MathDB

2

some classical combinatorial geometry

100

points are marked in the plane so that no three of marked points are collinear. One of marked points is red, and the others are blue. A triangle with vertices at blue points is called good if the red point lies inside it. Determine if it is possible that the number of good triangles is not less than the half of the total number of traingles with vertices at blue points.

hard combinatorics

Given a table in a form of the regular

1000

-gon with sidelength

1

. A Beetle initially is in one of its vertices. All

1000

vertices are numbered in some order by numbers

1

,

2

,

\ldots

,

1000

so that initially the Beetle is in the vertex

1

. The Beetle can move only along the edges of

1000

-gon and only clockwise. He starts to move from vertex

1

and he is moving without stops until he reaches vertex

2

where he has a stop. Then he continues his journey clockwise from vertex

2

until he reaches the vertex

3

where he has a stop, and so on. The Beetle finishes his journey at vertex

1000

. Find the number of ways to enumerate all vertices so that the total length of the Beetle's journey is equal to

2017

.

7

2

vieta theorem for small child :)

10

distinct numbers are given. Professor Odd had calculated all possible products of

1

,

3

,

5

,

7

,

9

numbers among given numbers, and wrote down the sum of all these products. Similarly, Professor Even had calculated all possible products of

2

,

4

,

6

,

8

,

10

numbers among given numbers, and wrote down the sum of all these products. It appears that Odd's sum is greater than Even's sum by

1

. Prove that one of

10

given numbers is equal to

1

.

folklore 3d geomtery inequality

8

ants are placed on the edges of the unit cube. Prove that there exists a pair of ants at a distance not exceeding

1

.

6

2

nonstandart geometry

A triangle is cut by

3

cevians from its

3

vertices into

7

pieces:

4

triangles and

3

quadrilaterals. Determine if it is possible that all

3

quadrilaterals are inscribed.

inequality with (not only) geometric solution

Given real numbers

a

,

b

,

c

satisfy inequality

\left| \frac{a^2+b^2-c^2}{ab} \right|<2

. Prove that they also satisfy equalities

\left| \frac{b^2+c^2-a^2}{bc} \right|<2

and

\left| \frac{c^2+a^2-b^2}{ca} \right| <2

.

5

2

II Caucasus Mathematical Olympiad, 9.5

In a football tournament

20

teams participated, each pair of teams played exactly one game. For the win the team is awarded

3

points, for the draw --

1

point, for the lose no points awarded. The total number of points of all teams in the tournament is

554

. Prove that there exist

7

teams each having at least one draw.

copy of existing problem

In a football tournament

20

teams participated, each pair of teams played exactly one game. For the win the team is awarded

3

points, for the draw --

1

point, for the lose no points awarded. The total number of points of all teams in the tournament is

554

. Prove that there exist

7

teams each having at least one draw.

4

2

II Caucasus Mathematical Olympiad, 9.4

In an acute traingle

ABC

with

AB< BC

let

BH_b

be its altitude, and let

O

be the circumcenter. A line through

H_b

parallel to

CO

meets

BO

at

X

. Prove that

X

and the midpoints of

AB

and

AC

are collinear.

number theory construction

Determine if there exist

101

positive integers (not necessarily distinct) such that their product is equal to the sum of all their pairwise LCM.

3

2

pretty number theory

Find the least positive integer

n

satisfying the following statement: for eash pair of positive integers

a

and

b

such that

36

divides

a+b

and

n

divides

ab

it follows that

36

divides both

a

and

b

.

copy of existing problem

In an acute traingle

ABC

with

AB< BC

let

BH_b

be its altitude, and let

O

be the circumcenter. A line through

H_b

parallel to

CO

meets

BO

at

X

. Prove that

X

and the midpoints of

AB

and

AC

are collinear.

2

easy but interesting sets problem

On Mars a basketball team consists of 6 players. The coach of the team Mars can select any line-up of 6 players among 100 candidates. The coach considers some line-ups as appropriate while the other line-ups are not (there exists at least one appropriate line-up). A set of 5 candidates is called perspective if one more candidate could be added to it to obtain an appropriate line-up. A candidate is called universal if he completes each perspective set of 5 candidates (not containing him) upto an appropriate line-up. The coach has selected a line-up of 6 universal candidates. Determine if it follows that this line-up is appropriate.

copy of existing problem

On Mars a basketball team consists of 6 players. The coach of the team Mars can select any line-up of 6 players among 100 candidates. The coach considers some line-ups as appropriate while the other line-ups are not (there exists at least one appropriate line-up). A set of 5 candidates is called perspective if one more candidate could be added to it to obtain an appropriate line-up. A candidate is called universal if he completes each perspective set of 5 candidates (not containing him) upto an appropriate line-up. The coach has selected a line-up of 6 universal candidates. Determine if it follows that this line-up is appropriate.

1

2

easy problem

Basil needs to solve an exercise on summing two fractions

\dfrac{a}{b}

and

\dfrac{c}{d}

, where

a

,

b

,

c

,

d

are some non-zero real numbers. But instead of summing he performed multiplication (correctly). It appears that Basil's answer coincides with the correct answer to given exercise. Find the value of

\dfrac{b}{a} + \dfrac{d}{c}

.

another easy problem

Two points

A

and

B

lie on two branches of hyperbola given by equation

y=\frac1x

. Let

A_x

and

A_y

be projections of

A

onto coordinate axis, similarly, let

B_x

and

B_y

be projections of

B

onto coordinate axis. Prove that triangles

AB_xB_y

and

BA_xA_y

have equal areas.

2017 Caucasus Mathematical Olympiad

Subcontests

some classical combinatorial geometry

hard combinatorics

vieta theorem for small child :)

folklore 3d geomtery inequality

nonstandart geometry

inequality with (not only) geometric solution

II Caucasus Mathematical Olympiad, 9.5

copy of existing problem

II Caucasus Mathematical Olympiad, 9.4

number theory construction

pretty number theory

copy of existing problem

easy but interesting sets problem

copy of existing problem

easy problem

another easy problem