MathDB

1

Tuymaada 2017 P1 Day1 Senior League

f

and

g

are defined on the set of all integers in the interval

[-100; 100]

and take integral values. Prove that for some integral

k

the number of solutions of the equation

f(x)-g(y)=k

is odd.\\ ( A. Golovanov)

8

2

4 cops and a robber

We consider the graph with vertices

A_1,A_2,\dots A_{2015}

,

B_1,B_2,\dots B_{2015}

and edges

A_iA _{i+1}, A_iB_i, B_iB_{i+17}

, taken cyclicaly. Is it true that 4 cops can catch a robber on this graph for every initial position?( First the 4 cops make a move, then the robber makes a move, then the cops make a move etc. A move consists of jumping from the vertex you stay on an adiacent vertex or by staying on your current vertex. Everyone knows the position of everyone everytime. The cops can coordinate their moves. The robber is caught when he shares the same vertex with a cop.)
Tuymaada 2017 Q8 Juniors

locus with coordinates from distances about 2 points

Two points

A

and

B

are given in the plane. A point

X

is called their preposterous midpoint if there is a Cartesian coordinate system in the plane such that the coordinates of

A

and

B

in this system are non-negative, the abscissa of

X

is the geometric mean of the abscissae of

A

and

B

, and the ordinate of

X

is the geometric mean of the ordinates of

A

and

B

. Find the locus of all the preposterous midpoints of

A

and

B

.
(K. Tyschu)

7

2

Line crossing triangles

An equilateral triangle with side

20

is divided by there series of parallel lines into

400

equilateral triangles with side

1

. What maximum number of these small triangles can be crossed (internally) by one line?
Tuymaada 2017 Q7 Juniors

Tuymaada 2017 Seniors Q7

A point

E

lies on the extension of the side

AD

of the rectangle

ABCD

over

D

. The ray

EC

meets the circumcircle

\omega

of

ABE

at the point

F\ne E

. The rays

DC

and

AF

meet at

P

.

H

is the foot of the perpendicular drawn from

C

to the line

\ell

going through

E

and parallel to

AF

. Prove that the line

PH

is tangent to

\omega

.
(A. Kuznetsov)

6

2

Sum of divisors

Let

\sigma(n)

denote the sum of positive divisors of a number

n

. A positive integer

N=2^rb

is given,where

r

and

b

are positive integers and

b

is odd. It is known that

\sigma(N)=2N-1

. Prove that

b

and

\sigma (b)

are coprime.
Tuymaada Q6 Juniors

sum of positive divisors coprime with the number

Let

\sigma(n)

denote the sum of positive divisors of a number

n

. A positive integer

N=2^r b

is given, where

r

and

b

are positive integers and

b

is odd. It is known that

\sigma(N)=2N-1

. Prove that

b

and

\sigma(b)

are coprime.
(J. Antalan, J. Dris)

5

2

Congruence

BL

is the bisector of an isosceles triangle

ABC

. A point

D

is chosen on the Base

BC

and a point

E

is chosen on the lateral side

AB

so that

AE=\frac {1}{2}AL=CD

. Prove that

LE=LD

.
Tuymaada 2017 Q5 Juniors

quadratic trinomial with integers 2 of it's coefficients

Does there exist a quadratic trinomial

f(x)

such that

f(1/2017)=1/2018

,

f(1/2018)=1/2017

, and two of its coefficients are integers?
(A. Khrabrov)

4

2

Geometry or number theory

A right triangle has all its sides rational numbers and the area

S

. Prove that there exists a right triangle, different from the original one, such that all its sides are rational numbers and its area is

S

.
Tuymaada 2017 Q4 Juniors

Tuymaada 2017 P4 Day 1 Senior League

There are 25 masks of different colours. k sages play the following game. They are shown all the masks. Then the sages agree on their strategy. After that the masks are put on them so that each sage sees the masks on the others but can not see who wears each mask and does not see his own mask. No communication is allowed. Then each of them simultaneously names one colour trying to guess the colour of his mask. Find the minimum k for which the sages can agree so that at least one of them surely guesses the colour of his mask. ( S. Berlov )

3

1

Clique, cluque, claque

In a country every 2 cities are connected either by a direct bus route or a direct plane flight. A

clique

is a set of cities such that every 2 cities in the set are connected by a direct flight. A

cluque

is a set of cities such that every 2 cities in the set are connected by a direct flight, and every 2 cities in the set are connected to the same number of cities by a bus route. A

claque

is a set of cities such that every 2 cities in the set are connected by a direct flight, and every 2 numbers of bus routes from a city in the set are different. Prove that the number of cities of any clique is at most the product of the biggest possible number of cities in a cluque and the the biggest possible number of cities in a claque.
Tuymaada 2017 Q3 Juniors

2

1

Geometrical inequality

ABCD

is a cyclic quadrilateral such that the diagonals

AC

and

BD

are perpendicular and their intersection is

P

. Point

Q

on the segment

CP

is such that

CQ=AP

. Prove that the perimeter of triangle

BDQ

is at least

2AC

.
Tuymaada 2017 Q2 Juniors

2017 Tuymaada Olympiad

Subcontests

Tuymaada 2017 P1 Day1 Senior League

4 cops and a robber

locus with coordinates from distances about 2 points

Line crossing triangles

Tuymaada 2017 Seniors Q7

Sum of divisors

sum of positive divisors coprime with the number

Congruence

quadratic trinomial with integers 2 of it's coefficients

Geometry or number theory

Tuymaada 2017 P4 Day 1 Senior League

Clique, cluque, claque

Geometrical inequality