MathDB

1

Party of 8 where each knows exactly 3 others

In a party of eight people, each pair of people either know each other or do not know each other. Each person knows exactly three of the others. Determine whether the following two conditions can be satisfied simultaneously:
– for any three people, at least two do not know each other; – for any four people there are at least two who know each other.

16

1

Number of n-trønder walks

A

n

-trønder walk is a walk starting at

(0, 0)

, ending at

(2n, 0)

with no self intersection and not leaving the first quadrant, where every step is one of the vectors

(1, 1)

,

(1, -1)

or

(-1, 1)

. Find the number of

n

-trønder walks.

15

1

Unit square cut into m quadrilaterals inequality

A unit square is cut into

m

quadrilaterals

Q_1,\ldots ,Q_m

. For each

i=1,\ldots ,m

let

S_i

be the sum of the squares of the four sides of

Q_i

. Prove that

S_1+\ldots +S_m\ge 4

14

1

n paiwise non-similar triangles

For which

n\ge 2

is it possible to find

n

pairwise non-similar triangles

A_1, A_2,\ldots , A_n

such that each of them can be divided into

n

A_1,A_2 ,\ldots ,A_n

?

12

1

AP is perpendicular to BQ

In a quadrilateral

ABCD

we have

AB||CD

and

AB=2CD

. A line

\ell

is perpendicular to

CD

and contains the point

C

. The circle with centre

D

and radius

DA

intersects the line

\ell

at points

P

and

Q

. Prove that

AP\perp BQ

.

11

1

Projection of M onto the line l is on PK

Let

M

be the midpoint of the side

AC

of a triangle

ABC

, and let

K

be a point on the ray

BA

beyond

A

. The line

KM

intersects the side

BC

at the point

L

.

P

is the point on the segment

BM

such that

PM

is the bisector of the angle

LPK

. The line

\ell

passes through

A

and is parallel to

BM

. Prove that the projection of the point

M

onto the line

\ell

belongs to the line

PK

.

8

1

Extension of a classic - product of elements of subsets

Determine all positive integers

n

for which there exists a partition of the set

\{n,n+1,n+2,\ldots ,n+8\}

into two subsets such that the product of all elements of the first subset is equal to the product of all elements of the second subset.

9

1

Simple difference of two squares problem

Determine all positive integers

n

for which

2^{n+1}-n^2

is a prime number.

10

1

Expression with d(n)

Let

d(k)

denote the number of positive divisors of a positive integer

k

. Prove that there exist infinitely many positive integers

M

that cannot be written as

M=\left(\frac{2\sqrt{n}}{d(n)}\right)^2

for any positive integer

n

.

13

1

concurrent problem

The point

H

is the orthocentre of a triangle

ABC

, and the segments

AD,BE,CF

are its altitudes. The points

I_1,I_2,I_3

are the incentres of the triangles

EHF,FHD,DHE

respectively. Prove that the lines

AI_1,BI_2,CI_3

intersect at a single point.

7

1

Baltic way 2009 /7

Suppose that for a prime number

p

and integers

a,b,c

the following holds: 6\mid p+1, p\mid a+b+c, p\mid a^4+b^4+c^4. Prove that

p\mid a,b,c

.

20

1

Baltic Way 2009 problem 20

In the future city Baltic Way there are sixteen hospitals. Every night exactly four of them must be on duty for emergencies. Is it possible to arrange the schedule in such a way that after twenty nights every pair of hospitals have been on common duty exactly once?

6

1

Coefficient of a cubic equations

Let

a

and

b

be integers such that the equation x^3\minus{}ax^2\minus{}b\equal{}0 has three integer roots. Prove that b\equal{}dk^2, where

d

and

k

are integers and

d

divides

a

.

4

1

Find n so that inequality holds

Determine all integers

n>1

for which the inequality x_1^2\plus{}x_2^2\plus{}\ldots\plus{}x_n^2\ge(x_1\plus{}x_2\plus{}\ldots\plus{}x_{n\minus{}1})x_n holds for all real

x_1,x_2,\ldots,x_n

.

3

1

Choosing signs for sum of squares

Let

n

be a given positive integer. Show that we can choose numbers c_k\in\{\minus{}1,1\} (

i\le k\le n

) such that 0\le\sum_{k\equal{}1}^nc_k\cdot k^2\le4.

17

1

Find the largest n

Find the largest integer

n

for which there exist

n

different integers such that none of them are divisible by either of

7,11

or

13

, but the sum of any two of them is divisible by at least one of

7,11

and

13

.

5

1

equation

Let

f_0=f_1=1

and

f_{i+2}=f_{i+1}+f_i

for all

n\ge 0

. Find all real solutions to the equation

x^{2010}=f_{2009}\cdot x+f_{2008}

18

1

Smallest value of m

Let

n>2

be an integer. In a country there are

n

cities and every two of them are connected by a direct road. Each road is assigned an integer from the set

\{1, 2,\ldots ,m\}

(different roads may be assigned the same number). The priority of a city is the sum of the numbers assigned to roads which lead to it. Find the smallest

m

for which it is possible that all cities have a different priority.

2

1

Inequality

Let

a_1,a_{2},\ldots ,a_{100}

be nonnegative integers satisfying the inequality

a_1\cdot (a_1-1)\cdot\ldots\cdot (a_1-20)+a_2\cdot (a_2-1)\cdot\ldots\cdot (a_2-20)+\\ \ldots+a_{100}\cdot (a_{100}-1)\cdot\ldots\cdot (a_{100}-20)\le 100\cdot 99\cdot 98\cdot\ldots\cdot 79.

Prove that

a_1+a_2+\ldots+a_{100}\le 9900

.

1

Values of p(1)

A polynomial

p(x)

of degree

n\ge 2

has exactly

n

real roots, counted with multiplicity. We know that the coefficient of

x^n

is

1

, all the roots are less than or equal to

1

, and

p(2)=3^n

. What values can

p(1)

take?

2009 Baltic Way

Subcontests

Party of 8 where each knows exactly 3 others

Number of n-trønder walks

Unit square cut into m quadrilaterals inequality

n paiwise non-similar triangles

AP is perpendicular to BQ

Projection of M onto the line l is on PK

Extension of a classic - product of elements of subsets

Simple difference of two squares problem

Expression with d(n)

concurrent problem

Baltic way 2009 /7

Baltic Way 2009 problem 20

Coefficient of a cubic equations

Find n so that inequality holds

Choosing signs for sum of squares

Find the largest n

equation

Smallest value of m

Inequality

Values of p(1)