MathDB

1

Estonian Math Competitions 2005/2006

n \ge 2

be a fixed integer and let

a_{i,j} (1 \le i < j \le n)

be some positive integers. For a sequence

x_1, ... , x_n

of reals, let

K(x_1, .... , x_n)

be the product of all expressions (x_i \minus{} x_j)^{a_{i,j}} where

1 \le i < j \le n

. Prove that if the inequality

K(x_1, .... , x_n) \ge 0

holds independently of the choice of the sequence

x_1, ... , x_n

then all integers

a_{i,j}

are even.

9

1

Estonian Math Competitions 2005/2006

In the sequence

(a_n)

with general term a_n \equal{} n^3 \minus{} (2n \plus{} 1)^2, does there exist a term that is divisible by 2006?

8

1

Estonian Math Competitions 2005/2006

Four points

A, B, C, D

are chosen on a circle in such a way that arcs

AB, BC,

and

CD

are of the same length and the

arc DA

is longer than these three. Line

AD

and the line tangent to the circle at

B

intersect at

E

. Let

F

be the other endpoint of the diameter starting at

C

of the circle. Prove that triangle

DEF

is equilateral.

7

1

Estonian Math Competitions 2005/2006

A real-valued function

f

satisfies for all reals

x

and

y

the equality f (xy) \equal{} f (x)y \plus{} x f (y). Prove that this function satisfies for all reals

x

and

y \ne 0

the equality f\left(\frac{x}{y}\right)\equal{}\frac{f (x)y \minus{} x f (y)}{y^2}

6

1

Estonian Math Competitions 2005/2006

Kati cut two equal regular n\minus{}gons out of paper. To the vertices of both n\minus{}gons, she wrote the numbers 1 to

n

in some order. Then she stabbed a needle through the centres of these n\minus{}gons so that they could be rotated with respect to each other. Kati noticed that there is a position where the numbers at each pair of aligned vertices are different. Prove that the n\minus{}gons can be rotated to a position where at least two pairs of aligned vertices contain equal numbers.

5

1

Estonian Math Competitions 2005/2006

Two players A and B play the following game. Initially, there are

m

equal positive integers

n

written on a blackboard. A begins and the players move alternately. The player to move chooses one of the non-zero numbers on the board. If this number k is the smallest among all positive integers on the board, the player replaces it with k\minus{}1; if not, the player replaces it with the smallest positive number on the board. The player who first turns all the numbers into zeroes, wins. Who wins if both players use their best strategies?

4

1

Estonian Math Competitions 2005/2006

Martin invented the following algorithm. Let two irreducible fractions

\frac{s_1}{t_1}

and

\frac{s_2}{t_2}

be given as inputs, with the numerators and denominators being positive integers. Divide

s_1

and

s_2

by their greatest common divisor

c

and obtain

a_1

and

a_2

, respectively. Similarily, divide

t_1

and

t_2

by their greatest common divisor

d

and obtain

b_1

and

b_2

, respectively. After that, form a new fraction \frac{a_1b_2 \plus{} a_2b_1}{t_1b_2}, reduce it, and multiply the numerator of the result by

c

. Martin claims that this algorithm always finds the sum of the original fractions as an irreducible fraction. Is his claim correct?

3

1

Estonian Math Competitions 2005/2006

Let

ABC

be an acute triangle and choose points

A_1, B_1

and

C_1

on sides

BC, CA

and

AB

, respectively. Prove that if the quadrilaterals

ABA_1B_1, BCB_1C_1

and

CAC_1A_1

are cyclic then their circumcentres lie on the sides of

ABC

.

2

1

Estonian Math Competitions 2005/2006

After the schoolday is over, Juku must attend an extra math class. The teacher writes a quadratic equation x^2\plus{} p_1x\plus{}q_1 \equal{} 0 with integer coefficients on the blackboard and Juku has to find its solutions. If they are not both integers, Jukumay go home. If the solutions are integers, then the teacher writes a new equation x^2 \plus{} p_2x \plus{} q_2 \equal{} 0, where

p_2

and

q_2

are the solutions of the previous equation taken in some order, and everything starts all over. Find all possible values for

p_1

and

q_1

such that the teacher can hold Juku at school forever.

1

Estonian Math Competitions 2005/2006

All the streets in a city run in one of two perpendicular directions, forming unit squares. Organizers of a car race want to mark down a closed race track in the city in such a way that it would not go through any of the crossings twice and that the track would turn 90◦ right or left at every crossing. Find all possible values of the length of the track.

2006 Estonia Math Open Senior Contests

Subcontests

Estonian Math Competitions 2005/2006

Estonian Math Competitions 2005/2006

Estonian Math Competitions 2005/2006

Estonian Math Competitions 2005/2006

Estonian Math Competitions 2005/2006

Estonian Math Competitions 2005/2006

Estonian Math Competitions 2005/2006

Estonian Math Competitions 2005/2006

Estonian Math Competitions 2005/2006

Estonian Math Competitions 2005/2006