MathDB

Easy polynomial problem

Source: Tuymaada 2012, Problem 2, Day 1, Seniors

7/20/2012

Let

P(x)

be a real quadratic trinomial, so that for all

x\in \mathbb{R}

the inequality

P(x^3+x)\geq P(x^2+1)

holds. Find the sum of the roots of

P(x)

.
Proposed by A. Golovanov, M. Ivanov, K. Kokhas

algebrapolynomialquadraticsinequalitiessum of rootsalgebra unsolved

Both cyclic and circumscribed quadrilateral

Source: Tuymaada 2012, Problem 6, Day 2, Seniors

7/21/2012

Quadrilateral

ABCD

is both cyclic and circumscribed. Its incircle touches its sides

AB

and

CD

at points

X

and

Y

, respectively. The perpendiculars to

AB

and

CD

drawn at

A

and

D

, respectively, meet at point

U

; those drawn at

X

and

Y

meet at point

V

, and finally, those drawn at

B

and

C

meet at point

W

. Prove that points

U

,

V

and

W

are collinear.
Proposed by A. Golovanov

geometryratiotrapezoidtrigonometrytrig identitiesLaw of Sinesgeometry proposed

Angle bisector in a rectangle

Source: Tuymaada 2012, Problem 2, Day 1, Juniors

7/21/2012

A rectangle

ABCD

is given. Segment

DK

is equal to

BD

and lies on the half-line

DC

.

M

is the midpoint of

BK

. Prove that

AM

is the angle bisector of

\angle BAC

.
Proposed by S. Berlov

geometryrectanglesymmetryangle bisectorgeometry proposed

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