MathDB

1

sum-products of integers

2011 non-zero integers are given. It is known that the sum of any one of them with the product of the remaining 2010 numbers is negative. Prove that if all numbers are split arbitrarily into two groups, the sum of the two products will also be negative. (Authors: N. Agahanov & I. Bogdanov)

11.4

1

Relocating cement across storage buildings

2011 storage buildings are connected by roads so that it is possible to reach any building from any other building, possibly using multiple roads. The buildings contain

x_1,\dots,x_{2011}

kilogram of cement. In one move, it is possible to relocate any quantity of cement from one building to any other building that is connected to it. The target is to have

y_1,\dots,y_{2011}

redistributed across storage buildings and

x_1+x_2+\dots+x_{2011}=y_1+y_2+\dots+y_{2011}.

What is the minimal number of moves that the redistribution can take regardless of values of

x_i

and

y_i

and of the road plan? (Author: P. Karasev)

10.6

1

Smallest number of zeros

2011 numbers are written on a board. For any three numbers, their sum is also among numbers written on the board. What is the smallest number of zeros among all 2011 numbers? (Author: I. Bogdanov)

10.1

1

Runners

Two runners started a race simultaneously. Initially they ran on the street toward the stadium and then 3 laps on the stadium. Both runners covered the whole distance at their own constant speed. During the whole race the first runner passed the second runner exactly twice. Prove that the speed of the first runner is at least double the speed of the second runner. (Author: I. Rubanov)

10.3

1

combinatorics of the last two digits

a_1,a_2,\dots,a_{14}

are different positive integers. All 196 numbers of the form

a_k+a_l

with

1\leq k,l\leq 14

are written on a board. Is it possible that for any two-digit combination, there exists a number among all 196 that ends with that combination (i.e., there exist numbers ending with

00, 01, \dots, 99

)? (Author: P. Kozhevnikov)

9.8

1

cutting sticks

Straight rod of 2 meter length is cut into

N

sticks. The length of each piece is an integer number of centimeters. For which smallest

N

can one guarantee that it is possible to form the contour of some rectangle, while using all sticks and not breaking them further? (Author: A. Magazinov)

11.7

1

All common tangents of all circle pairs

Basil drew several circles on the plane and drew all common tangent lines for all pairs of circles. It turned out that the lines contain all sides of a regular polygon with 2011 vertices. What is the smallest possible number of circles? (Author: N. Agahanov)

11.8

1

Problem 8 of Russian Regional Olympiad 2011, grade 11

b

and

c

are positive. Prove the inequality

\left(b-c\right)^{2011}\left(b+c\right)^{2011}\left(c-b\right)^{2011} \geq \left(b^{2011}-c^{2011}\right)\left(b^{2011}+c^{2011}\right)\left(c^{2011}-b^{2011}\right).

(Author: V. Senderov)

10.4

1

Problem 4 of Russian Regional Olympiad 2011, grade 10

Non-zero real numbers

a

,

b

and

c

are such that any two of the three equations

ax^{11}+bx^4+c=0

,

bx^{11}+cx^4+a=0

,

cx^{11}+ax^4+b=0

have a common root. Prove that all three equations have a common root. (Author: I. Bogdanov)

11.1

1

Problem 1 of Russian Regional Olympiad 2011, grade 11

Is there a real number

\alpha

such that

\cos\alpha

is irrational but

\cos 2\alpha

,

\cos 3\alpha

,

\cos 4\alpha

,

\cos 5\alpha

are all rational? (Author: V. Senderov)

11.6

1

Equality for acute triangle, and tangents of circumcircle

\omega

is the circumcirle of an acute triangle

ABC

. The tangent line passing through

A

intersects the tangent lines passing through points

B

and

C

at points

K

and

L

, respectively. The line parallel to

AB

through

K

and the line parallel to

AC

through

L

intersect at point

P

. Prove that

BP=CP

. (Author: P. Kozhevnikov)

11.3

1

Rectangle, two circumcircles and centroid

Point

K

lies on the circumcircle of a rectangle

ABCD

. Line

CK

intersects line segment

AD

at point

M

so that

AM:MD=2

.

O

is the center the rectangle. Prove that the centroid of triangle

OKD

belongs to the circumcircle of triangle

COD

. (Author: V. Shmarov)

10.7

1

Rhombus formed with circumcenter and orthocenter

Points

C_0

and

B_0

are the respective midpoints of sides

AB

and

AC

of a non-isosceles acute triangle

ABC

,

O

is its circumscenter and

H

is the orthocenter. Lines

BH

and

OC_0

intersect at

P

, while lines

CH

and

OB_0

intersect at

Q

.

OPHQ

is rhombus. Prove that points

A

,

P

and

Q

are collinear. (Author: L. Emelyanov)

10.2

1

Concyclic circumcenters

ABC

is an acute triangle. Points

M

and

K

on side

AC

are such that

\angle ABM = \angle CBK

. Prove that the circumcenters of triangles

ABM

,

ABK

,

CBM

and

CBK

are concyclic. (Author: T. Emelyanova)

9.6

1

Points symmetric with respect to perpendicular bisector

Initially, there are three different points on the plane. Every minute, three points are chosen, for example

A

,

B

and

C

, and a new point

D

is generated which is symmetric to

A

with respect to the perpendicular bisector of line segment

BC

. 24 hours later, it turns out that among all the points that were generated, there exist three collinear points. Prove that the three initial points were also collinear. (Author: V. Shmarov)

10.5

1

A bigger product made integer

Find all

a

such that for any positive integer

n

, the number

an(n+2)(n+3)(n+4)

is an integer. (Author: O. Podlipski) [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=57&t=427802](similar to Problem 5 of grade 9) Same problem for grades 10 and 11

9.7

1

Fourth powers of three primes

Find all prime numbers

p

,

q

and

r

such that the fourth power of any of them minus one is divisible by the product of the other two. (Author: V. Senderov)

9.5

1

A product made an integer

Find all

a

such that for any positive integer

n

, the number

an(n+2)(n+4)

is an integer. (Author: O. Podlipski)

9.4

1

Inequality in three fractions

x

,

y

and

z

are positive real numbers. Prove the inequality

\frac{x+1}{y+1}+\frac{y+1}{z+1}+\frac{z+1}{x+1}\leq\frac{x}{y}+\frac{y}{z}+\frac{z}{x}.

(Authors: A. Khrabrov, B. Trushin)

9.3

1

Polygonal chain on a chess board

A closed non-self-intersecting polygonal chain is drawn through the centers of some squares on the

8\times 8

chess board. Every link of the chain connects the centers of adjacent squares either horizontally, vertically or diagonally, where the two squares are adjacent if they share an edge or a corner. For the interior polygon bounded by the chain, prove that the total area of black pieces equals the total area of white pieces. (Author: D. Khramtsov)

9.2

1

A problem about isosceles triangle and related circumcirles

Consider an isosceles triangle

ABC

with

AB=AC

. Point

D

is on the smaller arc

AB

of its circumcirle. Point

E

lies on the continuation of segment

AD

beyond point

D

so that both

A

and

E

lie in the same half-plane relative to

BC

. The circumcirle of triangle

BDE

intersects side

AB

at point

F

. Prove that lines

EF

and

BC

are parallel. (Author: R. Zhenodarov)

9.1

1

Problem 1 of Russian Regional Olympiad 2011, grade 9

Three positive numbers are such that the sum of any one of them with the sum of squares of the remaining two numbers is the same. Is it true that all numbers are the same? (Author: L. Emelyanov)

2011 All-Russian Olympiad Regional Round

Subcontests

sum-products of integers

Relocating cement across storage buildings

Smallest number of zeros

Runners

combinatorics of the last two digits

cutting sticks

All common tangents of all circle pairs

Problem 8 of Russian Regional Olympiad 2011, grade 11

Problem 4 of Russian Regional Olympiad 2011, grade 10

Problem 1 of Russian Regional Olympiad 2011, grade 11

Equality for acute triangle, and tangents of circumcircle

Rectangle, two circumcircles and centroid

Rhombus formed with circumcenter and orthocenter

Concyclic circumcenters

Points symmetric with respect to perpendicular bisector

A bigger product made integer

Fourth powers of three primes

A product made an integer

Inequality in three fractions

Polygonal chain on a chess board

A problem about isosceles triangle and related circumcirles

Problem 1 of Russian Regional Olympiad 2011, grade 9