2

Part of 2018 Brazil National Olympiad

Problems(2)

Brazilian Mathematical Olympiad N2 (7th-8th grade) Day 1 P2

Source:

We say that a quadruple

(A,B,C,D)

is dobarulho when

A,B,C

are non-zero algorisms and

D

is a positive integer such that:

1.

A \leq 8

2.

D>1

3.

D

divides the six numbers

\overline{ABC}

\overline{BCA}

\overline{CAB}

\overline{(A+1)CB}

\overline{CB(A+1)}

\overline{B(A+1)C}

. Find all such quadruples.

Operations on a blackboard

Source: Brazilian Mathematical Olympiad 2018 - Q2

Azambuja writes a rational number

q

on a blackboard. One operation is to delete

q

and replace it by

q+1

q-1

\frac{q-1}{2q-1}

q \neq \frac{1}{2}

. The final goal of Azambuja is to write the number

\frac{1}{2018}

after performing a finite number of operations. a) Show that if the initial number written is

0

, then Azambuja cannot reach his goal. b) Find all initial numbers for which Azambuja can achieve his goal.

number theoryBrazilian Math OlympiadBrazilian Math Olympiad 2018games