MathDB

1

figure on the sheet of squares cut into exactly in F_n$ ways

Let us call a figure on a sheet of squares an arbitrary finite set of connected squares, i.e. a set of squares in which it is possible to go from any square to any other by walking only on the squares of this figure. Prove that for every natural n there exists such a figure on the sheet of squares that it can be cut into "corners" (Fig. 1) exactly in

F_n

ways, where

F_n

s the

n

-th Fibonacci number (in the series of Fibonacci numbers

F_1 = F_2 = 1

and for each

i > 1

holds

F_{i+2} = F_{i+1} + F_i

). For example, a rectangle of

2 \times 3

squares can be cut at the corners in exactly two ways (Fig.

2

). https://cdn.artofproblemsolving.com/attachments/6/5/c82340623ff5f92a410bd73755ba8cbdc501ff.png

10

1

hat in 1 of 100 colors, 1 girl is known to 2015 boys, 1 boy knows 2015 girls

Is it true that for all natural

n

, it is always possible to give each of the

n

children a hat painted in one of

100

colors so that if a girl is known to more than

2015

boys, then not all of these boys have hats of the same color, and, if a boy is acquainted with more than

2015

girls, don't all these girls have hats of the same color?
[hide=original wording]Vai tiesa, ka visiem naturāliem n vienmēr iespējams katram no n bērniem iedot pa cepurei, kas nokrāsota vienā no 100 krāsām tā, ka, ja kāda meitene ir pazīstama ar vairāk nekā 2015 zēniem, tad ne visiem šiem zēniem cepures ir vienā krāsā, un, ja kāds zēns ir pazīštams ar vairāk nekā 2015 meitenēm, tad ne visām šīm meitenēm cepures ir vienā krāsā?

4

1

diagonals in a convex 2014-gon

Can you draw some diagonals in a convex

2014

-gon so that they do not intersect, the whole

2014

-gon is divided into triangles and each vertex belongs to an odd number of these triangles?

2

1

f(2x - f (x)) = x, f(x) > x, f(x) > f(y) for all real x > y

It is known about the function

f : R \to R

that

\bullet

f(x) > f(y)

for all real

x > y

\bullet

f(x) > x

for all real

x

\bullet

f(2x - f (x)) = x

for all real

x

. Prove that

f(x) = x + f(0)

for all real numbers

x

.

1

x^4 y^2 + y^4 + 2 x^3 y + 6 x^2 y + x^2 + 8 <= 0

Given real numbers

x

and

y

, such that

x^4 y^2 + y^4 + 2 x^3 y + 6 x^2 y + x^2 + 8 \le 0 .

Prove that

x \ge - \frac16

3

1

integer P, x^m + x + 2 = P(P(x)) , for all integers x

Prove that there does not exist a polynomial

P (x)

with integer coefficients and a natural number

m

such that

x^m + x + 2 = P(P(x))

holds for all integers

x

.

12

1

sum a^{2014}/(1 + 2 bc) >=3/ (ab+bc+ca) for a,b,c<0 with abc = 1

For real positive numbers

a, b, c

, the equality

abc = 1

holds. Prove that

\frac{a^{2014}}{1 + 2 bc}+\frac{b^{2014}}{1 + 2ac}+\frac{c^{2014}}{1 + 2ab} \ge \frac{3}{ab+bc+ca}.

14

1

S (n^2)/S (n) = R, sum of digits

Let

S(a)

denote the sum of the digits of the number

a

. Given a natural

R

can one find a natural

n

such that

\frac{S (n^2)}{S (n)}= R

?

15

1

w(n) < w(n + 1) < w(n + 2), no of different primes that divide n

Let

w (n)

denote the number of different prime numbers by which

n

is divisible. Prove that there are infinitely many natural numbers

n

such that

w(n) < w(n + 1) < w(n + 2)

.

13

1

[an+b] is prime for all natural n

Are there positive real numbers

a

and

b

such that

[an+b]

is prime for all natural values of

n

?

[x]

denotes the integer part of the number

x

, the largest integer that does not exceed

x

.

9

1

coloring squares on NxN board in turns, no 2 squares lie on same diagonal

Two players play the following game on a square of

N \times N

squares. They color one square in turn so that no two colored squares are on the same diagonal. A player who cannot make a move loses. For what values of

N

does the first player have a winning strategy?

8

1

charismatic numbers x= prod (q+i)^{a_i}

Given a fixed rational number

q

. Let's call a number

x

charismatic if we can find a natural number

n

and integers

a_1, a_2,.., a_n

such that

x = (q + 1)^{a_1} \cdot (q + 2)^{a_2} \cdot ... \cdot(q + n)^{a_n} .

i) Prove that one can find a

q

such that all positive rational numbers are charismatic. ii) Is it true that for all

q

, if the number

x

is charismatic, then

x + 1

is also charismatic?

7

1

O_1O_2 __|_ AB, starting with intersecting circles

Two circle

\Gamma_1

and

\Gamma_2

intersect at points

A

and

B

, point

P

is not on the line

AB

. Line

AP

intersects again

\Gamma_1

and

\Gamma_2

at points

K

and

L

respectively, line

BP

intersects again

\Gamma_1

and

\Gamma_2

at points

M

and

N

respectively and all the points mentioned so far are different. The centers of the circles circumscribed around the triangles

KMP

and

LNP

are

O_1

and

O_2

respectively. Prove that

O_1O_2

is perpendicular to

AB

.

6

1

median bisects B_1C_1, projections of vertices on angle bisectors

AM

is the median of triangle

ABC

. A perpendicular

CC_1

is drawn from point

C

on the bisector of angle

\angle CMA

, a perpendicular

BB_1

is drawn from point

B

on the bisector of angle

\angle BMA

. Prove that line

AM

intersects segment

B_1C_1

at its midpoint.

5

1

EF // AB wanted, line tangent to (ABC)

BE

is the altitude of acute triangle

ABC

. The line

\ell

touches the circumscribed circle of the triangle

ABC

at point

B

. A perpendicular

CF

is drawn from

C

on line

\ell

. Prove that the lines

EF

and

AB

are parallel.

16

1

AB=CD wanted, 3 circles of diameters XZ, XY , YZ related where X-Y-Z (collinear)

Points

X

,

Y

,

Z

lie on a line

k

in this order. Let

\omega_1

,

\omega_2

,

\omega_3

be three circles of diameters

XZ

,

XY

,

YZ

, respectively. Line

\ell

passing through point

Y

intersects

\omega_1

at points

A

and

D

,

\omega_2

at

B

and

\omega_3

at

C

in such manner that points

A, B, Y, X, D

lie on

\ell

in this order. Prove that

AB =CD

.

2015 Latvia Baltic Way TST

Subcontests

figure on the sheet of squares cut into exactly in F_n$ ways

hat in 1 of 100 colors, 1 girl is known to 2015 boys, 1 boy knows 2015 girls

diagonals in a convex 2014-gon

f(2x - f (x)) = x, f(x) > x, f(x) > f(y) for all real x > y

x^4 y^2 + y^4 + 2 x^3 y + 6 x^2 y + x^2 + 8 <= 0

integer P, x^m + x + 2 = P(P(x)) , for all integers x

sum a^{2014}/(1 + 2 bc) >=3/ (ab+bc+ca) for a,b,c<0 with abc = 1

S (n^2)/S (n) = R, sum of digits

w(n) < w(n + 1) < w(n + 2), no of different primes that divide n

[an+b] is prime for all natural n

coloring squares on NxN board in turns, no 2 squares lie on same diagonal

charismatic numbers x= prod (q+i)^{a_i}

O_1O_2 __|_ AB, starting with intersecting circles

median bisects B_1C_1, projections of vertices on angle bisectors

EF // AB wanted, line tangent to (ABC)

AB=CD wanted, 3 circles of diameters XZ, XY , YZ related where X-Y-Z (collinear)

2015 Latvia Baltic Way TST

Subcontests

figure on the sheet of squares cut into exactly in F_n$ ways

hat in 1 of 100 colors, 1 girl is known to 2015 boys, 1 boy knows 2015 girls

diagonals in a convex 2014-gon

f(2x - f (x)) = x, f(x) &gt; x, f(x) &gt; f(y) for all real x &gt; y

x^4 y^2 + y^4 + 2 x^3 y + 6 x^2 y + x^2 + 8 &lt;= 0

integer P, x^m + x + 2 = P(P(x)) , for all integers x

sum a^{2014}/(1 + 2 bc) &gt;=3/ (ab+bc+ca) for a,b,c&lt;0 with abc = 1

S (n^2)/S (n) = R, sum of digits

w(n) &lt; w(n + 1) &lt; w(n + 2), no of different primes that divide n

[an+b] is prime for all natural n

coloring squares on NxN board in turns, no 2 squares lie on same diagonal

charismatic numbers x= prod (q+i)^{a_i}

O_1O_2 __|_ AB, starting with intersecting circles

median bisects B_1C_1, projections of vertices on angle bisectors

EF // AB wanted, line tangent to (ABC)

AB=CD wanted, 3 circles of diameters XZ, XY , YZ related where X-Y-Z (collinear)

f(2x - f (x)) = x, f(x) > x, f(x) > f(y) for all real x > y

x^4 y^2 + y^4 + 2 x^3 y + 6 x^2 y + x^2 + 8 <= 0

sum a^{2014}/(1 + 2 bc) >=3/ (ab+bc+ca) for a,b,c<0 with abc = 1

w(n) < w(n + 1) < w(n + 2), no of different primes that divide n