MathDB

1

Inequality in a quadrilateral

A

and

C

of a convex quadrilateral

ABCD

is less than

180^{\circ} .

Prove that

AB \cdot CD + AD \cdot BC < AC(AB + AD).

14

1

Concyclic points

Let

ABCD

be a convex quadrilateral such that the line

BD

bisects the angle

ABC.

The circumcircle of triangle

ABC

intersects the sides

AD

and

CD

in the points

P

and

Q,

respectively. The line through

D

and parallel to

AC

intersects the lines

BC

and

BA

at the points

R

and

S,

respectively. Prove that the points

P, Q, R

and

S

lie on a common circle.

13

1

Triangles with equal areas

Let

ABCD

be a square inscribed in a circle

\omega

and let

P

be a point on the shorter arc

AB

of

\omega

. Let

CP\cap BD = R

and

DP \cap AC = S.

Show that triangles

ARB

and

DSR

have equal areas.

12

1

Equal segments again

Triangle

ABC

is given. Let

M

be the midpoint of the segment

AB

and

T

be the midpoint of the arc

BC

not containing

A

of the circumcircle of

ABC.

The point

K

inside the triangle

ABC

is such that

MATK

is an isosceles trapezoid with

AT\parallel MK.

Show that

AK = KC.

11

1

Equal segments

Let

\Gamma

be the circumcircle of an acute triangle

ABC.

The perpendicular to

AB

from

C

meets

AB

at

D

and

\Gamma

again at

E.

The bisector of angle

C

meets

AB

at

F

and

\Gamma

again at

G.

The line

GD

meets

\Gamma

again at

H

and the line

HF

meets

\Gamma

again at

I.

Prove that

AI = EB.

20

1

Periodic sequence

Consider a sequence of positive integers

a_1, a_2, a_3, . . .

such that for

k \geq 2

we have

a_{k+1} =\frac{a_k + a_{k-1}}{2015^i},

where

2015^i

is the maximal power of

2015

that divides

a_k + a_{k-1}.

Prove that if this sequence is periodic then its period is divisible by

3.

17

1

Equation on rationals

Do there exist pairwise distinct rational numbers

x, y

and

z

such that

\frac{1}{(x - y)^2}+\frac{1}{(y - z)^2}+\frac{1}{(z - x)^2}= 2014?

18

1

Number of quadruples and divisibility

Let

p

be a prime number, and let

n

be a positive integer. Find the number of quadruples

(a_1, a_2, a_3, a_4)

with

a_i\in \{0, 1, \ldots, p^n - 1\}

for

i = 1, 2, 3, 4

, such that

p^n \mid (a_1a_2 + a_3a_4 + 1).

19

1

Possible values of gcd

Let

m

and

n

be relatively prime positive integers. Determine all possible values of

\gcd(2^m - 2^n, 2^{m^2+mn+n^2}- 1).

16

1

Is 712! + 1 prime?

Determine whether

712! + 1

is a prime number.

10

1

Airline connections

In a country there are

100

airports. Super-Air operates direct flights between some pairs of airports (in both directions). The traffic of an airport is the number of airports it has a direct Super-Air connection with. A new company, Concur-Air, establishes a direct flight between two airports if and only if the sum of their traffics is at least

100.

It turns out that there exists a round-trip of Concur-Air flights that lands in every airport exactly once. Show that then there also exists a round-trip of Super-Air flights that lands in every airport exactly once.

9

1

Marking cells on a board

What is the least posssible number of cells that can be marked on an

n \times n

board such that for each

m >\frac{ n}{2}

both diagonals of any

m \times m

sub-board contain a marked cell?

8

1

Winning strategy

Albert and Betty are playing the following game. There are

100

blue balls in a red bowl and

100

red balls in a blue bowl. In each turn a player must make one of the following moves: a) Take two red balls from the blue bowl and put them in the red bowl. b) Take two blue balls from the red bowl and put them in the blue bowl. c) Take two balls of different colors from one bowl and throw the balls away. They take alternate turns and Albert starts. The player who first takes the last red ball from the blue bowl or the last blue ball from the red bowl wins. Determine who has a winning strategy.

7

1

Number of permutations

Let

p_1, p_2, . . . , p_{30}

be a permutation of the numbers

1, 2, . . . , 30.

For how many permutations does the equality

\sum^{30}_{k=1}|p_k - k| = 450

hold?

6

1

Coloring seats

In how many ways can we paint

16

seats in a row, each red or green, in such a way that the number of consecutive seats painted in the same colour is always odd?

3

1

Inequality, Baltic Way 2014, Problem 3

Positive real numbers

a, b, c

satisfy

\frac{1}{a} +\frac{1}{b} +\frac{1}{c} = 3.

Prove the inequality

\frac{1}{\sqrt{a^3+ b}}+\frac{1}{\sqrt{b^3 + c}}+\frac{1}{\sqrt{c^3 + a}}\leq \frac{3}{\sqrt{2}}.

5

1

Possible values of an expression

Given positive real numbers

a, b, c, d

that satisfy equalities

a^2 + d^2 - ad = b^2 + c^2 + bc \ \ \text{and} \ \ a^2 + b^2 = c^2 + d^2

find all possible values of the expression

\frac{ab+cd}{ad+bc}.

1

Trigonometric equality

Show that

\cos(56^{\circ}) \cdot \cos(2 \cdot 56^{\circ}) \cdot \cos(2^2\cdot 56^{\circ})\cdot . . . \cdot \cos(2^{23}\cdot 56^{\circ}) = \frac{1}{2^{24}} .

2

1

Sequence with non-positive terms

Let

a_0, a_1, . . . , a_N

be real numbers satisfying

a_0 = a_N = 0

and

a_{i+1} - 2a_i + a_{i-1} = a^2_i

for

i = 1, 2, . . . , N - 1.

Prove that

a_i\leq 0

for

i = 1, 2, . . . , N- 1.

4

1

Functional equation

Find all functions

f

defined on all real numbers and taking real values such that

f(f(y)) + f(x - y) = f(xf(y) - x),

for all real numbers

x, y.

2014 Baltic Way

Subcontests

Inequality in a quadrilateral

Concyclic points

Triangles with equal areas

Equal segments again

Equal segments

Periodic sequence

Equation on rationals

Number of quadruples and divisibility

Possible values of gcd

Is 712! + 1 prime?

Airline connections

Marking cells on a board

Winning strategy

Number of permutations

Coloring seats

Inequality, Baltic Way 2014, Problem 3

Possible values of an expression

Trigonometric equality

Sequence with non-positive terms

Functional equation