MathDB

3

$5\times 7$ covered by triomino

5\times 7

checkerboard be covered by L's (figures formed from a

2\times2

square by removing one of its four

1\times1

corners), not crossing its borders, in several layers so that each square of the board is covered by the same number of L's?
M. Evdokimov

Ten $x^2 +px+q$

Goodnik writes 10 numbers on the board, then Nogoodnik writes 10 more numbers, all 20 of the numbers being positive and distinct. Can Goodnik choose his 10 numbers so that no matter what Nogoodnik writes, he can form 10 quadratic trinomials of the form

x^2 +px+q

, whose coeficients

p

and

q

run through all of the numbers written, such that the real roots of these trinomials comprise exactly 11 values?
I. Rubanov

How find the order of all numbers?

The numbers from 1 to 100 are written in an unknown order. One may ask about any 50 numbers and find out their relative order. What is the fewest questions needed to find the order of all 100 numbers?
S. Tokarev

7

3

Two piles of coins

Two piles of coins lie on a table. It is known that the sum of the weights of the coins in the two piles are equal, and for any natural number

k

, not exceeding the number of coins in either pile, the sum of the weights of the

k

heaviest coins in the first pile is not more than that of the second pile. Show that for any natural number

x

, if each coin (in either pile) of weight not less than

x

is replaced by a coin of weight

x

, the first pile will not be lighter than the second.
D. Fon-der-Flaas

A convex polygon without two parallel sides

A convex polygon is given, no two of whose sides are parallel. For each side we consider the angle the side subtends at the vertex farthest from the side. Show that the sum of these angles equals

180^\circ

.
M. Smurov

finite set M of nonzero real numbers

Does there exist a finite set

M

of nonzero real numbers, such that for any natural number

n

a polynomial of degree no less than

n

with coeficients in

M

, all of whose roots are real and belong to

M

?
E. Malinnikova

6

3

Prove that $ID$ and $AC$ are parallel

In the isosceles triangle

ABC

(

AC = BC

) point

O

is the circumcenter,

I

the incenter, and

D

lies on

BC

so that lines

OD

and

BI

are perpendicular. Prove that

ID

and

AC

are parallel.
M. Sonkin

Sergeants and solders

Three sergeants and several solders serve in a platoon. The sergeants take turns on duty. The commander has given the following orders: (a) Each day, at least one task must be issued to a soldier. (b) No soldier may have more than two task or receive more than one tasks in a single day. (c) The lists of soldiers receiving tasks for two different days must not be the same. (d) The first sergeant violating any of these orders will be jailed. Can at least one of the sergeants, without conspiring with the others, give tasks according to these rules and avoid being jailed?
M. Kulikov

perpendicular to bisector

In isosceles triangle

ABC

(

AB = BC

) one draws the angle bisector

CD

. The perpendicular to

CD

through the center of the circumcircle of

ABC

intersects

BC

at

E

. The parallel to

CD

through

E

meets

AB

at

F

. Show that

BE

=

FD

.
M. Sonkin

5

3

2

two subsets with no fewer than four common elements.

In the Duma there are 1600 delegates, who have formed 16000 committees of 80 persons each. Prove that one can find two committees having no fewer than four common members.
A. Skopenkov

$a_1 + a_22^k + a_33^k + \dots + a_mm^k = 0$

Show that if the integers

a_1

;

\dots

a_m

are nonzero and for each

k =0; 1; \dots ;n

(

n < m - 1

),

a_1 + a_22^k + a_33^k + \dots + a_mm^k = 0

; then the sequence

a_1, \dots, a_m

contains at least

n+1

pairs of consecutive terms having opposite signs.
O. Musin

3

If $x^n + y^n = p^k$, $n$ is a power of $p$ (Russia 1996)

Let

x, y, p, n

, and

k

be positive integers such that

x^n + y^n = p^k

. Prove that if

n > 1

is odd, and

p

is an odd prime, then

n

is a power of

p

.
A. Kovaldji, V. Senderov

$3^n =x^k + y^k$

Find all natural numbers

n

, such that there exist relatively prime integers

x

and

y

and an integer

k > 1

satisfying the equation

3^n =x^k + y^k

.
A. Kovaldji, V. Senderov

a cross-section of a pyramid

Show that for

n\ge 5

, a cross-section of a pyramid whose base is a regular

n

-gon cannot be a regular

(n + 1)

-gon.
N. Agakhanov, N. Tereshin

2

3

three circles of equal radius

The centers

O_1

;

O_2

;

O_3

of three nonintersecting circles of equal radius are positioned at the vertices of a triangle. From each of the points

O_1

;

O_2

;

O_3

one draws tangents to the other two given circles. It is known that the intersection of these tangents form a convex hexagon. The sides of the hexagon are alternately colored red and blue. Prove that the sum of the lengths of the red sides equals the sum of the lengths of the blue sides.
D. Tereshin

Prove that any two counters can be made to coincide

On a coordinate plane are placed four counters, each of whose centers has integer coordinates. One can displace any counter by the vector joining the centers of two of the other counters. Prove that any two preselected counters can be made to coincide by a finite sequence of moves.
Р. Sadykov

sum of pairwise distances is decreasing

Several hikers travel at fixed speeds along a straight road. It is known that over some period of time, the sum of their pairwise distances is monotonically decreasing. Show that there is a hiker, the sum of whose distances to the other hikers is monotonically decreasing over the same period.
A. Shapovalov

1

3

the sum of the square and cube

Which are there more of among the natural numbers from 1 to 1000000, inclusive: numbers that can be represented as the sum of a perfect square and a (positive) perfect cube, or numbers that cannot be?
A. Golovanov

Prove that $\angle FAC = \angle EDB$

Points

E

and

F

are given on side

BC

of convex quadrilateral

ABCD

(with

E

closer than

F

to

B

). It is known that

\angle BAE = \angle CDF

and

\angle EAF = \angle FDE

. Prove that

\angle FAC = \angle EDB

.
M. Smurov

read left-to-right and right-to-left

Can the number obtained by writing the numbers from 1 to

n

in order (

n > 1

) be the same when read left-to-right and right-to-left?
N. Agakhanov

1996 All-Russian Olympiad

Subcontests

$5\times 7$ covered by triomino

Ten $x^2 +px+q$

How find the order of all numbers?

Two piles of coins

A convex polygon without two parallel sides

finite set M of nonzero real numbers

Prove that $ID$ and $AC$ are parallel

Sergeants and solders

perpendicular to bisector

arithmetic progression with infinitely many powers of 10.

cube with numbers

three natural numbers

two subsets with no fewer than four common elements.

$a_1 + a_22^k + a_33^k + \dots + a_mm^k = 0$

If $x^n + y^n = p^k$, $n$ is a power of $p$ (Russia 1996)

$3^n =x^k + y^k$

a cross-section of a pyramid

three circles of equal radius

Prove that any two counters can be made to coincide

sum of pairwise distances is decreasing

the sum of the square and cube

Prove that $\angle FAC = \angle EDB$

read left-to-right and right-to-left