MathDB

IMO ShortList 1999, number theory problem 3

Source: IMO ShortList 1999, number theory problem 3

11/13/2004

Prove that there exists two strictly increasing sequences

(a_{n})

and

(b_{n})

such that

a_{n}(a_{n}+1)

divides

b^{2}_{n}+1

for every natural n.

modular arithmeticnumber theoryInteger sequenceDivisibilitySequenceIMO ShortlistHi

IMO ShortList 1999, algebra problem 3

Source: IMO ShortList 1999, algebra problem 3

11/14/2004

A game is played by

n

girls (

n \geq 2

), everybody having a ball. Each of the

\binom{n}{2}

pairs of players, is an arbitrary order, exchange the balls they have at the moment. The game is called nice nice if at the end nobody has her own ball and it is called tiresome if at the end everybody has her initial ball. Determine the values of

n

for which there exists a nice game and those for which there exists a tiresome game.

modular arithmeticsymmetryrotationcountingcombinatoricsgraph theoryIMO Shortlist

Recurrence giving periods of rest before catching flies

Source: IMO ShortList 1999, combinatorics problem 3

11/14/2004

A biologist watches a chameleon. The chameleon catches flies and rests after each catch. The biologist notices that: [*]the first fly is caught after a resting period of one minute; [*]the resting period before catching the

2m^\text{th}

fly is the same as the resting period before catching the

m^\text{th}

fly and one minute shorter than the resting period before catching the

(2m+1)^\text{th}

fly; [*]when the chameleon stops resting, he catches a fly instantly.
[*]How many flies were caught by the chameleon before his first resting period of

9

minutes in a row? [*]After how many minutes will the chameleon catch his

98^\text{th}

fly? [*]How many flies were caught by the chameleon after 1999 minutes have passed?

combinatoricsIMO ShortlistRecurrenceSequencescounting

3

Problems(3)