MathDB

1

Sixth Power and Seventh Power

Find all integer solutions of the equation

2x^6 + y^7 = 11

.

19

1

Infinitely Composite

Show that

n^n + (n + 1)^{n + 1}

is composite for infinitely many positive integers

n

.

18

1

20122012 as Sum of Squares

Find all triples

(a,b,c)

of integers satisfying

a^2 + b^2 + c^2 = 20122012

.

17

1

Number of Divisors in the Exponent

Let

d(n)

denote the number of positive divisors of

n

. Find all triples

(n,k,p)

, where

n

and

k

are positive integers and

p

is a prime number, such that

n^{d(n)} - 1 = p^k.

16

1

Product of Three Consecutive Integers

Let

n

,

m

, and

k

be positive integers satisfying

(n - 1)n(n + 1) = m^k

. Prove that

k = 1

.

15

1

Parallelogram from Cyclic Quadrilateral

The circumcentre

O

of a given cyclic quadrilateral

ABCD

lies inside the quadrilateral but not on the diagonal

AC

. The diagonals of the quadrilateral intersect at

I

. The circumcircle of the triangle

AOI

meets the sides

AD

and

AB

at points

P

and

Q

, respectively; the circumcircle of the triangle

COI

meets the sides

CB

and

CD

at points

R

and

S

, respectively. Prove that

PQRS

is a parallelogram.

14

1

Equal Angles at Tangent Points

Given a triangle

ABC

, let its incircle touch the sides

BC

,

CA

,

AB

at

D

,

E

,

F

, respectively. Let

G

be the midpoint of the segment

DE

. Prove that

\angle EFC = \angle GFD

.

13

1

Areas of Two Triangles

Let

ABC

be an acute triangle, and let

H

be its orthocentre. Denote by

H_A

,

H_B

, and

H_C

the second intersection of the circumcircle with the altitudes from

A

,

B

, and

C

respectively. Prove that the area of triangle

H_A H_B H_C

does not exceed the area of triangle

ABC

.

12

1

Minimal Sum with Cyclic Octagon

Let

P_0

,

P_1

,

\dots

,

P_8 = P_0

be successive points on a circle and

Q

be a point inside the polygon

P_0 P_1 \dotsb P_7

such that

\angle P_{i - 1} QP_i = 45^\circ

for

i = 1

,

\dots

, 8. Prove that the sum

\sum_{i = 1}^8 P_{i - 1} P_i^2

is minimal if and only if

Q

is the centre of the circle.

11

1

Equality with Fermat Point

Let

ABC

be a triangle with

\angle A = 60^\circ

. The point

T

lies inside the triangle in such a way that

\angle ATB = \angle BTC = \angle CTA = 120^\circ

. Let

M

be the midpoint of

BC

. Prove that

TA + TB + TC = 2AM

.

10

1

Game with Primes

Two players

A

and

B

play the following game. Before the game starts,

A

chooses 1000 not necessarily different odd primes, and then

B

chooses half of them and writes them on a blackboard. In each turn a player chooses a positive integer

n

, erases some primes

p_1

,

p_2

,

\dots

,

p_n

from the blackboard and writes all the prime factors of

p_1 p_2 \dotsm p_n - 2

instead (if a prime occurs several times in the prime factorization of

p_1 p_2 \dotsm p_n - 2

, it is written as many times as it occurs). Player

A

starts, and the player whose move leaves the blackboard empty loses the game. Prove that one of the two players has a winning strategy and determine who.
Remark: Since 1 has no prime factors, erasing a single 3 is a legal move.

9

1

Moves on a 5 x 5 board

Zeroes are written in all cells of a

5 \times 5

board. We can take an arbitrary cell and increase by 1 the number in this cell and all cells having a common side with it. Is it possible to obtain the number 2012 in all cells simultaneously?

8

1

Directed graph with no cycles

A directed graph does not contain directed cycles. The number of edges in any directed path does not exceed 99. Prove that it is possible to colour the edges of the graph in 2 colours so that the number of edges in any single-coloured directed path in the graph will not exceed 9.

7

1

Numbers in a 2012 x 2012 board

On a

2012 \times 2012

board, some cells on the top-right to bottom-left diagonal are marked. None of the marked cells is in a corner. Integers are written in each cell of this board in the following way. All the numbers in the cells along the upper and the left sides of the board are 1's. All the numbers in the marked cells are 0's. Each of the other cells contains a number that is equal to the sum of its upper neighbour and its left neighbour. Prove that the number in the bottom right corner is not divisible by 2011.

6

1

Game with Lamps

There are 2012 lamps arranged on a table. Two persons play the following game. In each move the player flips the switch of one lamp, but he must never get back an arrangement of the lit lamps that has already been on the table. A player who cannot move loses. Which player has a winning strategy?

5

1

Functional Equation

Find all functions

f : \mathbb{R} \rightarrow \mathbb{R}

for which

f(x + y) = f(x - y) + f(f(1 - xy))

holds for all real numbers

x

and

y

.

4

1

Roots with product 1

Prove that for infinitely many pairs

(a,b)

of integers the equation

x^{2012} = ax + b

has among its solutions two distinct real numbers whose product is 1.

3

1

Equation with Floor

(a) Show that the equation

\lfloor x \rfloor (x^2 + 1) = x^3,

where

\lfloor x \rfloor

denotes the largest integer not larger than

x

, has exactly one real solution in each interval between consecutive positive integers.
(b) Show that none of the positive real solutions of this equation is rational.

2

1

Inequality with Max

Let

a

,

b

,

c

be real numbers. Prove that

ab + bc + ca + \max\{|a - b|, |b - c|, |c - a|\} \le 1 + \frac{1}{3} (a + b + c)^2.

1

3 by 3 Magic Square

The numbers from 1 to 360 are partitioned into 9 subsets of consecutive integers and the sums of the numbers in each subset are arranged in the cells of a

3 \times 3

square. Is it possible that the square turns out to be a magic square?
Remark: A magic square is a square in which the sums of the numbers in each row, in each column and in both diagonals are all equal.

2012 Baltic Way

Subcontests

Sixth Power and Seventh Power

Infinitely Composite

20122012 as Sum of Squares

Number of Divisors in the Exponent

Product of Three Consecutive Integers

Parallelogram from Cyclic Quadrilateral

Equal Angles at Tangent Points

Areas of Two Triangles

Minimal Sum with Cyclic Octagon

Equality with Fermat Point

Game with Primes

Moves on a 5 x 5 board

Directed graph with no cycles

Numbers in a 2012 x 2012 board

Game with Lamps

Functional Equation

Roots with product 1

Equation with Floor

Inequality with Max

3 by 3 Magic Square