MathDB

1

Sum of digits of 12-digit number is not 76

12

-digit positive integer consisting only of digits

1,5

and

9

is divisible by

37

. Prove that the sum of its digits is not equal to

76

.

19

1

n^2 divides sum of every n consecutive elements

Does there exist a sequence

a_1,a_2,a_3,\ldots

of positive integers such that the sum of every

n

consecutive elements is divisible by

n^2

for every positive integer

n

?

18

1

A sequence of the last digit of n^(n^n) is periodic

For a positive integer

n

let

a_n

denote the last digit of

n^{(n^n)}

. Prove that the sequence

(a_n)

is periodic and determine the length of the minimal period.

16

1

Adding 2006 to the product of any two numbers gives a square

Are there

4

distinct positive integers such that adding the product of any two of them to

2006

yields a perfect square?

15

1

The line t through the centroid of triangle ABC

Let the medians of the triangle

ABC

intersect at point

M

. A line

t

through

M

intersects the circumcircle of

ABC

at

X

and

Y

so that

A

and

C

lie on the same side of

t

. Prove that

BX\cdot BY=AX\cdot AY+CX\cdot CY

.

14

1

2006 congruent pieces contain each one of 2006 points

There are

2006

points marked on the surface of a sphere. Prove that the surface can be cut into

2006

congruent pieces so that each piece contains exactly one of these points inside it.

13

1

A,D,F,E are concylic

In a triangle

ABC

, points

D,E

lie on sides

AB,AC

respectively. The lines

BE

and

CD

intersect at

F

. Prove that if
\color{white}\ . \ \color{black}\ BC^2=BD\cdot BA+CE\cdot CA,
then the points

A,D,F,E

lie on a circle.

10

1

Pluses and minuses in a 30 x 30 table

162

pluses and

144

minuses are placed in a

30\times 30

table in such a way that each row and each column contains at most

17

signs. (No cell contains more than one sign.) For every plus we count the number of minuses in its row and for every minus we count the number of pluses in its column. Find the maximum of the sum of these numbers.

6

1

Maximal size of a set of naturals with 5 conditions

Determine the maximal size of a set of positive integers with the following properties:

1.

The integers consist of digits from the set

\{ 1,2,3,4,5,6\}

.

2.

No digit occurs more than once in the same integer.

3.

The digits in each integer are in increasing order.

4.

Any two integers have at least one digit in common (possibly at different positions).

5.

There is no digit which appears in all the integers.

5

1

Professor's claims on a binary operation

An occasionally unreliable professor has devoted his last book to a certain binary operation

*

. When this operation is applied to any two integers, the result is again an integer. The operation is known to satisfy the following axioms:

\text{a})\ x*(x*y)=y

for all

x,y\in\mathbb{Z}

;

\text{b})\ (x*y)*y=x

for all

x,y\in\mathbb{Z}

.
The professor claims in his book that

1.

The operation

*

is commutative:

x*y=y*x

for all

x,y\in\mathbb{Z}

.

2.

The operation

*

is associative:

(x*y)*z=x*(y*z)

for all

x,y,z\in\mathbb{Z}

.
Which of these claims follow from the stated axioms?

4

1

Maximal value when a+b+c+d+e+f=6

Let

a,b,c,d,e,f

be non-negative real numbers satisfying

a+b+c+d+e+f=6

. Find the maximal possible value of
\color{white}\ . \ \color{black}\ abc+bcd+cde+def+efa+fab
and determine all

6

-tuples

(a,b,c,d,e,f)

for which this maximal value is achieved.

2

1

Sum of squares of a_i less than 2006

Suppose that the real numbers

a_i\in [-2,17],\ i=1,2,\ldots,59,

satisfy

a_1+a_2+\ldots+a_{59}=0.

Prove that

a_1^2+a_2^2+\ldots+a_{59}^2\le 2006

17

1

divisibility by n^2

Determine all positive integers

n

such that

3^{n}+1

is divisible by

n^{2}

.

9

1

numbers on a regular pentagon

To every vertex of a regular pentagon a real number is assigned. We may perform the following operation repeatedly: we choose two adjacent vertices of the pentagon and replace each of the two numbers assigned to these vertices by their arithmetic mean. Is it always possible to obtain the position in which all five numbers are zeroes, given that in the initial position the sum of all five numbers is equal to zero?

8

1

decreasing finite sequence.

The director has found out that six conspiracies have been set up in his department, each of them involving exactly

3

persons. Prove that the director can split the department in two laboratories so that none of the conspirative groups is entirely in the same laboratory.

7

1

let's take some photos

A photographer took some pictures at a party with

10

people. Each of the

45

possible pairs of people appears together on exactly one photo, and each photo depicts two or three people. What is the smallest possible number of photos taken?

1

classic sequence

For a sequence

(a_{n})_{n\geq 1}

of real numbers it is known that

a_{n}=a_{n-1}+a_{n+2}

for

n\geq 2

. What is the largest number of its consecutive elements that can all be positive?

12

1

segment equality

Let

ABC

be a triangle, let

B_{1}

be the midpoint of the side

AB

and

C_{1}

the midpoint of the side

AC

. Let

P

be the point of intersection, other than

A

, of the circumscribed circles around the triangles

ABC_{1}

and

AB_{1}C

. Let

P_{1}

be the point of intersection, other than

A

, of the line

AP

with the circumscribed circle around the triangle

AB_{1}C_{1}

. Prove that

2AP=3AP_{1}

.

11

1

Simple Geometry Problem

The altitudes of a triangle are

12

,

15

, and

20

. What is the area of this triangle?

3

1

polynomial as sum of 3rd degree

Prove that for every polynomial

P(x)

with real coefficients there exist a positive integer

m

and polynomials

P_{1}(x),\ldots , P_{m}(x)

with real coefficients such that

P(x) = (P_{1}(x))^{3}+\ldots +(P_{m}(x))^{3}

2006 Baltic Way

Subcontests

Sum of digits of 12-digit number is not 76

n^2 divides sum of every n consecutive elements

A sequence of the last digit of n^(n^n) is periodic

Adding 2006 to the product of any two numbers gives a square

The line t through the centroid of triangle ABC

2006 congruent pieces contain each one of 2006 points

A,D,F,E are concylic

Pluses and minuses in a 30 x 30 table

Maximal size of a set of naturals with 5 conditions

Professor's claims on a binary operation

Maximal value when a+b+c+d+e+f=6

Sum of squares of a_i less than 2006

divisibility by n^2

numbers on a regular pentagon

decreasing finite sequence.

let's take some photos

classic sequence

segment equality

Simple Geometry Problem

polynomial as sum of 3rd degree