MathDB

IMO ShortList 1999, number theory problem 6

Source: IMO ShortList 1999, number theory problem 6

11/13/2004

Prove that for every real number

M

there exists an infinite arithmetic progression such that: - each term is a positive integer and the common difference is not divisible by 10 - the sum of the digits of each term (in decimal representation) exceeds

M

.

algebramodular arithmeticarithmetic sequenceDivisibilitysum of digitsIMO Shortlist

IMO ShortList 1999, algebra problem 6

Source: IMO ShortList 1999, algebra problem 6

11/14/2004

For

n \geq 3

and

a_{1} \leq a_{2} \leq \ldots \leq a_{n}

given real numbers we have the following instructions: - place out the numbers in some order in a ring; - delete one of the numbers from the ring; - if just two numbers are remaining in the ring: let

S

be the sum of these two numbers. Otherwise, if there are more the two numbers in the ring, replace Afterwards start again with the step (2). Show that the largest sum

S

which can result in this way is given by the formula

S_{max}= \sum^n_{k=2} \begin{pmatrix} n -2 \\ [\frac{k}{2}] - 1\end{pmatrix}a_{k}.

matrixalgebrabinomial coefficientscountingcombinatoricsIMO Shortlist

IMO ShortList 1999, combinatorics problem 6

Source: IMO ShortList 1999, combinatorics problem 6

11/14/2004

Suppose that every integer has been given one of the colours red, blue, green or yellow. Let

x

and

y

be odd integers so that

|x| \neq |y|

. Show that there are two integers of the same colour whose difference has one of the following values:

x,y,x+y

or

x-y

.

functionlinear algebracombinatoricsIMO ShortlistRamsey Theory

6

Problems(3)