MathDB

Circle + triangle = hexagon ==> two parallel lines.

Source: Tuymaada 2002, day 1, problem 3. - Author : S. Berlov.

5/3/2007

A circle having common centre with the circumcircle of triangle

ABC

meets the sides of the triangle at six points forming convex hexagon

A_{1}A_{2}B_{1}B_{2}C_{1}C_{2}

(

A_{1}

and

A_{2}

lie on

BC

,

B_{1}

and

B_{2}

lie on

AC

,

C_{1}

and

C_{2}

lie on

AB

). If

A_{1}B_{1}

is parallel to the bisector of angle

B

, prove that

A_{2}C_{2}

is parallel to the bisector of angle

C

.
Proposed by S. Berlov

geometrycircumcircleincentergeometry proposed

trinomial with integer coefficients, and powers of two as natural values

Source: Tuymaada Junior 2002 p3

5/11/2019

Is there a quadratic trinomial with integer coefficients, such that all values which are natural to be natural powers of two?

algebrapolynomialInteger Polynomialtrinomialquadratic trinomialpower of 2

Acute triangle, circumcircle, isoceles triangle, altitudes.

Source: Tuymaada 2002, day 2, problem 3. - Author : D. Shiryaev.

5/3/2007

The points

D

and

E

on the circumcircle of an acute triangle

ABC

are such that

AD=AE = BC

. Let

H

be the common point of the altitudes of triangle

ABC

. It is known that

AH^{2}=BH^{2}+CH^{2}

. Prove that

H

lies on the segment

DE

.
Proposed by D. Shiryaev

geometrycircumcircletrigonometrypower of a pointradical axisgeometry proposed

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