2018 Silk Road

Part of Silk Road

Subcontests

(4)

1

word distance problem, number of digits words are different

Given the natural

n

. We shall call word sequence from

n

letters of the alphabet, and distance

\rho(A, B)

A=a_1a_2\dots a_n

B=b_1b_2\dots b_n

, the number of digits in which they differ (that is, the number of such

i

a_i\ne b_i

). We will say that the word

C

lies between words

A

B

\rho (A,B)=\rho(A,C)+\rho(C,B)

. What is the largest number of words you can choose so that among any three, there is a word lying between the other two?

1

f (x+1 )=1+f(x) and f (x^4-x^2 )=f^4(x)-f^2(x)

Find all functions

f:\ \mathbb{R}\rightarrow\mathbb{R}

such that for any real number

x

the equalities are true:

f\left(x+1\right)=1+f(x)

f\left(x^4-x^2\right)=f^4(x)-f^2(x).

[url=http://matol.kz/comments/3373/show]source

1

silk road 2018 geometry

In an acute-angled triangle

ABC

AB

BC

AC

H

L

K

CH \perp AB

HL \parallel AC

HK \parallel BC

P

Q

feet of altitudes of a triangle

HBL

, drawn from the vertices

H

B

respectively. Prove that the feet of the altitudes of the triangle

AKH

, drawn from the vertices

A

H

lie on the line

PQ

1

Exactly once sequence

Does there exist a sequence of positive integers

a_1,a_2,...

such that every positive integer occurs exactly once and that the number

\tau (na_{n+1}^n+(n+1)a_n^{n+1})

is divisible by

n

for all positive integer.
Here

\tau (n)

denotes the number of positive divisor of

n