MathDB

1

| a_ {i +1} - a_ i |= 1given, wanted a _ 1 - a _ {2n} = n iff 1<= a _ {2k} <= n

n

be a natural number. Consider all permutations

({{a} _ {1}}, \ \ldots, \ {{a} _ {2n}})

of the first

2n

natural numbers such that the numbers

| {{a} _ {i +1}} - {{a} _ {i}} |, \ i = 1, \ \ldots, \ 2n-1,

are pairwise different. Prove that

{{a} _ {1}} - {{a} _ {2n}} = n

if and only if

1 \le {{a} _ {2k}} \le n

for all

k = 1, \ \ldots, \ n

.

11

1

S (x^ 3) - S^2(x) <1/4 (P ^2 (x^3) +Q(x^3) , where S (x) = P (x) Q (x)

Let

P (x)

and

Q (x)

be polynomials with real coefficients such that

P (0)> 0

and all coefficients of the polynomial

S (x) = P (x) \cdot Q (x)

are integers. Prove that for any positive

x

the inequality holds:

S ({{x} ^ {2}}) - {{S} ^ {2}} (x) \le \frac {1} {4} ({{P} ^ {2}} ({{ x} ^ {3}}) + Q ({{x} ^ {3}})).

8

1

any natural number can be given as a_i+a_j, a _n>n^2 /16, increasing

Is there an increasing sequence of integers

0 = {{a} _{0}} <{{a} _{1}} <{{a} _{2}} <\ldots

for which the following two conditions are satisfied simultaneously: 1) any natural number can be given as

{{a} _{i}} + {{a} _{j}}

for some (possibly equal)

i \ge 0

,

j \ge 0

, 2)

{{a} _ {n}}> \tfrac {{{n} ^ {2}}} {16}

for all natural

n

?

4

1

a_i+ a_i* =a _ {i-1} +a_ {i + 1}

Suppose an ordered set of

({{a} _{1}}, \ {{a} _{2}},\ \ldots,\ {{a} _{n}})

real numbers,

n \ge 3

. It is possible to replace the number

{{a} _ {i}}

,

i = \overline {2, \ n-1}

by the number

a_ {i} ^ {*}

that

{{a} _ {i}} + a_ {i} ^ {*} = {{a} _ {i-1}} + {{a} _ {i + 1}}

. Let

({{b} _ {1}},\ {{b} _ {2}}, \ \ldots, \ {{b} _ {n}})

be the set with the largest sum of numbers that can be obtained from this, and

({{c} _ {1}},\ {{c} _ {2}}, \ \ldots, \ {{c} _ {n}})

n \ge 3

and set

(1,\ 3, \ \ldots, \ n, \ 2, \ 4, \ \ldots,\ n-1)

find the values of the expressions

{{b} _ {1}} + {{b} _ {2}} + \ldots + {{b} _ {n}}

and

{{c} _ {1}} + {{c} _ {2}} + \ldots + {{c} _ {n}}

.

3

1

n different integers in (k^n,(k+1)^n), product of which is n-th power of integer

Given a positive integer

n> 2

. Prove that there exists a natural

K

such that for all integers

k \ge K

on the open interval

({{k} ^{n}}, \ {{(k + 1)} ^{n}})

there are

n

different integers, the product of which is the

n

-th power of an integer.

1

sum \sin (<A _i MA_i)> sin 2 \pi/n + (n-2) sin \pi /n

Given a right

n

-angle

{{A} _ {1}} {{A} _ {2}} \ldots {{A} _ {n}}

,

n \ge 4

, and a point

M

inside it. Prove the inequality

\sin (\angle {{A} _ {1}} M {{A} _ {2}}) + \sin (\angle {{A} _ {2}} M {{A} _ {3}} ) + \ldots + \sin (\angle {{A} _ {n}} M {{A} _ {1}})> \sin \frac{2 \pi}{n} + (n-2) sin \frac{\pi}{n}

10

1

concurrency wanted, <APE =<BAC, <CQF = < BCA, orthocenter,

Let

H

be the point of intersection of the altitudes

AP

and

CQ

of the acute-angled triangle

ABC

. The points

E

and

F

are marked on the median

BM

such that

\angle APE = \angle BAC

,

\angle CQF = \angle BCA

, with point

E

lying inside the triangle

APB

and point

F

is inside the triangle

CQB

. Prove that the lines

AE, CF

, and

BH

intersect at one point.

9

1

circle tangent to 2 lines and 2 circumcircles, cyclic, <PBC = <PDA, <PCB =<PAD

Inside the inscribed quadrilateral

ABCD

, a point

P

is marked such that

\angle PBC = \angle PDA

,

\angle PCB = \angle PAD

. Prove that there exists a circle that touches the straight lines

AB

and

CD

, as well as the circles circumscribed by the triangles

ABP

and

CDP

.

6

1

3 collinear leads to 4 collinear , incircle related

The circle

\omega

inscribed in triangle

ABC

touches its sides

AB, BC, CA

at points

K, L, M

respectively. In the arc

KL

of the circle

\omega

that does not contain the point

M

, we select point

S

. Denote by

P, Q, R, T

the intersection points of straight

AS

and

KM, ML

and

SC, LP

and

KQ, AQ

and

PC

respectively. It turned out that the points

R, S

and

M

are collinear. Prove that the point

T

also lies on the line

SM

.

2011 Ukraine Team Selection Test

Subcontests

| a_ {i +1} - a_ i |= 1given, wanted a _ 1 - a _ {2n} = n iff 1<= a _ {2k} <= n

S (x^ 3) - S^2(x) <1/4 (P ^2 (x^3) +Q(x^3) , where S (x) = P (x) Q (x)

any natural number can be given as a_i+a_j, a _n>n^2 /16, increasing

a_i+ a_i* =a _ {i-1} +a_ {i + 1}

n different integers in (k^n,(k+1)^n), product of which is n-th power of integer

sum \sin (<A _i MA_i)> sin 2 \pi/n + (n-2) sin \pi /n

concurrency wanted, <APE =<BAC, <CQF = < BCA, orthocenter,

circle tangent to 2 lines and 2 circumcircles, cyclic, <PBC = <PDA, <PCB =<PAD

3 collinear leads to 4 collinear , incircle related

2011 Ukraine Team Selection Test

Subcontests

| a_ {i +1} - a_ i |= 1given, wanted a _ 1 - a _ {2n} = n iff 1&lt;= a _ {2k} &lt;= n

S (x^ 3) - S^2(x) &lt;1/4 (P ^2 (x^3) +Q(x^3) , where S (x) = P (x) Q (x)

any natural number can be given as a_i+a_j, a _n&gt;n^2 /16, increasing

a_i+ a_i* =a _ {i-1} +a_ {i + 1}

n different integers in (k^n,(k+1)^n), product of which is n-th power of integer

sum \sin (&lt;A _i MA_i)&gt; sin 2 \pi/n + (n-2) sin \pi /n

concurrency wanted, &lt;APE =&lt;BAC, &lt;CQF = &lt; BCA, orthocenter,

circle tangent to 2 lines and 2 circumcircles, cyclic, &lt;PBC = &lt;PDA, &lt;PCB =&lt;PAD

3 collinear leads to 4 collinear , incircle related

| a_ {i +1} - a_ i |= 1given, wanted a _ 1 - a _ {2n} = n iff 1<= a _ {2k} <= n

S (x^ 3) - S^2(x) <1/4 (P ^2 (x^3) +Q(x^3) , where S (x) = P (x) Q (x)

any natural number can be given as a_i+a_j, a _n>n^2 /16, increasing

sum \sin (<A _i MA_i)> sin 2 \pi/n + (n-2) sin \pi /n

concurrency wanted, <APE =<BAC, <CQF = < BCA, orthocenter,

circle tangent to 2 lines and 2 circumcircles, cyclic, <PBC = <PDA, <PCB =<PAD