2009 Silk Road

Part of Silk Road

Subcontests

(4)

1

a tourist visits Complant (silk road combo)

A tourist going to visit the Complant, found that: a) in this country

1024

cities, numbered by integers from

0

1023

, b) two cities with numbers

m

n

are connected by a straight line if and only if the binary entries of numbers

m

n

they differ exactly in one digit, c) during the stay of a tourist in that country

8

roads will be closed for scheduled repairs. Prove that a tourist can make a closed route along the existing roads of Complant, passing through each of its cities exactly once.

1

(x^2+p t^2)(y^2+p t^2)(z^2+p t^2) is a perfect square when (x, y, z, t) distinct

Prove that for any prime number

p

there are infinitely many fours

(x, y, z, t)

pairwise distinct natural numbers such that the number

(x^2+p t^2)(y^2+p t^2)(z^2+p t^2)

is a perfect square.

1

SRMC 2009

Prove that, abc≤1 and a,b,c>0

\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq 1+ \frac{6}{a+b+c}

1

Geometry

Bisectors of triangle ABC of an angles A and C intersect with BC and AB at points A1 and C1 respectively. Lines AA1 and CC1 intersect circumcircle of triangle ABC at points A2 and C2 respectively. K is intersection point of C1A2 and A1C2. I is incenter of ABC. Prove that the line KI divides AC into two equal parts.