MathDB

1

xcos x <= pi^2/ 16 if x \in [0, \pi/2]

x\cos x \le \frac{\pi^2}{16}

for

0 \le x \le \frac{\pi}{2}

11.2

1

X' - O in 10 x10 board

In some cells of the table

10\times 10

arranged several

X

's and a few

O

's. It is known that there is no line (row or column) completely filled with identical symbols (crosses or zeros). However, if in any empty If you place any icon in a cell, this condition will be violated. What is the minimum number of icons that can appear in a table?

11.1

1

(f(x))^5 - f(x) has exactly three real roots, trinomial

Square trinomial

f(x)

is such that the polynomial (f(x))^5 - f(x) has exactly three real roots. Find the ordinate of the vertex of the graph of this trinomial.

11.5

1

intersection points of lines

We drew several straight lines on the plane and marked all of them intersection points. How many lines could be drawn? if one point is marked on one of the drawn lines, on the other - three, and on the third - five? Find all possible options and prove that there are no others.

11.6

1

CL= BC if AB=AD

Point

D

on side

BC

of acute triangle ABC is such that

AB=AD

. The circumcircle of triangle

ABD

intersects side

AC

at points

A

and

K

. Line

DK

intersects the perpendicular drawn from

B

on

AC

, at the point

L

. Prove that

CL= BC

11.4

1

point symmetric to circumcneter of (HOA&#039; ) lies on midline

In an acute non-isosceles triangle

ABC

, the altitude

AA'

is drawn and point

H

is the intersection point of the altitudes and and

O

is the center of the circumscribed circle. Prove that the point symmetric to the circumcenter of triangle

HOA'

wrt straight line

HO

, lies on a midline of triangle

ABC

.

10.7

1

x_k^2 - x_k x_{k+1}+x^2_{n+1}=c

Positive numbers

x_1, x_2, . . ., x_{2009}

satisfy the equalities

x^2_1 - x_1x_2 +x^2_2 =x^2_2 -x_2x_3+x^2_3=x^2_3 -x_3x_4+x^2_4= ...= x^2_{2008}- x_{2008}x_{2009}+x^2_{2009}= x^2_{2009}-x_{2009}x_1+x^2_1

. Prove that the numbers

x_1, x_2, . . ., x_{2009}

are equal.

10.6

1

B_2C_2 diameter of incircle wanted

Circle

\omega

inscribed in triangle

ABC

touches sides

BC

,

CA

,

AB

at points

A_1

,

B_1

and

C_1

respectively. On the extension of segment

AA_1

, point

A

is taken as point D such that

AD= AC_1

. Lines

DB_1

and

DC_1

intersect a second time circle

\omega

at points

B_2

and

C_2

. Prove that

B_2C_2

is the diameter of circle of

\omega

.

10.8

1

20 people be seated at 4 tables

At a party, a group of

20

people needs to be seated at

4

tables. The seating arrangement is called successful if any two people at the same table are friends. It turned out that successful seating arrangements exist. In a successful seating arrangement, exactly

5

people sit at each table. What is the greatest possible number of pairs of friends in this companies?

10.4

1

collinear, ext. tangent circles related

Circles

\omega_1

and

\omega_2

touch externally at the point

O

. Points

A

and

B

on the circle

\omega_1

and points

C

and

D

on the circle

\omega_2

are such that

AC

and

BD

are common external tangents to circles. Line

AO

intersects segment

CD

at point

M

and straight line

CO

intersexts

\omega_1

again at point

N

. Prove that the points

B

,

M

and

N

lie on the same straight line.

10.3

1

2 sets of 17 coines, 5 fakes

Kostya had two sets of

17

coins: in one set all the coins were real, and in the other set there were exactly

5

fakes (all the coins look the same; all real coins weigh the same, all fake coins also weigh the same, but it is unknown lighter or heavier than real ones). Kostya gave away one of the sets friend, and subsequently forgot which of the two sets had stayed. With the help of two weighings, can Kostya on a cup scale without weights, find out which of the two did he give away the sets?

10.2

1

product of n consecutives = product of n + 100 consecutives

Prove that there is a natural number

n > 1

such that the product of some

n

consecutive natural numbers is equal to the product of some

n + 100

consecutive natural numbers.

10.1

1

f(x)) ^3- f(x)$ has exactly three real roots, , trinomial

Square trinomial

f(x)

is such that the polynomial

(f(x)) ^3- f(x)

has exactly three real roots. Find the ordinate of the vertex of the graph of this trinomial.

9.5

1

11 phrases written on 11 pieces of paper

There are

11

phrases written on

11

pieces of paper (one per sheet):
1) To the left of this sheet there are no sheets with false statements. 2) Exactly one sheet to the left of this one contains a false statement. 3) Exactly

2

sheets to the left of this one contain false statements ... 11) Exactly

10

sheets to the left of this one contain false statements.
The sheets of paper were laid out in some order in a row, going from left to right. After this, some of the written statements became true and some became false. What is the greatest possible number of true statements?

9.4

1

numbers 1-25 into 25 small triangles

The picture shows a triangle divided into

25

smaller triangles, numbered

1

to

25

. Is it possible to place the same numbers in the square cells 5

\times 5

so that any two numbers written in adjacent triangles were are also written in adjacent cells of the square? (The cells of a square are considered adjacent if they have a common side.) https://cdn.artofproblemsolving.com/attachments/4/3/758fe5531ab3e576ef4712c095b393f8dff397.png

9.3

1

right angle wanted, 3 altitudes

In an acute triangle

ABC

the altitudes

AA_1

,

BB_1

,

CC_1

are drawn. A line perpendicular to side

AC

and passing through a point

A_1

, intersects the line

B_1C_1

at point

D

. Prove that angle

ADC

is right.

9.2

1

1-ab=rational^2 if a^3b+ab^3+2a^2b2+2a + 2b + 1 = 0

Rational numbers

a

and

b

satisfy the equality

a^3b+ab^3+2a^2b^2+2a + 2b + 1 = 0.

Prove that the number

1-ab

is the square of the rational numbers.

9.1

1

good and bad mushrooms

A mushroom is called bad if it contains at least

10

worms. A basket contains

90

bad and

10

good mushrooms. Can all mushrooms become good after some worms crawl from bad mushrooms to good ones?
[hide=original wording]Гриб называется плохим, если в нем не менее 10 червей. В лукошке 90 плохих и 10 хороших грибов. Могут ли все грибы стать хорошими после того, как некоторые черви переползут из плохих грибов в хорошие?

9.8

1

8 chess players

8 chess players participated in the chess tournament and everyone played exactly one game with everyone else. It is known that any two chess players who play a draw with each other ended up scoring different numbers of points. Find the greatest possible number of draws in this tournament. (For winning the game the chess player is awarded

1

point, for a draw

1/2

points, for defeat

0

points.)

9.7

1

circumcentet of (DEF) lies on (ABC)

Given a parallelogram

ABCD

, in which the angle

\angle ABC

is obtuse. Line

AD

intersects the circle a second time

\omega

circumscribed around triangle

ABC

, at the point

E

. Line

CD

intersects second time circle

\omega

at point

F

. Prove that the circumcenter of triangle

DEF

lies on the circle

\omega

.

9.6

1

Last digit of a power of 8

Positive integer

m

is such that the sum of decimal digits of

8^m

equals 8. Can the last digit of

8^m

be equal 6?
(Author: V. Senderov)
(compare with http://www.artofproblemsolving.com/Forum/viewtopic.php?f=57&t=431860)

10.5

1

Last digit of a power of 2

Positive integer

m

is such that the sum of decimal digits of

2^m

equals 8. Can the last digit of

2^m

be equal 6?
(Author: V. Senderov)

11.7

1

Sum of all 3 perfect powers not divisible by a power of 2

a

,

b

and

c

are positive integers with

\textrm{gcd}(a,b,c)=1

. Is it true that there exist a positive integer

n

such that

a^k+b^k+c^k

is not divisible by

2^n

for all

k

?

11.8

1

11 integers on a circle

11 integers are placed along the circle. It is known that any two neighbors differ by at least 20 and sum of any two neighbors is no more than 100. Find the minimal possible sum of all numbers.

2009 All-Russian Olympiad Regional Round

Subcontests

xcos x <= pi^2/ 16 if x \in [0, \pi/2]

X' - O in 10 x10 board

(f(x))^5 - f(x) has exactly three real roots, trinomial

intersection points of lines

CL= BC if AB=AD

point symmetric to circumcneter of (HOA&#039; ) lies on midline

x_k^2 - x_k x_{k+1}+x^2_{n+1}=c

B_2C_2 diameter of incircle wanted

20 people be seated at 4 tables

collinear, ext. tangent circles related

2 sets of 17 coines, 5 fakes

product of n consecutives = product of n + 100 consecutives

f(x)) ^3- f(x)$ has exactly three real roots, , trinomial

11 phrases written on 11 pieces of paper

numbers 1-25 into 25 small triangles

right angle wanted, 3 altitudes

1-ab=rational^2 if a^3b+ab^3+2a^2b2+2a + 2b + 1 = 0

good and bad mushrooms

8 chess players

circumcentet of (DEF) lies on (ABC)

Last digit of a power of 8

Last digit of a power of 2

Sum of all 3 perfect powers not divisible by a power of 2

11 integers on a circle

2009 All-Russian Olympiad Regional Round

Subcontests

xcos x &lt;= pi^2/ 16 if x \in [0, \pi/2]

X&#039; - O in 10 x10 board

(f(x))^5 - f(x) has exactly three real roots, trinomial

intersection points of lines

CL= BC if AB=AD

point symmetric to circumcneter of (HOA&amp;#039; ) lies on midline

x_k^2 - x_k x_{k+1}+x^2_{n+1}=c

B_2C_2 diameter of incircle wanted

20 people be seated at 4 tables

collinear, ext. tangent circles related

2 sets of 17 coines, 5 fakes

product of n consecutives = product of n + 100 consecutives

f(x)) ^3- f(x)$ has exactly three real roots, , trinomial

11 phrases written on 11 pieces of paper

numbers 1-25 into 25 small triangles

right angle wanted, 3 altitudes

1-ab=rational^2 if a^3b+ab^3+2a^2b2+2a + 2b + 1 = 0

good and bad mushrooms

8 chess players

circumcentet of (DEF) lies on (ABC)

Last digit of a power of 8

Last digit of a power of 2

Sum of all 3 perfect powers not divisible by a power of 2

11 integers on a circle

xcos x <= pi^2/ 16 if x \in [0, \pi/2]

X' - O in 10 x10 board

point symmetric to circumcneter of (HOA' ) lies on midline