MathDB

1

angles wanted, starting with 120-30-30- All-Russian MO 2006 Regional (R4) 8.7

AA_0

of triangle

ABC

is drawn from point

A_0

perpendicular to side

BC

to the outer side of the triangle. Let's denote the second end of the constructed segment as

A_1

. Points

B_1

and

C_1

are constructed similarly. Find the angles of triangle

A_1B_1C_1

if the angles of the triangle

ABC

are

30^o

,

30^o

and

120^o

.
[hide=original wording]Медиану AA0 треугольника ABC отложили от точки A0 перпендикулярно стороне BC во внешнюю сторону треугольника. Обозначим второй конец построенного отрезка через A1. Аналогично строятся точки B1 и C1. Найдите углы треугольника A1B1C1, если углы треугольника ABC равны 30^o, 30^o и 120^o.

10.8

1

max no of vertices that meet 3 edges in convex polyhedron

A convex polyhedron has

2n

faces (

n\ge 3

), and all faces are triangles. What is the largest number of vertices at which converges exactly

3

edges at a such a polyhedron ?

10.6

1

similar trianglex,3 circles through H - All-Russian MO 2006 Regional (R4) 10.6

Through the point of intersection of the altitudes of an acute triangle

ABC

three circles pass through, each of which touches one of the sides triangle at the foot of the altitude . Prove that the second intersection points of the circles are the vertices of a triangle similar to the original one.

11.6

1

parallel planes wanted, tetrahedron related

In the tetrahedron

ABCD

, perpendiculars

AB'

,

AC'

,

AD'

are dropped from vertex

A

, on the plane dividing the dihedral angles at the edges

CD

,

BD

,

BC

in half. Prove that the plane

(B'C'D' )

is parallel to the plane

(BCD)

.

11.4

1

PI // AC, intersections of angle bisectors with (ABC)

The bisectors of angles

A

and

C

of triangle

ABC

intersect its sides at points

A_1

and

C_1

, and the circumcircle of this triangle is at points

A_0

and

C_0

, respectively. Lines

A_1C_1

and

A_0C_0

intersect at point P. Prove that the segment connecting

P

to the center of the incircle of triangle

ABC

is parallel to

AC

.

11.8

1

black sqθαres in 300x300 - All-Russian MO 2006 Regional (R4) 11.8

What is the minimum number of cells that can be painted black in white square

300 \times 300

so that no three black cells form a corner, and after painting any white cell this condition was it violated?

11.7

1

N=a^2+b^2+c^2=K^2+l^2+m^2 - All-Russian MO 2006 Regional (R4) 11.7

Prove that if a natural number

N

is represented in the form as the sum of three squares of integers divisible by

3

, then it is also represented as the sum of three squares of integers not divisible by

3

.

11.3

1

12 stages and n participants - All-Russian MO 2006 Regional (R4) 11.3

A racing tournament has

12

stages and

n

participants. After each stage, all participants, depending on their place

k

, receive points

a_k

(numbers

a_k

are natural numbers and

a_1 > a_2 >... > a_n

). At what smallest

n

can the organizer of the tournament choose numbers

a_1

,

...

,

a_n

so that after the penultimate stage for any possible distribution of places at least two participants had a chance to take first place?

11.2

1

product of square trinomials - All-Russian MO 2006 Regional (R4) 11.2

Product of square trinomials

x^2 - a_1x + b_1

,

x^2 - a_2x + b_2

,

...

,

x^2-a_nx + b_n

is equal to the polynomial

P(x) = x^{2n} +c_1x^{2n-1} +c_2x^{2n-2} +...+ c_{2n-1}x + c_{2n}

, where the coefficients are

c_1

,

c_2

,

...

,

c_{2n}

are positive. Show that for some

k

(

1\le k \le n

) the coefficients

a_k

and

b_k

are positive.

10.7

1

a + b and a^n + b^n are integers - All-Russian MO 2006 Regional (R4) 10.7

For what positive integers

n

are there positive rational, but not integer, numbers

a

and

b

such that both numbers

a + b

and

a^n + b^n

are integers?

10.5

1

| sin nx| >= \sqrt3 / 2 - All-Russian MO 2006 Regional (R4) 10.5

Prove that for every

x

such that

\sin x \ne 0

, there is such natural

n

, which

| \sin nx| \ge \frac{\sqrt3}{2}.

10.2

1

good 3-coloring of 8x8 board - All-Russian MO 2006 Regional (R4) 10.2

We call a coloring of an

8\times 8

board in three colors good if in any corner of five cells contains cells of all three colors. (A five-square corner is a shape made from a

3 \times 3

square by cutting square

2\times 2

.) Prove that the number of good colorings is not less than than

68

.

10.4

1

n monic trinomials x^2-a_i x + b_i - All-Russian MO 2006 Regional (R4) 10.4

Given

n > 1

monic square trinomials

x^2 - a_1x + b_1

,

...

,

x^2-a_nx + b_n

, and all

2n

numbers are

a_1

,

...

,

a_n

,

b_1

,

...

,

b_n

are different. Can it happen that each of the numbers

a_1

,

...

,

a_n

,

b_1

,

...

, $b_n is the root of one of these trinomials?

10.1

1

50 sets.with naturals 1-200 - All-Russian MO 2006 Regional (R4) 10.1

Natural numbers from

1

to

200

were divided into

50

sets. Prove that one of them contains three numbers that are the lengths of the sides some triangle.

9.4

1

PB tangent to (ABC) - All-Russian MO 2006 Regional (R4) 9.4

The bisectors of angles

A

and

C

of triangle

ABC

intersect the circumcircle of this triangle at points

A_0

and

C_0

, respectively. A straight line passing through the center of the inscribed circle of a triangle

ABC

is parallel to side

AC

and intersects line

A_0C_0

at point

P

. Prove that line

PB

is tangent to the circumcircle of the triangle

ABC

.

9.6

1

<CED>45^o , bisector AD, altitude BE - All-Russian MO 2006 Regional (R4) 9.6

In an acute triangle

ABC

, the angle bisector

AD

and altitude

BE

are drawn. Prove that angle

CED

is greater than

45^o

.

9.8

1

N=a^2+b^2+c^2=K^2+l^2+m^2 - All-Russian MO 2006 Regional (R4) 9.8

A number

N

that is not divisible by

81

can be represented as a sum of squares of three integers divisible by

3

. Prove that it is also representable as the sum of the squares of three integers not divisible by

3

.

9.3

1

prod x_1<=1/2 id sum x_i^2=6, sum x_i=0 - All-Russian MO 2006 Regional (R4) 9.3

It is known that

x^2_1+ x^2_2+...+ x^2_6= 6

and

x_1 + x_2 +....+ x_6 = 0.

Prove that

x_1x_2....x_6 \le \frac12

. .

9.2

1

1-4 on infinitecheckered plane - All-Russian MO 2006 Regional (R4) 9.2

Each cell of the infinite checkered plane contains one from the numbers

1, 2, 3, 4

so that each number appears at least once. Let's call a cell correct if the number of distinct numbers written in four adjacent (side) cells to it, equal to the number written in this cell. Can all the cells of the plane be correct?

8.8

1

N coins - All-Russian MO 2006 Regional (R4) 8.8

When making a batch of

N \ge 5

coins, a worker mistakenly made two coins from a different material (all coins look the same). The boss knows that there are exactly two such coins, that they weigh the same, but differ in weight from the others. The employee knows what coins these are and that they are lighter than others. He needs, after carrying out two weighings on cup scales without weights, to convince his boss that the coins are counterfeit easier than real ones, and in which coins are counterfeit. Can he do it?

8.6

1

10 colors in 101x101 - All-Russian MO 2006 Regional (R4) 8.6

In a checkered square

101 \times 101

, each cell of the inner square

99 \times 99

is painted in one of ten colors (cells adjacent to the border of the square, not painted). Could it turn out that in every in a

3\times 3

square, is exactly one more cell painted the same color as the central cell?

8.5

1

max no of even among 100 numbers - All-Russian MO 2006 Regional (R4) 8.5

The product

a_1 \cdot a_2 \cdot ... \cdot a_{100}

is written on the board , where

a_1

,

a_2

,

...

,

a_{100}

, are natural numbers. Let's consider

99

expressions, each of which is obtained by replacing one of the multiplication signs with an addition sign. It is known that the values of exactly

32

of these expressions are even. What is the largest number of even numbers among

a_1

,

a_2

,

...

,

a_{100}

could it be?

8.4

1

frame 2x2x2 and 1x1x1 - All-Russian MO 2006 Regional (R4) 8.4

Each detail of the “Young Solderer” instructor is a bracket in the shape of the letter

\Pi

, consisting of three single segments. Is it possible from the parts of this constructor are soldered together, a complete wire frame of the cube

2 \times 2 \times 2

, divided into

1 \times 1 \times 1

cubes? (The frame consists of 27 points, connected by single segments; any two adjacent points must be connected by exactly one piece of wire.)
=original wording]Каждая деталько нструктора ''Юный паяльщик'' — это скобка в виде буквы П, остоящая из трех единичных отрезков. Можно ли издеталей этого конструктора спаятьполный роволочный каркас куба 2 × × 2 × 2, разбитого на кубики 1 × 1 × 1? (Каркас состоит из 27 точек,соединенных единичными отрезками; любые две соседние точки должны бытьсоединены ровно одним проволочным отрезком.)

8.3

1

round-robin racing - All-Russian MO 2006 Regional (R4) 8.3

Four drivers took part in the round-robin racing. Their cars started simultaneously from one point and moved at constant speeds. It is known that after the start of the race, for any three cars there was a moment when they met. Prove that after the start of the race there will be a moment when all 4 cars meet. (We consider races to be infinitely long in time.)

8.2

1

4 numbers in a circle, 2 players - All-Russian MO 2006 Regional (R4) 8.2

Two people play this game. At the beginning there are numbers 1, 2, 3, 4 in a circle. With each move, the first one adds 1 to two adjacent numbers, and the second swaps any two adjacent numbers. The first one wins if all numbers become equal. Can the second one interfere with him?

8.1

1

9 digit number - All-Russian MO 2006 Regional (R4) 8.1

Find some nine-digit number

N

, consisting of different digits, such that among all the numbers obtained from

N

by crossing out seven digits, there would be no more than one prime. Prove that the number found is correct. (If the number obtained by crossing out the digits starts at zero, then the zero is crossed out.)

2006 All-Russian Olympiad Regional Round

Subcontests

angles wanted, starting with 120-30-30- All-Russian MO 2006 Regional (R4) 8.7

max no of vertices that meet 3 edges in convex polyhedron

similar trianglex,3 circles through H - All-Russian MO 2006 Regional (R4) 10.6

parallel planes wanted, tetrahedron related

PI // AC, intersections of angle bisectors with (ABC)

black sqθαres in 300x300 - All-Russian MO 2006 Regional (R4) 11.8

N=a^2+b^2+c^2=K^2+l^2+m^2 - All-Russian MO 2006 Regional (R4) 11.7

12 stages and n participants - All-Russian MO 2006 Regional (R4) 11.3

product of square trinomials - All-Russian MO 2006 Regional (R4) 11.2

a + b and a^n + b^n are integers - All-Russian MO 2006 Regional (R4) 10.7

| sin nx| >= \sqrt3 / 2 - All-Russian MO 2006 Regional (R4) 10.5

good 3-coloring of 8x8 board - All-Russian MO 2006 Regional (R4) 10.2

n monic trinomials x^2-a_i x + b_i - All-Russian MO 2006 Regional (R4) 10.4

50 sets.with naturals 1-200 - All-Russian MO 2006 Regional (R4) 10.1

PB tangent to (ABC) - All-Russian MO 2006 Regional (R4) 9.4

<CED>45^o , bisector AD, altitude BE - All-Russian MO 2006 Regional (R4) 9.6

N=a^2+b^2+c^2=K^2+l^2+m^2 - All-Russian MO 2006 Regional (R4) 9.8

prod x_1<=1/2 id sum x_i^2=6, sum x_i=0 - All-Russian MO 2006 Regional (R4) 9.3

1-4 on infinitecheckered plane - All-Russian MO 2006 Regional (R4) 9.2

N coins - All-Russian MO 2006 Regional (R4) 8.8

10 colors in 101x101 - All-Russian MO 2006 Regional (R4) 8.6

max no of even among 100 numbers - All-Russian MO 2006 Regional (R4) 8.5

frame 2x2x2 and 1x1x1 - All-Russian MO 2006 Regional (R4) 8.4

round-robin racing - All-Russian MO 2006 Regional (R4) 8.3

4 numbers in a circle, 2 players - All-Russian MO 2006 Regional (R4) 8.2

9 digit number - All-Russian MO 2006 Regional (R4) 8.1

2006 All-Russian Olympiad Regional Round

Subcontests

angles wanted, starting with 120-30-30- All-Russian MO 2006 Regional (R4) 8.7

max no of vertices that meet 3 edges in convex polyhedron

similar trianglex,3 circles through H - All-Russian MO 2006 Regional (R4) 10.6

parallel planes wanted, tetrahedron related

PI // AC, intersections of angle bisectors with (ABC)

black sq&theta;&alpha;res in 300x300 - All-Russian MO 2006 Regional (R4) 11.8

N=a^2+b^2+c^2=K^2+l^2+m^2 - All-Russian MO 2006 Regional (R4) 11.7

12 stages and n participants - All-Russian MO 2006 Regional (R4) 11.3

product of square trinomials - All-Russian MO 2006 Regional (R4) 11.2

a + b and a^n + b^n are integers - All-Russian MO 2006 Regional (R4) 10.7

| sin nx| &gt;= \sqrt3 / 2 - All-Russian MO 2006 Regional (R4) 10.5

good 3-coloring of 8x8 board - All-Russian MO 2006 Regional (R4) 10.2

n monic trinomials x^2-a_i x + b_i - All-Russian MO 2006 Regional (R4) 10.4

50 sets.with naturals 1-200 - All-Russian MO 2006 Regional (R4) 10.1

PB tangent to (ABC) - All-Russian MO 2006 Regional (R4) 9.4

&lt;CED&gt;45^o , bisector AD, altitude BE - All-Russian MO 2006 Regional (R4) 9.6

N=a^2+b^2+c^2=K^2+l^2+m^2 - All-Russian MO 2006 Regional (R4) 9.8

prod x_1&lt;=1/2 id sum x_i^2=6, sum x_i=0 - All-Russian MO 2006 Regional (R4) 9.3

1-4 on infinitecheckered plane - All-Russian MO 2006 Regional (R4) 9.2

N coins - All-Russian MO 2006 Regional (R4) 8.8

10 colors in 101x101 - All-Russian MO 2006 Regional (R4) 8.6

max no of even among 100 numbers - All-Russian MO 2006 Regional (R4) 8.5

frame 2x2x2 and 1x1x1 - All-Russian MO 2006 Regional (R4) 8.4

round-robin racing - All-Russian MO 2006 Regional (R4) 8.3

4 numbers in a circle, 2 players - All-Russian MO 2006 Regional (R4) 8.2

9 digit number - All-Russian MO 2006 Regional (R4) 8.1

black sqθαres in 300x300 - All-Russian MO 2006 Regional (R4) 11.8

| sin nx| >= \sqrt3 / 2 - All-Russian MO 2006 Regional (R4) 10.5

<CED>45^o , bisector AD, altitude BE - All-Russian MO 2006 Regional (R4) 9.6

prod x_1<=1/2 id sum x_i^2=6, sum x_i=0 - All-Russian MO 2006 Regional (R4) 9.3