MathDB

Nice polynomial

Source: Romanian TST 1 2006, Problem 2

4/19/2006

Let

p

a prime number,

p\geq 5

. Find the number of polynomials of the form x^p + px^k + p x^l + 1, k > l, k, l \in \left\{1,2,\dots,p-1\right\}, which are irreducible in

\mathbb{Z}[X]

.
Valentin Vornicu

algebrapolynomialalgebra proposed

Equilateral triangles with vertices in circles

Source: Romanian IMO TST 2006, day 2, problem 2

4/22/2006

Let

ABC

be a triangle with

\angle B = 30^{\circ }

. We consider the closed disks of radius

\frac{AC}3

, centered in

A

,

B

,

C

. Does there exist an equilateral triangle with one vertex in each of the 3 disks?
Radu Gologan, Dan Schwarz

geometryinequalitiesromania

Metric relation concerning points and a circle

Source: Romanian IMO TST 2006, day 3, problem 2

5/16/2006

Let

A

be point in the exterior of the circle

\mathcal C

. Two lines passing through

A

intersect the circle

\mathcal C

in points

B

and

C

(with

B

between

A

and

C

) respectively in

D

and

E

(with

D

between

A

and

E

). The parallel from

D

to

BC

intersects the second time the circle

\mathcal C

in

F

. Let

G

be the second point of intersection between the circle

\mathcal C

and the line

AF

and

M

the point in which the lines

AB

and

EG

intersect. Prove that

\frac 1{AM} = \frac 1{AB} + \frac 1{AC}.

geometrygeometric transformationreflectionprojective geometrycircumcirclepower of a pointcyclic quadrilateral

Sets of chain divisors

Source: Romanian IMO TST 2006, day 5, problem 2

5/23/2006

Let

m

and

n

be positive integers and

S

be a subset with

(2^m-1)n+1

elements of the set

\{1,2,3,\ldots, 2^mn\}

. Prove that

S

contains

m+1

distinct numbers

a_0,a_1,\ldots, a_m

such that

a_{k-1} \mid a_{k}

for all

k=1,2,\ldots, m

.

inductioncombinatorics proposedcombinatorics

2

Problems(4)