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an altitude of an acute triangle

Source: All-Russian 2007

5/4/2007

Let

ABC

be an acute triangle. The points

M

and

N

are midpoints of

AB

and

BC

respectively, and

BH

is an altitude of

ABC

. The circumcircles of

AHN

and

CHM

meet in

P

where

P\ne H

. Prove that

PH

passes through the midpoint of

MN

. V. Filimonov

geometrycircumcircleAsymptotepower of a pointradical axisperpendicular bisectorgeometry proposed

two circles and their common tangents

Source: All-Russian 2007

5/4/2007

Two circles

\omega_{1}

and

\omega_{2}

intersect in points

A

and

B

. Let

PQ

and

RS

be segments of common tangents to these circles (points

P

and

R

lie on

\omega_{1}

, points

Q

and

S

lie on

\omega_{2}

). It appears that

RB\parallel PQ

. Ray

RB

intersects

\omega_{2}

in a point

W\ne B

. Find

RB/BW

. S. Berlov

geometryparallelogramgeometry proposed

polynomial has exactly $n$ integral roots?

Source: All-Russian 2007

5/4/2007

Do there exist non-zero reals

a

,

b

,

c

such that, for any

n>3

, there exists a polynomial

P_{n}(x) = x^{n}+\dots+a x^{2}+bx+c

, which has exactly

n

(not necessary distinct) integral roots? N. Agakhanov, I. Bogdanov

algebrapolynomialcalculusintegrationquadraticsalgebra proposed

6

Problems(3)