MathDB

1

P(n)+(-1)^{a_1}P(n-1)+(-1)^{a_2}P(n-2)+...+(-1)^{a_{n-1}} P(1)+-1)^{a_n} = 0

P(n)

be the number of ways to split a natural number

n

to the sum of powers of two, when the order does not matter. For example

P(5) = 4

, as

5=4+1=2+2+1=2+1+1+1=1+1+1+1+1

. Prove that for any natural the identity

P(n) + (-1)^{a_1} P(n-1) + (-1)^{a_2} P(n-2) + \ldots + (-1)^{a_{n-1}} P(1) + (-1)^{a_n} = 0,

is true, where

a_k

is the number of units in the binary number record

k

.
[url=http://matol.kz/comments/2720/show]source

3

1

if f([{\sqrt n}]+ b)divides f(n+a) .. then f{{a}_{i} divides f({{a}_{i+1}})

Given natural numbers

a,b

and function

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

such that for any natural number

n, f\left( n+a \right)

is divided by

f\left( {\left[ {\sqrt n } \right] + b} \right)

. Prove that for any natural

n

exist

n

pairwise distinct and pairwise relatively prime natural numbers

{{a}_{1}}

,

{{a}_{2}}

,

\ldots

,

{{a}_{n}}

such that the number

f\left( {{a}_{i+1}} \right)

is divided by

f\left( {{a}_{i}} \right)

for each

i=1,2, \dots ,n-1

.
(Here

[x]

is the integer part of number

x

, that is, the largest integer not exceeding

x

.)

2

1

siilk road geometry 2016

Around the acute-angled triangle

ABC

(

AC>CB

) a circle is circumscribed, and the point

N

is midpoint of the arc

ACB

of this circle. Let the points

A_1

and

B_1

be the feet of perpendiculars on the straight line

NC

, drawn from points

A

and

B

respectively (segment

NC

lies inside the segment

A_1B_1

). Altitude

A_1A_2

of triangle

A_1AC

and altitude

B_1B_2

of triangle

B_1BC

intersect at a point

K

. Prove that

\angle A_1KN=\angle B_1KM

, where

M

is midpoint of the segment

A_2B_2

.

1

An easy problem from silk road 2016

Let

a,b

and

c

be real numbers such that

| (a-b) (b-c) (c-a) | = 1

. Find the smallest value of the expression

| a | + | b | + | c |

. (K.Satylhanov )

2016 Silk Road

Subcontests

P(n)+(-1)^{a_1}P(n-1)+(-1)^{a_2}P(n-2)+...+(-1)^{a_{n-1}} P(1)+-1)^{a_n} = 0

if f([{\sqrt n}]+ b)divides f(n+a) .. then f{{a}_{i} divides f({{a}_{i+1}})

siilk road geometry 2016

An easy problem from silk road 2016