4

Part of 2023/2024 Tournament of Towns

Problems(4)

area of A convex quadrilateral

Source: 45th International Tournament of Towns, Senior A-Level P4, Fall 2023

A convex quadrilateral

A B C D

S

is given. Inside each side of the quadrilateral a point is selected. These points are consecutively linked by segments, so that

A B C D

is split into a smaller quadrilateral and 4 triangles. Prove that the area of at least one triangle does not exceed

S / 8

. Mikhail Malkin

incircle and excircle tangent

Source: 45th International Tournament of Towns, Junior A-Level P4, Fall 2023

A B C

A

60^{\circ}

is given. Its incircle is tangent to side

A B

D

, while its excircle tangent to side

A C

, is tangent to the extension of side

A B

E

. Prove that the perpendicular to side

A C

, passing through point

D

, meets the incircle again at a point equidistant from points

E

C

. (The excircle is the circle tangent to one side of the triangle and to the extensions of two other sides.) Azamat Mardanov

Variants of Stainer

Source: 45th International Tournament of Towns, Senior O-Level P4, Fall 2023

4. Given is an acute-angled triangle

A B C, H

is its orthocenter. Let

P

be an arbitrary point inside (and not on the sides) of the triangle

A B C

that belongs to the circumcircle of the triangle

A B H

A^{\prime}, B^{\prime}

C^{\prime}

be projections of point

P

B C, C A, A B

. Prove that the circumcircle of the triangle

A^{\prime} B^{\prime} C^{\prime}

passes through the midpoint of segment

C P

. Alexey Zaslavsky

Can the sum of all these integers equal 2023

Source: 45th International Tournament of Towns, Junior O-Level P4, Fall 2023

4. There are several (at least two) positive integers written along the circle. For any two neighboring integers one is either twice as big as the other or five times as big as the other. Can the sum of all these integers equal 2023 ? Sergey Dvoryaninov