MathDB

1

Squiggle is composed of six equilateral triangles

A squiggle is composed of six equilateral triangles with side length

1

as shown in the figure below. Determine all possible integers

n

such that an equilateral triangle with side length

n

can be fully covered with squiggles (rotations and reflections of squiggles are allowed, overlappings are not).
[asy] import graph; size(100); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; draw((0,0)--(0.5,1),linewidth(2pt)); draw((0.5,1)--(1,0),linewidth(2pt)); draw((0,0)--(3,0),linewidth(2pt)); draw((1.5,1)--(2,0),linewidth(2pt)); draw((2,0)--(2.5,1),linewidth(2pt)); draw((0.5,1)--(2.5,1),linewidth(2pt)); draw((1,0)--(2,2),linewidth(2pt)); draw((2,2)--(3,0),linewidth(2pt)); dot((0,0),ds); dot((1,0),ds); dot((0.5,1),ds); dot((2,0),ds); dot((1.5,1),ds); dot((3,0),ds); dot((2.5,1),ds); dot((2,2),ds); clip((-4.28,-10.96)--(-4.28,6.28)--(16.2,6.28)--(16.2,-10.96)--cycle);[/asy]

20

1

ab is a perfect cube if ab(a-b) divides a^3+b^3+ab

Let

a

and

b

be positive integers,

b<a

, such that

a^3+b^3+ab

is divisible by

ab(a-b)

. Prove that

ab

is a perfect cube.

19

1

A very 'nice' problem

Let

r

and

k

be positive integers such that all prime divisors of

r

are greater than

50

.
A positive integer, whose decimal representation (without leading zeroes) has at least

k

digits, will be called nice if every sequence of

k

consecutive digits of this decimal representation forms a number (possibly with leading zeroes) which is a multiple of

r

.
Prove that if there exist infinitely many nice numbers, then the number

10^k-1

is nice as well.

18

1

Must a,b,c,d have the same sign?

Let

a,b,c,d

be non-zero integers, such that the only quadruple of integers

(x, y, z, t)

satisfying the equation

ax^2+by^2+cz^2+dt^2=0

is

x=y=z=t=0

. Does it follow that the numbers

a,b,c,d

have the same sign?

17

1

Inequality with the gcd of the integers x,y,z

Let

x,y,z

be positive integers such that

\frac{x+1}{y}+\frac{y+1}{z}+\frac{z+1}{x}

is an integer. Let

d

be the greatest common divisor of

x,y

and

z

. Prove that

d\le \sqrt[3]{xy+yz+zx}

.

16

1

s can be written as a fraction with denominator coprime to 6

Let

a

and

b

be rational numbers such that

s=a+b=a^2+b^2

. Prove that

s

can be written as a fraction where the denominator is relatively prime to

6

.

15

1

Determine the ratio AD/DC

The incircle of the triangle

ABC

touches the side

AC

at the point

D

. Another circle passes through

D

and touches the rays

BC

and

BA

, the latter at the point

A

. Determine the ratio

\frac{AD}{DC}

.

14

1

ABCD must be cyclic (with diameter AC)

In a convex quadrilateral

ABCD

we have

ADC = 90^{\circ}

. Let

E

and

F

be the projections of

B

onto the lines

AD

and

AC

, respectively. Assume that

F

lies between

A

and

C

, that

A

lies between

D

and

E

, and that the line

EF

passes through the midpoint of the segment

BD

. Prove that the quadrilateral

ABCD

is cyclic.

13

1

Projections of points and lines

Let

t_1,t_2,\ldots,t_k

be different straight lines in space, where

k>1

. Prove that points

P_i

on

t_i

,

i=1,\ldots,k

, exist such that

P_{i+1}

is the projection of

P_i

on

t_{i+1}

for

i=1,\ldots,k-1

, and

P_1

is the projection of

P_k

on

t_1

.

12

1

MNK is a right angle

Let

M

be a point on the arc

AB

of the circumcircle of the triangle

ABC

which does not contain

C

. Suppose that the projections of

M

onto the lines

AB

and

BC

lie on the sides themselves, not on their extensions. Denote these projections by

X

and

Y

, respectively. Let

K

and

N

be the midpoints of

AC

and

XY

, respectively. Prove that

\angle MNK=90^{\circ}

.

11

1

S must be the incentre of triangle ABC

In triangle

ABC

let

AD,BE

and

CF

be the altitudes. Let the points

P,Q,R

and

S

fulfil the following requirements: i)

P

is the circumcentre of triangle

ABC

. ii) All the segments

PQ,QR

and

RS

are equal to the circumradius of triangle

ABC

. iii) The oriented segment

PQ

has the same direction as the oriented segment

AD

. Similarly,

QR

has the same direction as

BE

, and

Rs

has the same direction as

CF

. Prove that

S

is the incentre of triangle

ABC

.

10

1

Changing the colour of all cells in a column or row

We are given an

18\times 18

table, all of whose cells may be black or white. Initially all the cells are coloured white. We may perform the following operation: choose one column or one row and change the colour of all cells in this column or row. Is it possible by repeating the operation to obtain a table with exactly

16

black cells?

9

1

10 person board can make everyone happy

A society has to elect a board of governors. Each member of the society has chosen

10

candidates for the board, but he will be happy if at least one of them will be on the board. For each six members of the society there exists a board consisting of two persons making all of these six members happy. Prove that a board consisting of

10

persons can be elected making every member of the society happy.

8

1

Non-isolated subsets

Call a set

A

of integers non-isolated, if for every

a\in A

at least one of the numbers

a-1

and

a+1

also belongs to

A

. Prove that the number of five-element non-isolated subsets of

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

is

(n-4)^2

.

6

1

Action on the pair (i,j)

Freddy writes down numbers

1, 2,\ldots ,n

in some order. Then he makes a list of all pairs

(i, j)

such that

1\le i<j\le n

and the

i

-th number is bigger than the

j

-th number in his permutation. After that, Freddy repeats the following action while possible: choose a pair

(i, j)

from the current list, interchange the

i

-th and the

j

-th number in the current permutation, and delete

(i, j)

from the list. Prove that Freddy can choose pairs in such an order that, after the process finishes, the numbers in the permutation are in ascending order.

5

1

Determine all functions f

A function

f

is defined on the set of all real numbers except

0

and takes all real values except

1

. It is also known that
\color{white}\ . \ \color{black}\ f(xy)=f(x)f(-y)-f(x)+f(y)
for any

x,y\not= 0

and that
\color{white}\ . \ \color{black}\ f(f(x))=\frac{1}{f(\frac{1}{x})}
for any

x\not\in\{ 0,1\}

. Determine all such functions

f

.

4

1

Cauchy-Schwarz prone

Let

a_1,a_2,\ldots ,a_n

be positive real numbers, and let

S=a_1+a_2 +\ldots +a_n

. Prove that

(2S+n)(2S+a_1a_2+a_2a_3+\ldots +a_na_1)\ge 9(\sqrt{a_1a_2}+\sqrt{a_2a_3}+\ldots +\sqrt{a_na_1})^2

3

1

3 conditions on the polynomials F,G,H

Suppose that

F,G,H

are polynomials of degree at most

2n+1

with real coefficients such that: i) For all real

x

we have

F(x)\le G(x)\le H(x)

. ii) There exist distinct real numbers

x_1,x_2,\ldots ,x_n

such that F(x_i)=H(x_i) \text{for}\ i=1,2,3,\ldots ,n. iii) There exists a real number

x_0

different from

x_1,x_2,\ldots ,x_n

such that

F(x_0)+H(x_0)=2G(x_0)

. Prove that

F(x)+H(x)=2G(x)

for all real numbers

x

.

2

1

Determine a_2007 in an 'exact' sequence

A sequence of integers

a_1,a_2,a_3,\ldots

is called exact if

a_n^2-a_m^2=a_{n-m}a_{n+m}

for any

n>m

. Prove that there exists an exact sequence with

a_1=1,a_2=0

and determine

a_{2007}

.

1

Partitions of {1,2,3...2n} into 2-element subsets inequality

For a positive integer

n

consider any partition of the set

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

into

n

two-element subsets

P_1,P_2\ldots,P_n

. In each subset

P_i

, let

p_i

be the product of the two numbers in

P_i

. Prove that

\frac{1}{p_1}+\frac{1}{p_2}+\ldots + \frac{1}{p_n}<1

2007 Baltic Way

Subcontests

Squiggle is composed of six equilateral triangles

ab is a perfect cube if ab(a-b) divides a^3+b^3+ab

A very 'nice' problem

Must a,b,c,d have the same sign?

Inequality with the gcd of the integers x,y,z

s can be written as a fraction with denominator coprime to 6

Determine the ratio AD/DC

ABCD must be cyclic (with diameter AC)

Projections of points and lines

MNK is a right angle

S must be the incentre of triangle ABC

Changing the colour of all cells in a column or row

10 person board can make everyone happy

Non-isolated subsets

Action on the pair (i,j)

Determine all functions f

Cauchy-Schwarz prone

3 conditions on the polynomials F,G,H

Determine a_2007 in an 'exact' sequence

Partitions of {1,2,3...2n} into 2-element subsets inequality