2003 Silk Road

Part of Silk Road

Subcontests

(4)

1

International competition SRMC 2003 P-2

s=\frac{AB+BC+AC}{2}

be half-perimeter of triangle

ABC

L

N

be a point's on ray's

AB

CB

AL=CN=s

K

is point, symmetric of point

B

by circumcenter of

ABC

. Prove, that perpendicular from

K

NL

passes through incenter of

ABC

.
Solution for problem [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=365714&p=2011659#p2011659]here

1

International competition SRMC 2003 P-1

a_1, a_2, ....., a_{2003}

be sequence of reals number. Call

a_k

leading

element, if at least one of expression

a_k; a_k+a_{k+1}; a_k+a_{k+1}+a_{k+2}; ....; a_k+a{k+1}+a_{k+2}+....+a_{2003}

is positive. Prove, that if exist at least one

leading

element, then sum of all

leading

's elements is positive.
Official solution [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=365714&p=2011659#p2011659]here

1

International competition SRMC 2003 P-3

0<a<b<1

be reals numbers and g(x)=\left\{\begin{array}{cc}x+1-a,&\mbox{ if } 0

n+1

0<x_0<x_1<...<x_n<1

g^{[n]}(x_i)=x_i \ (0 \leq i \leq n)

. Prove that there exists a positive integer

N

g^{[N]}(x)=x

0<x<1

g^{[n]}(x)= \underbrace{g(g(....(g(x))....))}_{\text{n times}}

)
Official solution [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=365714&p=2011659#p2011659]here

1

International competition SRMC 2003 P-4

\sum_{k \in A} \frac{1}{k-1}

A= \{ m^n : m,n \in \mathbb{Z} m,n \geq 2 \}

.
Problem was post earlier [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=67&t=29456&hilit=silk+road]here , but solution not gives and olympiad doesn't indicate, so I post it again :blush:
Official solution [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=365714&p=2011659#p2011659]here