2006 Silk Road

Part of Silk Road

Subcontests

(4)

1

International competition SRMC 2006 P-2

a,b,c

abc=1

prove the inequality:

4(\sqrt[3]{\frac{a}{b}}+\sqrt[3]{\frac{b}{c}}+\sqrt[3]{\frac{c}{a}}) \leq 3(2+a+b+c+\frac{1}{a}+\frac{1}{b}+ \frac{1}{c})^{\frac{2}{3}}

1

Find all functions f with f(x^2+xy+f(y))=f^2(x)+xf(y)+y

Found all functions

f: \mathbb{R} \to \mathbb{R}

, such that for any

x,y \in \mathbb{R}

f(x^2+xy+f(y))=f^2(x)+xf(y)+y.

1

A family l of 2006 lines

L

of 2006 lines on the plane is given in such a way that it doesn't contain parallel lines and it doesn't contain three lines with a common point.We say that the line

l_1\in L

is bounding the line

l_2\in L

,if all intersection points of the line

l_2

with other lines from

L

lie on the one side of the line

l_1

. Prove that in the family

L

there are two lines

l

l'

such that the following 2 conditions are satisfied simultaneously: 1) The line

l

is bounding the line

l'

l'

is not bounding the line

l

1

Deviation of subset s

S

M=\{1,2,.....,p-1\}

p

is a prime number of the kind

12n+11

,is essential,if the product

{\Pi}_s

of all elements of the subset is not less than the product

\bar{{\Pi}_s}

of all other elements of the set.The difference

\bigtriangleup_s=\Pi_s-\bar{{\Pi}_s}

is called the deviation of the subset

S

.Define the least possible remainder of division by

p

of the deviation of an essential subset,containing

\frac{p-1}{2}