MathDB

2

boxes of stones

There are three boxes of stones. Sisyphus moves stones one by one between the boxes. Whenever he moves a stone, Zeus gives him the number of coins that is equal to the difference between the number of stones in the box the stone was put in, and that in the box the stone was taken from (the moved stone does not count). If this difference is negative, then Sisyphus returns the corresponding amount to Zeus (if Sisyphus cannot pay, generous Zeus allows him to make the move and pay later). After some time all the stones lie in their initial boxes. What is the greatest possible earning of Sisyphus at that moment? I. Izmest’ev

tetrahedron

The altitudes of a tetrahedron intersect in a point. Prove that this point, the foot of one of the altitudes, and the points dividing the other three altitudes in the ratio

2 : 1

(measuring from the vertices) lie on a sphere. D. Tereshin

4

2

numbers from 1 to 81 be written in a $9×9$ board

Can the numbers from 1 to 81 be written in a 9×9 board, so that the sum of numbers in each 3×3 square is the same? S. Tokarev

all angles of a convex $n$-gon are equal

Prove that if all angles of a convex

n

-gon are equal, then there are at least two of its sides that are not longer than their adjacent sides. A. Berzin’sh, O. Musin

6

2

midpoints of the altitudes

In an acute-angled triangle ABC, points

A_2

,

B_2

,

C_2

are the midpoints of the altitudes

AA_1

,

BB_1

,

CC_1

, respectively. Compute the sum of angles

B_2A_1C_2

,

C_2B_1A_2

and

A_2C_1B_2

. D. Tereshin

cyclic shifts

A boy goes

n

times at a merry-go-round with

n

seats. After every time he moves in the clockwise direction and takes another seat, not making a full circle. The number of seats he passes by at each move is called the length of the move. For which

n

can he sit at every seat, if the lengths of all the

n-1

moves he makes have different lengths? V. New

2

chords of a circle

A chord

CD

of a circle with center

O

is perpendicular to a diameter

AB

. A chord

AE

bisects the radius

OC

. Show that the line

DE

bisects the chord

BC

V. Gordon

sum of two functions with axis of symmetry

Prove that every real function, defined on all of

\mathbb R

, can be represented as a sum of two functions whose graphs both have an axis of symmetry. D. Tereshin

8

2

board 2000×2000 with 1 and -1

Numbers 1 and −1 are written in the cells of a board 2000×2000. It is known that the sum of all the numbers in the board is positive. Show that one can select 1000 rows and 1000 columns such that the sum of numbers written in their intersection cells is at least 1000. D. Karpov

sum of the squares of the coeficients of the P(x)Q(x)

Let

P(x)

and

Q(x)

be monic polynomials. Prove that the sum of the squares of the coeficients of the polynomial

P(x)Q(x)

is not smaller than the sum of the squares of the free coefficients of

P(x)

and

Q(x)

. A. Galochkin, O. Ljashko

3

composition of quadratic polynomials

Can the equation

f(g(h(x))) = 0

, where

f

,

g

,

h

are quadratic polynomials, have the solutions

1, 2, 3, 4, 5, 6, 7, 8

?
S. Tokarev

k | a_1+\dots+a_k

Does there exist a sequence of natural numbers in which every natural number occurs exactly once, such that for each

k = 1, 2, 3, \dots

the sum of the first

k

terms of the sequence is divisible by

k

? A. Shapovalov

compass and ruler

Two points on the distance 1 are given in a plane. It is allowed to draw a line through two marked points, as well as a circle centered in a marked point with radius equal to the distance between some two marked points. By marked points we mean the two initial points and intersection points of two lines, two circles, or a line and a circle constructed so far. Let

C(n)

be the minimum number of circles needed to construct two points on the distance

n

if only a compass is used, and let

LC(n)

be the minimum total number of circles and lines needed to do so if a ruler and a compass are used, where

n

is a natural number. Prove that the sequence

C(n)/LC(n)

is not bounded. A. Belov

1

2

hours and minutes

A freight train departed from Moscow at

x

hours and

y

minutes and arrived at Saratov at

y

hours and

z

minutes. The length of its trip was

z

hours and

x

minutes. Find all possible values of

x

.
S. Tokarev

the numbers 1...100 be covered with 12 geom progressions

Can the numbers

1,2,3,\ldots,100

be covered with

12

geometric progressions?
A. Golovanov

5

3

three numbers with the same digits

We call natural numbers similar if they are written with the same (decimal) digits. For example, 112, 121, 211 are similar numbers having the digits 1, 1, 2. Show that there exist three similar 1995-digit numbers with no zero digits, such that the sum of two of them equals the third. S. Dvoryaninov

GCD

The sequence

a_1, a_2, ...

of natural numbers satisfies

GCD(a_i, a_j)=GCD(i, j)

for all

i \neq j

. Prove that

a_i=i

for all

i

.

a^2_1+a^_22+...+a^2_k is divisible by a_1+a_2+...+a_k

Prove that for every natural number

a_1>1

there exists an increasing sequence of natural numbers

a_n

such that

a^2_1+a^2_2+\cdots+a^2_k

is divisible by

a_1+a_2+\cdots+a_k

for all

k \geq 1

.
A. Golovanov

1995 All-Russian Olympiad

Subcontests

boxes of stones

tetrahedron

numbers from 1 to 81 be written in a $9×9$ board

all angles of a convex $n$-gon are equal

midpoints of the altitudes

cyclic shifts

chords of a circle

sum of two functions with axis of symmetry

board 2000×2000 with 1 and -1

sum of the squares of the coeficients of the P(x)Q(x)

composition of quadratic polynomials

k | a_1+\dots+a_k

compass and ruler

hours and minutes

the numbers 1...100 be covered with 12 geom progressions

three numbers with the same digits

GCD

a^2_1+a^_22+...+a^2_k is divisible by a_1+a_2+...+a_k