2013 Silk Road

Part of Silk Road

Subcontests

(4)

1

IA_2=IB_2=R where A_1, B_1 touchpoints of incircle (I), prove AA_2=BB_2=OI

Circle with center

I

, inscribed in a triangle

ABC

, touches the sides

BC

AC

A_1

B_1

respectively. On rays

A_1I

B_1I

, respectively, let be the points

A_2

B_2

IA_2=IB_2=R

R

is the radius of the circumscribed circle of the triangle

ABC

. Prove that: a)

AA_2 = BB_2 = OI

O

is the center of the circumscribed circle of the triangle

ABC

AA_2

BB_2

intersect on the circumcircle of the triangle

ABC

1

silk road castings

In the film there is

n

roles. For each

i

1 \le i \le n

), the role of number

i

a_i

a person, and one person can play only one role. Every day a casting is held, in which participate people for

n

roles, and from each role only one person. Let

p

be a prime number such that

p \ge a_1, \ldots, a_n, n

. Prove that you can have

p^k

castings such that if we take any

k

people who are tried in different roles, they together participated in some casting (

k

is a natural number not exceeding

n

1

SRMC 2013 P1

Determine all pairs of positive integers

m, n,

satisfying the equality

(2^{m}+1;2^n+1)=2^{(m;n)}+1

(a;b)

is the greatest common divisor

1

Functional equation from SRMC

Find all non-decreasing functions

f\,:\,\mathbb{N}\to\mathbb{N}

f(f(m)f(n)+m)=f(mf(n))+f(m)