2021/2022 Tournament of Towns

Part of Tournament Of Towns

Subcontests

(7)

2

Positive integer inequality

Prove that for any positive integers

a_1, a_2, \ldots , a_n

the following inequality holds true:

\left\lfloor\frac{a_1^2}{a_2}\right\rfloor+\left\lfloor\frac{a_2^2}{a_3}\right\rfloor+\cdots+\left\lfloor\frac{a_n^2}{a_1}\right\rfloor\geqslant a_1+a_2+\cdots+a_n.

43rd International Mathematics Tournament Of Towns

There were made 7 golden, 7 silver and 7 bronze for a tournament. All the medals of the same material should weigh the same and the medals of different materials should have different weight. However, it so happened that exactly one medal had a wrong weight. If this medal is golden, it is lighter than a standard golden medal; if it is bronze, it is heavier than a standard bronze one; if it is silver, it may be lighter or heavier than a standard silver one. Is it possible to find the nonstandard one for sure, using three weighings on a balance scale with no weights?

3

Which of the hexagons is bigger?

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

Chess tournament at ToT

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

Proving tangents

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.

6

3

A 2x2 grid covered in triangles

A checkered square of size

2\times2

is covered by two triangles. Is it necessarily true that:
[*]at least one of its four cells is fully covered by one of the triangles; [*]some square of size

1\times1

can be placed into one of these triangles?
Alexandr Shapovalov

2022 TTs SA7

A starship is located in a halfspace at the distance

a

from its boundary. The crew knows this but does not know which direction to move to reach the boundary plane. The starship may travel through the space by any path, may measure the way it has already travelled and has a sensor that signals when the boundary is reached. Is it possible to reach the boundary for sure, having passed no more than:

a)14a

b)13a

2022TTsJA7

p

be a prime number and let

M

be a convex polygon. Suppose that there are precisely

p

m

with equilateral triangles with side

1

and squares with side

1

. Show there is some side of

M

p-1

5

8

7