MathDB

Geo with incircles

Source: Russian TST 2014, Day 7 P2 (Group NG), P3 (Groups A & B)

4/21/2023

In an acute-angled triangle

ABC

, the point

H{}

is the orthocenter,

M{}

is the midpoint of the side

BC

and

\omega

is the circumcircle. The lines

AH, BH

and

CH{}

intersect

\omega

a second time at points

D, E

and

F{}

respectively. The ray

MH

intersects

\omega

at point

J{}

. The points

K{}

and

L{}

are the centers of the inscribed circles of the triangles

DEJ

and

DFJ

respectively. Prove that

KL\parallel BC

.

geometryincircle

Two polygons are homothetic

Source: Russian TST 2014, Day 8 P2 (Group NG)

1/8/2024

The polygon

M{}

is bicentric. The polygon

P{}

has vertices at the points of contact of the sides of

M{}

with the inscribed circle. The polygon

Q{}

is formed by the external bisectors of the angles of

M{}.

Prove that

P{}

and

Q{}

are homothetic.

geometrypolygonhomothety

Incenters coincide

Source: Russian TST 2014, Day 10 P2 (Group NG)

1/8/2024

A circle centered at

O{}

passes through the vertices

B{}

and

C{}

of the acute-angles triangle

ABC

and intersects the sides

AC{}

and

AB{}

at

D{}

and

E{}

respectively. The segments

CE

and

BD

intersect at

U{}.

The ray

OU

intersects the circumcircle of

ABC

at

P{}.

Prove that the incenters of the triangles

PEC

and

PBD

coincide.

geometryincenter

Cyclic quadrilateral geo

Source: Russian TST 2014, Day 10 P2 (Groups A & B)

1/8/2024

In the quadrilateral

ABCD

the angles

B{}

and

D{}

are straight. The lines

AB{}

and

DC{}

intersect at

E

and the lines

AD

and

BC

intersect at

F{}.

The line passing through

B{}

parallel to

C{}

D intersects the circumscribed circle

\omega

of

ABF{}

at

K{}

and the segment

KE{}

intersects

\omega

at

P{}.

Prove that the line

AP

divides the segment

CE

in half.

geometrycyclic quadrilateral

Equation with primes

Source: Russian TST 2014, Day 11 P2 (Groups A & B)

1/8/2024

Let

p,q

and

s{}

be prime numbers such that

2^sq =p^y-1

where

y > 1.

Find all possible values of

p.

number theoryprime numbers

P2

Problems(5)