MathDB

Vietnamese System Of Equation

Source: IMO Shortlist 1993, Vietnam 1

10/24/2005

Solve the following system of equations, in which

a

is a given number satisfying

|a| > 1

:

\begin{matrix} x_{1}^2 = ax_2 + 1 \\ x_{2}^2 = ax_3 + 1 \\ \ldots \\ x_{999}^2 = ax_{1000} + 1 \\ x_{1000}^2 = ax_1 + 1 \\ \end{matrix}

linear algebramatrixalgebrasystem of equationsIMO Shortlist

How do Spanish IMO Shortlist Geometry Problems look like?

Source: IMO Shortlist 1993, Spain 2; India TST 1994

3/15/2006

Given a triangle

ABC

, let

D

and

E

be points on the side

BC

such that

\angle BAD = \angle CAE

. If

M

and

N

are, respectively, the points of tangency of the incircles of the triangles

ABD

and

ACE

with the line

BC

, then show that

\frac{1}{MB}+\frac{1}{MD}= \frac{1}{NC}+\frac{1}{NE}.

geometryinradiusincentertrigonometryLaw of SinesIMO Shortlist

Easy-Moderate Discrete Mathematics Question

Source: IMO Shortlist 1993, Macedonia 3

3/25/2006

Let

n \geq 2, n \in \mathbb{N}

and

A_0 = (a_{01},a_{02}, \ldots, a_{0n})

be any

n-

tuple of natural numbers, such that

0 \leq a_{0i} \leq i-1,

for

i = 1, \ldots, n.

n-

tuples

A_1= (a_{11},a_{12}, \ldots, a_{1n}), A_2 = (a_{21},a_{22}, \ldots, a_{2n}), \ldots

are defined by:

a_{i+1,j} = Card \{a_{i,l}| 1 \leq l \leq j-1, a_{i,l} \geq a_{i,j}\},

for

i \in \mathbb{N}

and

j = 1, \ldots, n.

Prove that there exists

k \in \mathbb{N},

such that

A_{k+2} = A_{k}.

functioncombinatoricstupleSubsetsIMO Shortlist

For any finite we can find a set

Source: IMO Shortlist 1993, United Kingdom 3

10/24/2005

Show that for any finite set

S

of distinct positive integers, we can find a set

T \supseteq S

such that every member of

T

divides the sum of all the members of

T

.
Original Statement:
A finite set of (distinct) positive integers is called a DS-set if each of the integers divides the sum of them all. Prove that every finite set of positive integers is a subset of some DS-set.

inductionnumber theorySubsetsAdditive Number TheoryIMO ShortlistDivisibility

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Problems(4)