MathDB

1

Concurrency in a cyclic quadrilateral

ABCD

be a cyclic quadrilateral with

AB

and

CD

not parallel. Let

M

be the midpoint of

CD.

Let

P

be a point inside

ABCD

such that

P A = P B = CM.

Prove that

AB, CD

and the perpendicular bisector of

MP

are concurrent.

19

1

Transformations on lattice points

Consider triangles in the plane where each vertex has integer coordinates. Such a triangle can be legally transformed by moving one vertex parallel to the opposite side to a different point with integer coordinates. Show that if two triangles have the same area, then there exists a series of legal transformations that transforms one to the other.

18

1

Find an angle

Let

ABCD

be a parallelogram such that

\angle BAD = 60^{\circ}.

Let

K

and

L

be the midpoints of

BC

and

CD,

respectively. Assuming that

ABKL

is a cyclic quadrilateral, find

\angle ABD.

17

1

Inequality in quadrilateral

Let

ABCD

be a convex quadrilateral with

AB = AD.

Let

T

be a point on the diagonal

AC

such that

\angle ABT + \angle ADT = \angle BCD.

Prove that

AT + AC \geq AB + AD.

16

1

Parallel lines

In triangle

ABC,

the points

D

and

E

are the intersections of the angular bisectors from

C

and

B

with the sides

AB

and

AC,

respectively. Points

F

and

G

on the extensions of

AB

and

AC

beyond

B

and

C,

respectively, satisfy

BF = CG = BC.

Prove that

F G \parallel DE.

15

1

Connections between harbours

The Baltic Sea has

2016

harbours. There are two-way ferry connections between some of them. It is impossible to make a sequence of direct voyages

C_1 - C_2 - ... - C_{1062}

where all the harbours

C_1, . . . , C_{1062}

are distinct. Prove that there exist two disjoint sets

A

and

B

of

477

harbours each, such that there is no harbour in

A

with a direct ferry connection to a harbour in

B.

14

1

Combinatorics in a cube

A cube consists of

4^3

unit cubes each containing an integer. At each move, you choose a unit cube and increase by

1

all the integers in the neighbouring cubes having a face in common with the chosen cube. Is it possible to reach a position where all the

4^3

integers are divisible by

3,

no matter what the starting position is?

13

1

Numbers on blackboard

Let

n

numbers all equal to

1

be written on a blackboard. A move consists of replacing two numbers on the board with two copies of their sum. It happens that after

h

moves all

n

numbers on the blackboard are equal to

m.

Prove that

h \leq \frac{1}{2}n \log_2 m.

12

1

Tiling a hexagon

Does there exist a hexagon (not necessarily convex) with side lengths

1, 2, 3, 4, 5, 6

(not necessarily in this order) that can be tiled with a)

31

b)

32

equilateral triangles with side length

1?

11

1

PHP type problem

Set

A

consists of

2016

positive integers. All prime divisors of these numbers are smaller than

30.

Prove that there are four distinct numbers

a, b, c

and

d

in

A

such that

abcd

is a perfect square.

10

1

Sequence and inequality

Let

a_{0,1}, a_{0,2}, . . . , a_{0, 2016}

be positive real numbers. For

n\geq 0

and

1 \leq k < 2016

set

a_{n+1,k} = a_{n,k} +\frac{1}{2a_{n,k+1}} \ \ \text{and} \ \ a_{n+1,2016} = a_{n,2016} +\frac{1}{2a_{n,1}}.

Show that

\max_{1\leq k \leq 2016} a_{2016,k} > 44.

9

1

System of equations

Find all quadruples

(a, b, c, d)

of real numbers that simultaneously satisfy the following equations:

\begin{cases} a^3 + c^3 = 2 \\ a^2b + c^2d = 0 \\ b^3 + d^3 = 1 \\ ab^2 + cd^2 = -6.\end{cases}

8

1

Functional equation

Find all real numbers

a

for which there exists a non-constant function

f :\Bbb R \to \Bbb R

satisfying the following two equations for all

x\in \Bbb R:

i)

f(ax) = a^2f(x)

and ii)

f(f(x)) = a f(x).

7

1

Inequality and a constant

Find all positive integers

n

for which

3x^n + n(x + 2) - 3 \geq nx^2

holds for all real numbers

x.

6

1

Partition of {1,2,...,10} in three subsets

The set

\{1, 2, . . . , 10\}

is partitioned to three subsets

A, B

and

C.

For each subset the sum of its elements, the product of its elements and the sum of the digits of all its elements are calculated. Is it possible that

A

alone has the largest sum of elements,

B

alone has the largest product of elements, and

C

alone has the largest sum of digits?

5

1

Sequence with no constant subsequence mod p

Let

p > 3

be a prime such that

p\equiv 3 \pmod 4.

Given a positive integer

a_0

define the sequence

a_0, a_1, \ldots

of integers by

a_n = a^{2^n}_{n-1}

for all

n = 1, 2,\ldots.

Prove that it is possible to choose

a_0

such that the subsequence

a_N , a_{N+1}, a_{N+2}, \ldots

is not constant modulo

p

for any positive integer

N.

4

1

Divisibility

Let

n

be a positive integer and let

a, b, c, d

be integers such that

n | a + b + c + d

and

n | a^2 + b^2 + c^2 + d^2.

Show that

n | a^4 + b^4 + c^4 + d^4 + 4abcd.

3

1

Diophantine equation with parameter

For which integers

n = 1, \ldots , 6

does the equation

a^n + b^n = c^n + n

have a solution in integers?

2

1

Two conjectures

Prove or disprove the following hypotheses. a) For all

k \geq 2,

each sequence of

k

consecutive positive integers contains a number that is not divisible by any prime number less than

k.

b) For all

k\geq 2,

each sequence of

k

consecutive positive integers contains a number that is relatively prime to all other members of the sequence.

1

Classic diophantine equation

Find all pairs of primes

(p, q)

such that

p^3 - q^5 = (p + q)^2.

2016 Baltic Way

Subcontests

Concurrency in a cyclic quadrilateral

Transformations on lattice points

Find an angle

Inequality in quadrilateral

Parallel lines

Connections between harbours

Combinatorics in a cube

Numbers on blackboard

Tiling a hexagon

PHP type problem

Sequence and inequality

System of equations

Functional equation

Inequality and a constant

Partition of {1,2,...,10} in three subsets

Sequence with no constant subsequence mod p

Divisibility

Diophantine equation with parameter

Two conjectures

Classic diophantine equation