MathDB

1

Infinite subset A of the naturals with coprime condition

n

for which there exists an infinite subset

A

of the set

\mathbb{N}

of positive integers such that for all pairwise distinct

a_1,\ldots , a_n \in A

the numbers

a_1+\ldots +a_n

and

a_1a_2\ldots a_n

are coprime.

19

1

Sum of the squares of k primes

For which

k

do there exist

k

pairwise distinct primes

p_1,p_2,\ldots ,p_k

such that

p_1^2+p_2^2+\ldots +p_k^2=2010?

18

1

Maximum number of distinct elements in sequence of inverses

Let

p

be a prime number. For each

k

,

1\le k\le p-1

, there exists a unique integer denoted by

k^{-1}

such that

1\le k^{-1}\le p-1

and

k^{-1}\cdot k=1\pmod{p}

. Prove that the sequence 1^{-1}, 1^{-1}+2^{-1}, 1^{-1}+2^{-1}+3^{-1}, \ldots , 1^{-1}+2^{-1}+\ldots +(p-1)^{-1} (addition modulo

p

) contains at most

\frac{p+1}{2}

distinct elements.

17

1

Decimal representation of n^2 has only odd digits

Find all positive integers

n

such that the decimal representation of

n^2

consists of odd digits only.

16

1

Find the smallest odd amusing integer

For a positive integer

k

, let

d(k)

denote the number of divisors of

k

and let

s(k)

denote the digit sum of

k

. A positive integer

n

is said to be amusing if there exists a positive integer

k

such that

d(k)=s(k)=n

. What is the smallest amusing odd integer greater than

1

?

15

1

Find angle MXN

The points

M

and

N

are chosen on the angle bisector

AL

of a triangle

ABC

such that

\angle ABM=\angle ACN=23^{\circ}

.

X

is a point inside the triangle such that

BX=CX

and

\angle BXC=2\angle BML

. Find

\angle MXN

.

14

1

Midpoint of XY is foot of altitude from C

Assume that all angles of a triangle

ABC

are acute. Let

D

and

E

be points on the sides

AC

and

BC

of the triangle such that

A, B, D,

and

E

lie on the same circle. Further suppose the circle through

D,E,

and

C

intersects the side

AB

in two points

X

and

Y

. Show that the midpoint of

XY

is the foot of the altitude from

C

to

AB

.

13

1

Determine all possible values of angle CAB

In an acute triangle

ABC

, the segment

CD

is an altitude and

H

is the orthocentre. Given that the circumcentre of the triangle lies on the line containing the bisector of the angle

DHB

, determine all possible values of

\angle CAB

.

12

1

Sides of a quadrilateral in arithmetic progression

Let

ABCD

be a convex quadrilateral with precisely one pair of parallel sides.

(a)

Show that the lengths of its sides

AB,BC,CD, DA

(in this order) do not form an arithmetic progression.

(b)

Show that there is such a quadrilateral for which the lengths of its sides

AB ,BC,CD,DA

form an arithmetic progression after the order of the lengths is changed.

11

1

Centre of square is collinear with intersections of k and k'

Let

ABCD

be a square and let

S

be the point of intersection of its diagonals

AC

and

BD

. Two circles

k,k'

go through

A,C

and

B,D

; respectively. Furthermore,

k

and

k'

intersect in exactly two different points

P

and

Q

. Prove that

S

lies on

PQ

.

10

1

Least possible number of black triangles

Let

n

be an integer with

n\ge 3

. Consider all dissections of a convex

n

-gon into triangles by

n-3

non-intersecting diagonals, and all colourings of the triangles with black and white so that triangles with a common side are always of a different colour. Find the least possible number of black triangles.

9

1

Player who takes the last match wins

There is a pile of

1000

matches. Two players each take turns and can take

1

to

5

matches. It is also allowed at most

10

times during the whole game to take

6

matches, for example

7

exceptional moves can be done by the first player and

3

moves by the second and then no more exceptional moves are allowed. Whoever takes the last match wins. Determine which player has a winning strategy.

8

1

Receiving hats from fellow members

In a club with

30

members, every member initially had a hat. One day each member sent his hat to a different member (a member could have received more than one hat). Prove that there exists a group of

10

members such that no one in the group has received a hat from another one in the group.

7

1

Round trips that miss the capital cost the same

There are some cities in a country; one of them is the capital. For any two cities

A

and

B

there is a direct flight from

A

to

B

and a direct flight from

B

to

A

, both having the same price. Suppose that all round trips with exactly one landing in every city have the same total cost. Prove that all round trips that miss the capital and with exactly one landing in every remaining city cost the same.

6

1

Attacking rooks on a coloured chessboard

An

n\times n

board is coloured in

n

colours such that the main diagonal (from top-left to bottom-right) is coloured in the first colour; the two adjacent diagonals are coloured in the second colour; the two next diagonals (one from above and one from below) are coloured in the third colour, etc; the two corners (top-right and bottom-left) are coloured in the

n

-th colour. It happens that it is possible to place on the board

n

rooks, no two attacking each other and such that no two rooks stand on cells of the same colour. Prove that

n=0\pmod{4}

or

n=1\pmod{4}

.

5

1

Find all functions

Let

\mathbb{R}

denote the set of real numbers. Find all functions

f:\mathbb{R}\rightarrow\mathbb{R}

such that

f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y)

for all

x,y\in\mathbb{R}

.

4

1

Find all polynomials satisyfing product condition

Find all polynomials

P(x)

with real coefficients such that

(x-2010)P(x+67)=xP(x)

for every integer

x

.

3

1

Sum of fractions is less than 2n-1

Let

x_1, x_2, \ldots ,x_n(n\ge 2)

be real numbers greater than

1

. Suppose that

|x_i-x_{i+1}|<1

for

i=1, 2,\ldots ,n-1

. Prove that

\frac{x_1}{x_2}+\frac{x_2}{x_3}+\ldots +\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1}<2n-1

2

1

cos, sin, tan and cot inequality

Let

x

be a real number such that

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

. Prove that

\cos^2(x)\cot (x)+\sin^2(x)\tan (x)\ge 1

1

Symmetric simultaneous equations of degree 2010

Find all quadruples of real numbers

(a,b,c,d)

satisfying the system of equations

\begin{cases}(b+c+d)^{2010}=3a\\ (a+c+d)^{2010}=3b\\ (a+b+d)^{2010}=3c\\ (a+b+c)^{2010}=3d\end{cases}

2010 Baltic Way

Subcontests

Infinite subset A of the naturals with coprime condition

Sum of the squares of k primes

Maximum number of distinct elements in sequence of inverses

Decimal representation of n^2 has only odd digits

Find the smallest odd amusing integer

Find angle MXN

Midpoint of XY is foot of altitude from C

Determine all possible values of angle CAB

Sides of a quadrilateral in arithmetic progression

Centre of square is collinear with intersections of k and k'

Least possible number of black triangles

Player who takes the last match wins

Receiving hats from fellow members

Round trips that miss the capital cost the same

Attacking rooks on a coloured chessboard

Find all functions

Find all polynomials satisyfing product condition

Sum of fractions is less than 2n-1

cos, sin, tan and cot inequality

Symmetric simultaneous equations of degree 2010