1990 Nordic

Part of Nordic

Subcontests

(4)

1

sum from k=1 to (n-1)^p, of k^m

m, n,

p

be odd positive integers. Prove that the number

\sum\limits_{k=1}^{{{(n-1)}^{p}}}{{{k}^{m}}}

is divisible by

n

1

sqrt[3]{Σa_i^3} \le \sqrt{Σ_i^2}

a_1, a_2, . . . , a_n

be real numbers. Prove

\sqrt[3]{a_1^3+ a_2^3+ . . . + a_n^3} \le \sqrt{a_1^2+ a_2^2+ . . . + a_n^2}

(1) When does equality hold in (1)?

1

constructing every positive integer with 3 operations

It is possible to perform three operations

f, g

h

for positive integers:

f(n) = 10n, g(n) = 10n + 4

h(2n) = n

; in other words, one may write

0

4

in the end of the number and one may divide an even number by

2

. Prove: every positive integer can be constructed starting from

4

and performing a finite number of the operations

f, g,

h

1

find line so that perimeter of a triangle is minimum

ABC

be a triangle and let

P

be an interior point of

ABC

. We assume that a line

l

, which passes through

P

, but not through

A

AB

AC

(or their extensions over

B

C

Q

R

, respectively. Find

l

such that the perimeter of the triangle

AQR

is as small as possible.