2012 Saint Petersburg Mathematical Olympiad

Part of Saint Petersburg Mathematical Olympiad

Subcontests

(7)

2

Cities and airlines

Some cities of Russia are connected with some cities of Ukraine with international airlines. The Interstate Council for the Promotion of Migration intends to introduce a one-way traffic on each airline so that, by taking off from the city, it could no longer be returned in this city (using other one-way airlines). Prove that the number of ways to do this is not divided by

3

Breaking sticks

2012

sticks with integer length, and sum of length is

n

. We need to have sticks with lengths

1,2,....,2012

. For it we can break some sticks ( for example from stick with length

6

1

4

). For what minimal

n

it is always possible?

3

Light beam and mirrors

On the coordinate plane in the first quarter there are

100

non-intersecting single unit segments parallel to the coordinate axes. These segments aremirrors (on both sides), they reflect the light according to the rule. "The angle of incidence is equal to the angle of reflection." (If you hit the edge of the mirror, the beam of light does not change its direction.) From the point lying in the unit circle with the center at the origin, a ray of light in the direction of the bisector of the first coordinate angle. Prove that, that this initial point can be chosen so that the ray is reflected from the mirrors not more than

150

Interesting geometry

ABCD

is parallelogram. Line

l

is perpendicular to

BC

B

. Two circles passes through

D,C

l

is tangent in points

P

Q

M

AB

\angle DMP=\angle DMQ

Some geometry

ABC

is triangle. Point

L

ABC

and lies on bisector of

\angle B

K

BL

\angle KAB=\angle LCB= \alpha

P

inside triangle is such, that

AP=PC

\angle APC=2\angle AKL

\angle KPL=2\alpha

3

Number of Divisors

n \geq k

-two natural numbers.

S

-such natural, that have not less than

n

divisors. All divisors of

S

are written in descending order. What minimal number of divisors can have number from

k

Divisors of divisor

S

is natural, and

S=d_{1}>d_2>...>d_{1000000}=1

are all divisors of

S

. What minimal number of divisors can have

d_{250}

Numbers in square

100 \times 100

table in every cell there is natural number. All numbers in same row or column are different. Can be that for every square sum of numbers, that are in angle cells, is square number ?

2

Sequences

x_1,...,x_n

x_1^2+...+x_n^2=1

Prove, that exists such

y_1,...,y_n

z_1,...,z_n

|y_1|+...+|y_n| \leq 1

max(|z_1|,...,|z_n|) \leq 1

2x_i=y_i+z_i

i

Numbers on circle

Some notzero reals numbers are placed around circle. For every two neighbour numbers

a,b

a+b

\frac{1}{a}+\frac{1}{b}

are integer. Prove that there are not more than

4

different numbers.

3

Pyramid base

At the base of the pyramid

SABCD

lies a convex quadrilateral

ABCD

BC * AD = BD * AC

\angle ADS =\angle BDS ,\angle ACS =\angle BCS

. Prove that the plane

SAB

is perpendicular to the plane of the base.

Questions of exam.

25

students are on exams. Exam consists of some questions with

5

variants of answer. Every two students gave same answer for not more than

1

question. Prove, that there are not more than

6

questions in exam.

Some geometry

ABCD

is inscribed. Bisector of angle between diagonals intersect

AB

CD

X

Y

M,N

are midpoints of

AD,BC

XM=YM

XN=YN

3

Books about dogs

We have big multivolume encyclopaedia about dogs on the shelf in alphabetical order, each volume in its specially selected place. Near each place there is an instruction that prescribes one of four actions: to rearrange this volume is one or two places left or right. If you simultaneously run all instructions, volumes will be placed in the same places in another order. The cynologist Dima performs all the instructions every morning. Once he discovered, that the volume of "Bichons" stands still, which was initially occupied by the volume of "Terriers". Prove , that after some time the volume of "Mudies" will stand on the original place of the volume "Poodles".

Some Geometry

C,D

BE

ABE

BC=CD=DE

X,Y,Z,T

are circumcenters of

ABE,ABC,ADE,ACD

T

XYZ

Find minimal

a,b,c

>100

(a,b,c)=1

c|a+b,a|b+c

b

3

Common roots

a,b,c

are reals, such that every pair of equations of

x^3-ax^2+b=0,x^3-bx^2+c=0,x^3-cx^2+a=0

has common root. Prove

a=b=c

system of cubic

\begin{cases} x^3-ax^2+b^3=0 \\x^3-bx^2+c^3=0 \\ x^3-cx^2+a^3=0 \end{cases}

Prove that system hasn`t solutions if

a,b,c

Number of roots

Find all integer

b

[x^2]-2012x+b=0

has odd number of roots.