MathDB

2

touchpoint of incircle, orthocenter and midpoint of altitude collinear wanted

ABC

, the inscribed circle touches side

BC

at point

T, Q

is the midpoint of altitude

AK

,

P

is the orthocenter of the triangle formed by the bisectors of angles

B

and

C

and line

AK

. Prove that the points

P, Q

and

T

lie on the same line.
(D. Prokopenko)

equal faces of perpendicular edges in tetrahedron, concurrency related

In a tetrahedron, segments connecting the midpoints of heights with the orthocenters of the faces to which these heights are drawn intersect at one point. Prove that in such a tetrahedron all faces are equal or there are perpendicular edges.
(Yu. Blinkov)

5

2

locus of intersections of 2 lines, fixed circle and fixed line

Given a circle and a straight line

AB

passing through its center (points

A

and

B

are fixed,

A

is outside the circle, and

B

is inside). Find the locus of the intersection of lines

AX

and

BY

, where

XY

is an arbitrary diameter of the circle.
(A. Akopyan, A. Zaslavsky)

circumcircle of CGE tangent to AF wanted, internal tangent circles

Circle

\omega

is tangent to the interior of the circle

\Omega

at the point C. Chord

AB

of circle

\Omega

is tangent to

\omega

. Chords

CF

and

BG

of circle

\Omega

intersect at point

E

lying on

\omega

. Prove that the circumcircle of triangle

CGE

is tangent to straight line

AF

.
(I. Kukharchuk)

4

2

CK _|_ A'B' wanted, <BAC =60^o

In triangle

ABC

, angle

C

is equal to

60^o

. Bisectors

AA'

and

BB'

intersect at point

I

. Point

K

is symmetric to

I

with respect to line

AB

. Prove that lines

CK

and

A'B'

are perpendicular.
(D. Shvetsov, A. Zaslavsky)

orthocenter construction using a ruler at most with 6 lines

An acute-angled non-isosceles triangle

ABC

is drawn, a circumscribed circle and its center

O

are drawn. The midpoint of side

AB

is also marked. Using only a ruler (no divisions), construct the triangle's orthocenter by drawing no more than

6

lines.
(Yu. Blinkov)

3

2

AP = CQ wanted, axis of MN related, midpoints of AD,BC where AB=CD

In quadrilateral

ABCD

, sides

AB

and

CD

are equal (but not parallel), points

M

and

N

are the midpoints of

AD

and

BC

. The perpendicular bisector of

MN

intersects sides

AB

and

CD

at points

P

and

Q

, respectively. Prove that

AP = CQ

.
(M. Kungozhin)

intersection of diagonals of parallelogram lies on a diagonal of convex ABCD

Extensions of opposite sides of a convex quadrilateral

ABCD

intersect at points

P

and

Q

. Points are marked on the sides of

ABCD

(one per side), which are the vertices of a parallelogram with a side parallel to

PQ

. Prove that the intersection point of the diagonals of this parallelogram lies on one of the diagonals of quadrilateral

ABCD

.
(E. Bakaev)

2

triangle construction given intersectionf of <B,<C bisectors and axis of BC

Angle bisectors from vertices

B

and

C

and the perpendicular bisector of side

BC

are drawn in a non-isosceles triangle

ABC

. Next, three points of pairwise intersection of these three lines were marked (remembering which point is which), and the triangle itself was erased. Restore it according to the marked points using a compass and ruler.
(Yu. Blinkov)

concyclic wanted, tangents of circumcircle related

In an acute triangle

ABC

,

O

is the center of the circumscribed circle

\omega

,

P

is the point of intersection of the tangents to

\omega

through the points

B

and

C

, the median AM intersects the circle

\omega

at point

D

. Prove that points

A, D, P

and

O

lie on the same circle.
(D. Prokopenko)

1

2

circumcenter of BLD lies on circumcircle of trapezoid ABCD

Given an isosceles trapezoid

ABCD

. The bisector of angle

B

intersects the base

AD

at point

L

. Prove that the center of the circle circumscribed around triangle

BLD

lies on the circle circumscribed around the trapezoid.
(Yu. Blinkov)

one of XB_1, OA_1 and C_1Y is equal to sum of other two

In a circle with center

O

, chords

AB

and

AC

are drawn, both equal to the radius. Points

A_1

,

B_1

and

C_1

are projections of points

A, B

and

C

, respectively, onto an arbitrary diameter

XY

. Prove that one of the segments

XB_1

,

OA_1

and

C_1Y

is equal to the sum of the other two.
(A. Shklover)

2022 Oral Moscow Geometry Olympiad

Subcontests

touchpoint of incircle, orthocenter and midpoint of altitude collinear wanted

equal faces of perpendicular edges in tetrahedron, concurrency related

locus of intersections of 2 lines, fixed circle and fixed line

circumcircle of CGE tangent to AF wanted, internal tangent circles

CK _|_ A'B' wanted, <BAC =60^o

orthocenter construction using a ruler at most with 6 lines

AP = CQ wanted, axis of MN related, midpoints of AD,BC where AB=CD

intersection of diagonals of parallelogram lies on a diagonal of convex ABCD

triangle construction given intersectionf of <B,<C bisectors and axis of BC

concyclic wanted, tangents of circumcircle related

circumcenter of BLD lies on circumcircle of trapezoid ABCD

one of XB_1, OA_1 and C_1Y is equal to sum of other two

2022 Oral Moscow Geometry Olympiad

Subcontests

touchpoint of incircle, orthocenter and midpoint of altitude collinear wanted

equal faces of perpendicular edges in tetrahedron, concurrency related

locus of intersections of 2 lines, fixed circle and fixed line

circumcircle of CGE tangent to AF wanted, internal tangent circles

CK _|_ A'B' wanted, &lt;BAC =60^o

orthocenter construction using a ruler at most with 6 lines

AP = CQ wanted, axis of MN related, midpoints of AD,BC where AB=CD

intersection of diagonals of parallelogram lies on a diagonal of convex ABCD

triangle construction given intersectionf of &lt;B,&lt;C bisectors and axis of BC

concyclic wanted, tangents of circumcircle related

circumcenter of BLD lies on circumcircle of trapezoid ABCD

one of XB_1, OA_1 and C_1Y is equal to sum of other two

CK _|_ A'B' wanted, <BAC =60^o

triangle construction given intersectionf of <B,<C bisectors and axis of BC