P5

Part of 2021/2022 Tournament of Towns

Problems(3)

Which of the hexagons is bigger?

Source: 43rd International Tournament of Towns, Junior A-Level P5, Fall 2021

A parallelogram

ABCD

is split by the diagonal

BD

into two equal triangles. A regular hexagon is inscribed into the triangle

ABD

so that two of its consecutive sides lie on

AB

AD

and one of its vertices lies on

BD

. Another regular hexagon is inscribed into the triangle

CBD{}

so that two of its consecutive vertices lie on

CB

CD

and one of its sides lies on

BD

. Which of the hexagons is bigger?
Konstantin Knop

geometryhexagonTournament of Towns

Chess tournament at ToT

Source: 43rd International Tournament of Towns, Junior O-Level P5, Fall 2021

There were 20 participants in a chess tournament. Each of them played with each other twice: once as white and once as black. Let us say that participant

X{}

is no weaker than participant

Y{}

X{}

has won at least the same number of games playing white as

Y{}

and also has won at least the same number of games playing black as

Y{}

. Do there exist for sure two participants

A{}

B{}

A{}

is not weaker than

B{}

?
Boris Frenkin

combinatoricsgraph theoryTournament of Towns

Proving tangents

Source:

A quadrilateral ABCD is inscribed into a circle ω with center O. The circumcircle of the triangle AOC intersects the lines AB, BC, CD and DA the second time at the points M, N, K and L respectively. Prove that the lines MN, KL and the tangents to ω at the points A и C all touch the same circle.