MathDB

1

2 trinomials with same value at x=1234

a,b,c

are such that the values of the trinomials

bx^2+cx+a

and

cx^2+ax+b

at

x=1234

coincide. Can the first trinomial at

x = 1

take the value

2009

?

11.1

1

increase legs of right triangle by 1

Each leg of a right triangle is increased by one. Could its hypotenuse increase by more than

\sqrt2

?

11.8

1

1-10000 in 100x100 square

The numbers

1, 2,. . . , 10000,

were placed in the cells of a

100 \times 100

square, each once; in this case, numbers differing by

1

are written in cells adjacent to each side. After that we calculated distances between the centers of every two cells whose numbers differ by exactly

5000

. Let

S

be the minimum of these distances What is the largest value

S

can take?

11.6

1

equal sums of volumes of tetrahedra, pyramid with base #

At the base of the quadrangular pyramid

SABCD

lies the parallelogram

ABCD

. Prove that for any point

O

inside the pyramid, the sum of the volumes of the tetrahedra

OSAB

and

OSCD

is equal to the sum of the volumes of the tetrahedra

OSBC

and

OSDA

.

11.5

1

acute if sin A > cos B, sin B> cos C, sin C > cos A

The angles of the triangle

\alpha, \beta, \gamma

satisfy the inequalities

\sin \alpha > \cos \beta, \sin \beta > \cos \gamma, \sin \gamma > \cos \alpha.

Prove that the trαiangle is acute-angled.

11.4

1

square triple of natural numbers , abc =d^2

We call a triple of natural numbers

(a, b, c)

square if they form an arithmetic progression (in exactly this order), the number

b

is coprime to each of the numbers

a

and

c

, and the number

abc

is a perfect square. Prove that for any given a square triple, there is another square triple that has at least one common number with it.

11.3

1

right angle wanted, cyclic given

Quadrangle

ABCD

is inscribed in a circle with diameter

AC

. Points

K

and

M

are projections of vertices

A

and

C

, respectively, onto line

BD

. A line parallel to

BC

is drawn through point

K

and intersecting

AC

at point

P

. Prove that angle

KPM

is a right angle.

11.2

1

2009 integer weights <=1 kg

In a row of

2009

weights, the weight of each weight is an integer grams and does not exceed

1

kg. The weights of any two adjacent weights differ by exactly

1

g, and the total weight of all weights in grams is an even number. Prove that weights can be separated into two piles, the sums of the weights in which are equal.

9.3

1

2 groups for all numbers from 1 to k

Is it possible for some natural number

k

to divide all natural numbers from

1

to

k

into two groups and write down the numbers in each group in a row in some order so that you get two the same numbers?
[hide=original wording beacuse it doesn't make much sense]Можно ли при каком-то натуральном k разбить все натуральные числа от 1 до k на две группы и выписать числа в каждой группе подряд в некотором порядке так, чтобы получились два одинаковых числа?

10.8

1

different partitions of staircase of height n

Let's call it a staircase of height

n

, a figure consisting from all square cells

n\times n

lying no higher diagonals (the figure shows a staircase of height

4

). In how many different ways can a staircase of height

n

can be divided into several rectangles whose sides go along the grid lines, but the areas are different in pairs? https://cdn.artofproblemsolving.com/attachments/f/0/f66d7e9ada0978e8403fbbd8989dc1b201f2cd.png

10.7

1

a^{13}+b^{13}+c^{13}=d^2, a+b+c=0

Are there three pairwise distinct non-zero integers whose sum is zero and whose sum of thirteenth powers is the square of some natural number?

10.6

1

poles and polars candidate

The tangent lines to the circle

\omega

at points

B

and

D

intersect at point

P

. The line passing through

P

cuts out from circle chord

AC

. Through an arbitrary point on the segment

AC

a straight line parallel to

BD

is drawn. Prove that it divides the lengths of polygonal

ABC

and

ADC

in the same ratio.
[hide=last sentence was in Russian: ]Докажите, что она делит длины ломаных ABC и ADC в одинаковых отношениях.

10.5

1

cx^2-bx + a > cx-b if ax^2+bx+c > cx

Non-zero numbers

a, b, c

are such that

ax^2+bx+c > cx

for any

x

. Prove that

cx^2-bx + a > cx-b

for any

x

.

10.1

1

nine skiers now

Nine skiers left the start line in turn and covered the distance, each at their own constant speed. Could it turn out that each skier participated in exactly four overtakes? (In each overtaking, exactly two skiers participate - the one who is overtaking, and the one who is being overtaken.)

10.4

1

a^5 is divisible by b^2, a^2 is divisible by b

We call a natural number

b

lucky if for any natural number

a

such that

a^5

is divisible by

b^2

, the number

a^2

is divisible by

b

. Find the number of lucky natural numbers less than

2010

.

10.3

1

concyclic wanted, angle bisectors related

In triangle

ABC

, the angle bisectors

AD

,

BE

and

CF

are drawn, intersecting at point

I

. The perpendicular bisector of the segment

AD

intersects lines

BE

and

CF

at points

M

and

N

, respectively. Prove that points

A

,

I

,

M

and

N

lie on the same circle.

9.7

1

acquaintances among 7 people, round table

In a company of seven people, any six can sit at a round table so that every two neighbors turn out to be acquaintances. Prove that the whole company can be seated at a round table so that every two neighbors turn out to be acquaintances.

9.5

1

11 natural numbers in a circle.

Dunno wrote down

11

natural numbers in a circle. For every two adjacent numbers, he calculated their difference. As a result among the differences found there were four units, four twos and three threes. Prove that Dunno made a mistake somewhere an error.

9.4

1

point symmetric of vertex wrt line connecting feet of bisectors lies on side

In triangle

ABC

,

\angle A =60^o

. Let

BB_1

and

CC_1

be angle bisectors of this triangle. Prove that the point symmetrical to vertex

A

with respect to line

B_1C_1

lies on side

BC

.

9.6

1

Perpendiculars dropped from circumcircle's tangent

Let points

A

,

B

,

C

lie on a circle, and line

b

be the tangent to the circle at point

B

. Perpendiculars

PA_1

and

PC_1

are dropped from a point

P

on line

b

onto lines

AB

and

BC

respectively. Points

A_1

and

C_1

lie inside line segments

AB

and

BC

respectively. Prove that

A_1C_1

is perpendicular to

AC

.

9.1

1

Three quadratic polynomials

f_1(x) = x^2+2a_1x+b_1

,

f_2(x) = x^2+2a_2x+b_2

,

f_3(x) = x^2 + 2a_3x + b_3

are such that

a_1a_2a_3 = b_1b_2b_3 > 1

. Prove that at least one polynomial has two distinct roots.

9.8

1

sums of first n primes

For every positive integer

n

, let

S_n

be the sum of the first

n

prime numbers:

S_1 = 2, S_2 = 2 + 3 = 5, S_3 = 2 + 3 + 5 = 10

, etc. Can both

S_n

and

S_{n+1}

be perfect squares?

9.2

1

Number of ranks in a skiing race

This problem is given by my teacher. :wink:
Seven skiers numbered 1,2,3,4,5,6,7 set out in turn at the starting point,each one slides the same distance at a constant speed. During this period,everyone just had two "beyond" experience.(going beyond one skier or be went beyond by another skier is called a "beyond" experience). When the race ended,we would decide the rank according to the order that skiers reached the ending. Prove that:there are two different rank at most.

2010 All-Russian Olympiad Regional Round

Subcontests

2 trinomials with same value at x=1234

increase legs of right triangle by 1

1-10000 in 100x100 square

equal sums of volumes of tetrahedra, pyramid with base #

acute if sin A > cos B, sin B> cos C, sin C > cos A

square triple of natural numbers , abc =d^2

right angle wanted, cyclic given

2009 integer weights <=1 kg

2 groups for all numbers from 1 to k

different partitions of staircase of height n

a^{13}+b^{13}+c^{13}=d^2, a+b+c=0

poles and polars candidate

cx^2-bx + a > cx-b if ax^2+bx+c > cx

nine skiers now

a^5 is divisible by b^2, a^2 is divisible by b

concyclic wanted, angle bisectors related

acquaintances among 7 people, round table

11 natural numbers in a circle.

point symmetric of vertex wrt line connecting feet of bisectors lies on side

Perpendiculars dropped from circumcircle's tangent

Three quadratic polynomials

sums of first n primes

Number of ranks in a skiing race

2010 All-Russian Olympiad Regional Round

Subcontests

2 trinomials with same value at x=1234

increase legs of right triangle by 1

1-10000 in 100x100 square

equal sums of volumes of tetrahedra, pyramid with base #

acute if sin A &gt; cos B, sin B&gt; cos C, sin C &gt; cos A

square triple of natural numbers , abc =d^2

right angle wanted, cyclic given

2009 integer weights &lt;=1 kg

2 groups for all numbers from 1 to k

different partitions of staircase of height n

a^{13}+b^{13}+c^{13}=d^2, a+b+c=0

poles and polars candidate

cx^2-bx + a &gt; cx-b if ax^2+bx+c &gt; cx

nine skiers now

a^5 is divisible by b^2, a^2 is divisible by b

concyclic wanted, angle bisectors related

acquaintances among 7 people, round table

11 natural numbers in a circle.

point symmetric of vertex wrt line connecting feet of bisectors lies on side

Perpendiculars dropped from circumcircle's tangent

Three quadratic polynomials

sums of first n primes

Number of ranks in a skiing race

acute if sin A > cos B, sin B> cos C, sin C > cos A

2009 integer weights <=1 kg

cx^2-bx + a > cx-b if ax^2+bx+c > cx