MathDB

inequality in n variables

Source: Romanian I TST 2007

4/13/2007

If

a_{1}

,

a_{2}

,

\ldots

,

a_{n}\geq 0

are such that

a_{1}^{2}+\cdots+a_{n}^{2}=1,

then find the maximum value of the product

(1-a_{1})\cdots (1-a_{n})

.

inequalitiesinductionlogarithmscalculusderivativeinequalities proposed

Polynomial - odd or even coefficients

Source: Romanian TST 2 2007, Problem 1

4/15/2007

Let

f = X^{n}+a_{n-1}X^{n-1}+\ldots+a_{1}X+a_{0}

be an integer polynomial of degree

n \geq 3

such that

a_{k}+a_{n-k}

is even for all

k \in \overline{1,n-1}

and

a_{0}

is even. Suppose that

f = gh

, where

g,h

are integer polynomials and

\deg g \leq \deg h

and all the coefficients of

h

are odd. Prove that

f

has an integer root.

algebrapolynomialalgebra proposed

Prove that the function n^{2007}-n! is injective

Source: Romanian TST 4 2007, Problem 1

5/23/2007

Prove that the function

f : \mathbb{N}\longrightarrow \mathbb{Z}

defined by

f(n) = n^{2007}-n!

, is injective.

functioninequalitiesfloor functionfactorialnumber theory proposednumber theory

sum of OI^2 independent of triangulation

Source: romania TST 5 / 2007 problem 1

6/13/2007

In a circle with center

O

is inscribed a polygon, which is triangulated. Show that the sum of the squares of the distances from

O

to the incenters of the formed triangles is independent of the triangulation.

geometryincenterinductioncyclic quadrilateralgeometry proposed

equilateral triangles and parallelogram

Source: Romanian TST 5 2007, Problem 1

6/7/2007

Let

ABCD

be a parallelogram with no angle equal to

60^{\textrm{o}}

. Find all pairs of points

E, F

, in the plane of

ABCD

, such that triangles

AEB

and

BFC

are isosceles, of basis

AB

, respectively

BC

, and triangle

DEF

is equilateral.
Valentin Vornicu

geometryparallelogramgeometric transformationrotationreflectionanalytic geometryperpendicular bisector

1

Problems(5)