MathDB

1

combinatorics with floor function

\{a_n \}

is defined as follows:

a_0 = 1

and

{a_n} = \sum \limits_ {k = 1} ^ {[\sqrt n]} {{a_ {n - {k ^ 2 }}}}

for

n \ge 1.

Prove that among

a_1, a_2, \ldots, a_ {10 ^ 6}

there are at least

500

even numbers. (Here,

[x]

is the largest integer not exceeding

x

.)

3

1

\varphi (a ^ n + n) = 2 ^ n , euler function diophantine

Find all pairs of

(a, n)

natural numbers such that

\varphi (a ^ n + n) = 2 ^ n.

(

\varphi (n)

is the Euler function, that is, the number of integers from

1

up to

n

, relative prime to

n

)

2

1

(a_1 + 1)(a_2 + 2)... (a_ {99} +99)>=100! in positive real under conditions

Let

a_1,

a_2,

\ldots,

a_ {99}

be positive real numbers such that

ia_j + ja_i \ge i + j

for all

1 \le i <j \le 99.

Prove , that

(a_1 + 1) (a_2 + 2) \ldots (a_ {99} +99) \ge 100!

.

1

silk road angle chasing , perpendiculars given, equal angles wanted

The altitudes of the acute-angled non-isosceles triangle

ABC

intersect at the point

H

. On the segment

C_1H

, where

CC_1

is the altitude of the triangle, the point

K

is marked. Points

L

and

M

are the feet of perpendiculars from point

K

on straight lines

AC

and

BC

, respectively. The lines

AM

and

BL

intersect at

N

. Prove that

\angle ANK = \angle HNL

.

2019 Silk Road

Subcontests

combinatorics with floor function

\varphi (a ^ n + n) = 2 ^ n , euler function diophantine

(a_1 + 1)(a_2 + 2)... (a_ {99} +99)>=100! in positive real under conditions

silk road angle chasing , perpendiculars given, equal angles wanted

2019 Silk Road

Subcontests

combinatorics with floor function

\varphi (a ^ n + n) = 2 ^ n , euler function diophantine

(a_1 + 1)(a_2 + 2)... (a_ {99} +99)&gt;=100! in positive real under conditions

silk road angle chasing , perpendiculars given, equal angles wanted

(a_1 + 1)(a_2 + 2)... (a_ {99} +99)>=100! in positive real under conditions