2015 Tuymaada Olympiad

Part of Tuymaada Olympiad

Subcontests

(8)

2

Some another geometry

\triangle ABC

M,O

are midpoint of

AB

and circumcenter. It is true, that

OM=R-r

. Bisector of external

\angle A

BC

D

and bisector of external

\angle C

AB

E

. Find possible values of

\angle CED

Incircle and bisector

CL

\angle C

ABC

and intersect circumcircle at

K

I

ABC

IL=LK

CI=IK

2

Form of integer

0 \leq b \leq c \leq d \leq a

a>14

are integers. Prove, that there is such natural

n

that can not be represented as

n=x(ax+b)+y(ay+c)+z(az+d)

x,y,z

are some integers.
K. Kohas

Interesting sequence

Is there sequence

(a_n)

of natural numbers, such that differences

\{a_{n+1}-a_n\}

take every natural value and only one time and differences

\{a_{n+2}-a_n\}

take every natural value greater

2015

and only one time ?
A. Golovanov

1

Number and divisor

There is some natural number

n>1

on the board. Operation is adding to number on the board it maximal non-trivial divisor. Prove, that after some some operations we get number, that is divisible by

3^{2000}

2

Plane with parallelogramms

\frac{k(k+1)}{2}+1

points on the planes, some are connected by disjoint segments ( also point can not lies on segment, that connects two other points). It is true, that plane is divided to some parallelograms and one infinite region. What maximum number of segments can be drawn ?
A.Kupavski, A. Polyanski

Interesting problem

Four sages stand around a non-transparent baobab. Each of the sages wears red, blue, or green hat. A sage sees only his two neighbors. Each of them at the same time must make a guess about the color of his hat. If at least one sage guesses correctly, the sages win. They could consult before the game started. How should they act to win?

2

Factorial and square

n!=ab^2

a

is free from squares. Prove, that for every

\epsilon>0

for every big enough

n

it is true, that

2^{(1-\epsilon)n}<a<2^{(1+\epsilon)n}

Decimal representation

Prove that there exists a positive integer

n

such that in the decimal representation of each of the numbers

\sqrt{n}

\sqrt[3]{n},..., \sqrt[10]{n}

2015

stand immediately after the decimal point.
A.Golovanov

1

Polynomial

P(x,y)

is polynomial with real coefficients and

P(x+2y,x+y)=P(x,y)

. Prove that exists polynomial

Q(t)

P(x,y)=Q((x^2-2y^2)^2)

2

Perpendicular angle bisectors

D

AC

\triangle ABC

\angle ACB,\angle ABD

are perpendicular. Find max value for

\angle BAC

Funny numbers

We call number as funny if it divisible by sum its digits

+1

1+2+1|12

12

is funny) What is maximum number of consecutive funny numbers ?
O. Podlipski

2

Football team

On the football training there was

n

footballers - forwards and goalkeepers. They made

k

goals. Prove that main trainer can give for every footballer squad number from

1

n

such, that for every goal the difference between squad number of forward and squad number of goalkeeper is more than

n-k

Table with numbers

100

different real numbers. Prove, that we can put it in

10 \times 10

table, such that difference between two numbers in cells with common side are not equals

1