MathDB

Perpendicular

Source: Iranian National Olympiad (3rd Round) 2006

9/21/2006

ABC

is a triangle and

R,Q,P

are midpoints of

AB,AC,BC

. Line

AP

intersects

RQ

in

E

and circumcircle of

ABC

in

F

.

T,S

are on

RP,PQ

such that

ES\perp PQ,ET\perp RP

.

F'

is on circumcircle of

ABC

that

FF'

is diameter. The point of intersection of

AF'

and

BC

is

E'

.

S',T'

are on

AB,AC

that

E'S'\perp AB,E'T'\perp AC

. Prove that

TS

and

T'S'

are perpendicular.

geometrycircumcircleparallelogramrectanglegeometric transformationgeometry proposed

Z_2

Source: Iranian National Olympiad (3rd Round) 2006

8/26/2006

n

is a natural number that

\frac{x^{n}+1}{x+1}

is irreducible over

\mathbb Z_{2}[x]

. Consider a vector in

\mathbb Z_{2}^{n}

that it has odd number of

1

's (as entries) and at least one of its entries are

0

. Prove that these vector and its translations are a basis for

\mathbb Z_{2}^{n}

vectorgeometrygeometric transformationrotationalgebrapolynomialgroup theory

Find all polynomials

Source: Iranian National Olympiad (3rd Round) 2006

9/19/2006

Find all real polynomials that

p(x+p(x))=p(x)+p(p(x))

algebrapolynomialalgebra proposed

Linear map

Source: Iranian National Olympiad (3rd Round) 2006

9/21/2006

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

is a non-zero linear map. Prove that there is a base

\{v_{1},\dots,v_{n}m\}

for

\mathbb R^{n}

that the set

\{f(v_{1}),\dots,f(v_{n})\}

is linearly independent, after ommitting Repetitive elements.

linear algebralinear algebra unsolved

Infinite pipe

Source: Iranian National Math Olympiad (Final exam) 2006

9/14/2006

A liquid is moving in an infinite pipe. For each molecule if it is at point with coordinate

x

then after

t

seconds it will be at a point of

p(t,x)

. Prove that if

p(t,x)

is a polynomial of

t,x

then speed of all molecules are equal and constant.

analytic geometryalgebrapolynomialalgebra proposed

Number-Theory related

Source: Iranian National Olympiad (3rd Round) 2006

9/11/2006

Let

B

be a subset of

\mathbb{Z}_{3}^{n}

with the property that for every two distinct members

(a_{1},\ldots,a_{n})

and

(b_{1},\ldots,b_{n})

of

B

there exist

1\leq i\leq n

such that

a_{i}\equiv{b_{i}+1}\pmod{3}

. Prove that

|B| \leq 2^{n}

.

modular arithmeticvectorfunctionlinear algebracombinatorics proposedcombinatorics

2

Problems(6)