2009 Danube Mathematical Competition

Part of Danube Competition in Mathematics

Subcontests

(5)

1

sum_{k=i}^{j} f(\sigma (k)) <= 2, for permutations of {1,..,n}

\sigma, \tau

be two permutations of the quantity

\{1, 2,. . . , n\}

. Prove that there is a function

f: \{1, 2,. . . , n\} \to \{-1, 1\}

such that for any

1 \le i \le j \le n

\left|\sum_{k=i}^{j} f(\sigma (k)) \right| \le 2

\left|\sum_{k=i}^{j} f(\tau (k))\right| \le 2

1

all positive integer, besides power of 2, are sums of consecutive naturals

Prove that all the positive integer numbers , except for the powers of

2

, can be written as the sum of (at least two) consecutive natural numbers .

1

min no ef equilaterals of side 1 to cover surface of equilateral side n+1/2n

n

be a natural number. Determine the minimal number of equilateral triangles of side

1

to cover the surface of an equilateral triangle of side

n +\frac{1}{2n}

1

Danube Mathematical Competition 2009

\triangle ABC

A'

B'

C'

be the foot of perpendiculars from

A

B

C

respectively. The points

E

F

are on the sides

CB'

BC'

respectively, such that

B'E\cdot C'F = BF\cdot CE

AEA'F

1

square root

a,b,c

positive integers.Prove that

|a-b\sqrt{c}|<\frac{1}{2b}

is true if and only if

|a^{2}-b^{2}c|<\sqrt{c}