinatorics proposedcombinatorics

sequence of primes build recursively

Source: Tuymaada 2002

A positive integer

c

is given. The sequence

\{p_{k}\}

is constructed by the following rule:

p_{1}

is arbitrary prime and for

k\geq 1

p_{k+1}

is any prime divisor of

p_{k}+c

not present among the numbers

p_{1}

p_{2}

\dots

p_{k}

. Prove that the sequence

\{p_{k}\}

cannot be infinite.
Proposed by A. Golovanov

limitnumber theory proposednumber theory