2005 Silk Road

Part of Silk Road

Subcontests

(4)

1

International competition SRMC 2005 P-1

n \geq 2

be natural number. Prove, that

(1^{n-1}+2^{n-1}+....+(n-1)^{n-1})+1

n

iff for any prime divisor

p

n

p| \frac{n}{p}-1

(p-1)| \frac{n}{p}-1

1

ABC

A,B,C

are three collinear points that

B \in [AC]

AA'

BB'

are to parrallel lines that

A'

B'

C

are not collinear. Suppose

O_1

is circumcenter of circle passing through

A

A'

C

O_2

is circumcenter of circle passing through

B

B'

C

A'CB'

is equal to area of

O_1CO_2

, then find all possible values for

\angle CAA'

1

m*n table

(m,n) \in \mathbb{Z}^2

that we can color each unit square of

m \times n

with the colors black and white that for each unit square number of unit squares that have the same color with it and have at least one common vertex (including itself) is even.

1

a(2n) \leq 2a(n)

\{a(n) \}_{n=1}^{\infty}

is a sequence that:

a(n) =a(a(n-1))+a(n-a(n-1)) \ \ \ \forall \ n \geq 3

a(1)=a(2)=1

. Prove that for each

n \geq 1

a(2n) \leq 2a(n)