MathDB

Mongolian TST 2008

Source: Day2 Problem3

5/17/2008

Find the maximum number

C

such that for any nonnegative

x,y,z

the inequality x^3 \plus{} y^3 \plus{} z^3 \plus{} C(xy^2 \plus{} yz^2 \plus{} zx^2) \ge (C \plus{} 1)(x^2 y \plus{} y^2 z \plus{} z^2 x) holds.

inequalitiesalgebrapolynomialthree variable inequality

Mongolian Test2008

Source: Day 1, problem 3

5/13/2008

Given a circumscribed trapezium

ABCD

with circumcircle

\omega

and 2 parallel sides

AD,BC

(

BC<AD

). Tangent line of circle

\omega

at the point

C

meets with the line

AD

at point

P

.

PE

is another tangent line of circle

\omega

and

E\in\omega

. The line

BP

meets circle

\omega

at point

K

. The line passing through the point

C

paralel to

AB

intersects with

AE

and

AK

at points

N

and

M

respectively. Prove that

M

is midpoint of segment

CN

.

geometrytrapezoidcircumcirclecyclic quadrilateralgeometry proposed

Find the locus of circumcenter.

Source: Mongolian TST 2008 day4 problem3

5/28/2008

Let

\Omega

is circle with radius

R

and center

O

. Let

\omega

is a circle inside of the

\Omega

with center

I

radius

r

.

X

is variable point of

\omega

and tangent line of

\omega

pass through

X

intersect the circle

\Omega

at points

A,B

. A line pass through

X

perpendicular with

AI

intersect

\omega

at

Y

distinct with

X

.Let point

C

is symmetric to the point

I

with respect to the line

XY

.Find the locus of circumcenter of triangle

ABC

when

X

varies on

\omega

geometrycircumcirclepower of a pointradical axisgeometry proposed

Equation has finitely number of solution.

Source: Mongolian TST 2008 day3, problem3

5/23/2008

Given positive integers

m,n > 1

. Prove that the equation (x \plus{} 1)^n \plus{} (x \plus{} 2)^n \plus{} ... \plus{} (x \plus{} m)^n \equal{} (y \plus{} 1)^{2n} \plus{} (y \plus{} 2)^{2n} \plus{} ... \plus{} (y \plus{} m)^{2n} has finitely number of solutions

x,y \in N

algebrapolynomialalgebra proposed

3

Problems(4)