9.5

Part of I Soros Olympiad 1994-95 (Rus + Ukr)

Problems(3)

1/p +q/(q^r -1)=1 (I Soros Olympiad 1994-95 R1 9.5 10.3)

Source:

Find the triplets of natural numbers

(p,q,r)

that satisfy the equality

\frac{1}{p}+\frac{q}{q^r -1}=1.

number theoryDiophantine equationdiophantine

2 player game wit 8 numbered cards (I Soros Olympiad 1994-95 R2 9.5)

Source:

Kolya and Vasya each have

8

cards with numbers from

1

8

(each has all the numbers from

1

8

4

cards on the table, and Vasya put a card with a larger number on each of them. Now Vasya puts his remaining

4

cards on the table.
a) Can Kolya always put his own card with a larger number on each of Vasya’s cards?
b) Can Kolya always put on each of Vasya’s cards his own card with a number no less than on Vasya’s card?

number theorycombinatorics

1,995 soldiers lined up in a column

Source: I Soros Olympiad 1994-95 Ukraine R2 9.5 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

1,995

soldiers lined up in a column, and some of them stood correctly, and some turned backwards. Sergeant Smith remembers only the command "as". With this command, each soldier who sees an even number of faces facing him turns

180^o

, while the rest remain stationary. All movements on command are performed simultaneously. Prove that the sergeant can orient all the soldiers in one direction.