MathDB

Lattice

Source: Iranian National Olympiad (3rd Round) 2006

8/26/2006

L

is a fullrank lattice in

\mathbb R^{2}

and

K

is a sub-lattice of

L

, that

\frac{A(K)}{A(L)}=m

. If

m

is the least number that for each

x\in L

,

mx

is in

K

. Prove that there exists a basis

\{x_{1},x_{2}\}

for

L

that

\{x_{1},mx_{2}\}

is a basis for

K

.

abstract algebragroup theorycalculusintegrationinvariantnumber theory proposednumber theory

Find x,y,z

Source: Iranian National Olympiad (3rd Round) 2006

9/19/2006

Find all real

x,y,z

that

\left\{\begin{array}{c}x+y+zx=\frac12\\ \\ y+z+xy=\frac12\\ \\ z+x+yz=\frac12\end{array}\right.

algebrapolynomialalgebra proposed

Inner product

Source: Iranian National Olympiad (3rd Round) 2006

9/21/2006

Suppose

(u,v)

is an inner product on

\mathbb R^{n}

and

f: \mathbb R^{n}\longrightarrow\mathbb R^{n}

is an isometry, that

f(0)=0

. 1) Prove that for each

u,v

we have

(u,v)=(f(u),f(v)

2) Prove that

f

is linear.

linear algebralinear algebra unsolved

Geometric Inequality

Source: Iranian National Olympiad (3rd Round) 2006

9/21/2006

In triangle

ABC

, if

L,M,N

are midpoints of

AB,AC,BC

. And

H

is orthogonal center of triangle

ABC

, then prove that

LH^{2}+MH^{2}+NH^{2}\leq\frac14(AB^{2}+AC^{2}+BC^{2})

inequalitiestrigonometrygeometryincentergeometric transformationreflectioncircumcircle

Countable Zorn Lemma

Source: Iranian National Olympiad (3rd Round) 2006

9/11/2006

Let

C

be a (probably infinite) family of subsets of

\mathbb{N}

such that for every chain

C_{1}\subset C_{2}\subset \ldots

of members of

C

, there is a member of

C

containing all of them. Show that there is a member of

C

such that no other member of

C

contains it!

combinatorics proposedcombinatorics

Cantor set in Z

Source: Iranian National Math Olympiad (Final exam) 2006

9/14/2006

For

A\subset\mathbb Z

and

a,b\in\mathbb Z

. We define

aA+b: =\{ax+b|x\in A\}

. If

a\neq0

then we calll

aA+b

and

A

to similar sets. In this question the Cantor set

C

is the number of non-negative integers that in their base-3 representation there is no

1

digit. You see

C=(3C)\dot\cup(3C+2)\ \ \ \ \ \ (1)

(i.e.

C

is partitioned to sets

3C

and

3C+2

). We give another example

C=(3C)\dot\cup(9C+6)\dot\cup(3C+2)

. A representation of

C

is a partition of

C

to some similiar sets. i.e.

C=\bigcup_{i=1}^{n}C_{i}\ \ \ \ \ \ (2)

and

C_{i}=a_{i}C+b_{i}

are similar to

C

. We call a representation of

C

a primitive representation iff union of some of

C_{i}

is not a set similar and not equal to

C

. Consider a primitive representation of Cantor set. Prove that a)

a_{i}>1

. b)

a_{i}

are powers of 3. c)

a_{i}>b_{i}

d) (1) is the only primitive representation of

C

.

geometrygeometric transformationnumber theory proposednumber theory

3

Problems(6)