MathDB

Functional equation

Source: IMO Shortlist 2005 problem A4

3/12/2006

Find all functions

f: \mathbb{R}\to\mathbb{R}

such that

f(x+y)+f(x)f(y)=f(xy)+2xy+1

for all real numbers

x

and

y

.
Proposed by B.J. Venkatachala, India

algebrafunctional equationIMO Shortlist

Labelling edges of a complete graph - maximal # of labels?

Source: IMO Shortlist 2005 Combinatorics problem C4; 5th German TST 2006, 23 April 2006, problem 1

12/29/2006

Let

n\geq 3

be a fixed integer. Each side and each diagonal of a regular

n

-gon is labelled with a number from the set

\left\{1;\;2;\;...;\;r\right\}

in a way such that the following two conditions are fulfilled:
1. Each number from the set

\left\{1;\;2;\;...;\;r\right\}

occurs at least once as a label.
2. In each triangle formed by three vertices of the

n

-gon, two of the sides are labelled with the same number, and this number is greater than the label of the third side.
(a) Find the maximal

r

for which such a labelling is possible.
(b) Harder version (IMO Shortlist 2005): For this maximal value of

r

, how many such labellings are there?
[hide="Easier version (5th German TST 2006) - contains answer to the harder version"] Easier version (5th German TST 2006): Show that, for this maximal value of

r

, there are exactly

\frac{n!\left(n-1\right)!}{2^{n-1}}

possible labellings.
Proposed by Federico Ardila, Colombia

functionIMO Shortlistgraph theorycombinatoricsExtremal combinatorics

Factorial: n!|a^n+1

Source: IMO Shortlist 2005 N4, Iran preparation exam

4/24/2006

Find all positive integers

n

such that there exists a unique integer

a

such that

0\leq a < n!

with the following property: n!\mid a^n \plus{} 1
Proposed by Carlos Caicedo, Colombia

factorialmodular arithmeticnumber theoryDivisibilityexponentialIMO ShortlistChinese Remainder Theorem

4

Problems(3)