MathDB

Problem 5 of First round

Source: XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade

9/9/2022

Let

\Delta ABC

be an acute scalene triangle with

AC<BC

, an orthocenter

H

and altitudes

AE

,

BF

. The points

E'

and

F'

are symmetrical to

E

and

F

with respect to

A

and

B

respectively. Point

O

is the center of the circumscribed circle of

ABC

and

M

is the midpoint of

AB

. Let

N

be the midpoint of

OM

. Prove that the tangent through

H

to the circumscribed circle of

\Delta E'HF'

is perpendicular to line

CN

.

geometry

Problem 5 of Second round

Source: XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade

9/9/2022

Let

a

,

b

and

c

be given positive integers which are two by two coprime. A positive integer

n

is called sozopolian, if it can’t be written as

n=bcx+cay+abz

where

x

,

y

,

z

are also positive integers. Find the number of sozopolian numbers as a function of

a

,

b

and

c

.

number theory

Problem 5 of Third round

Source: XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade

9/9/2022

Prove that

\sum_{n=1}^{2022^{2022}} \frac{1}{\sqrt{n^3+2n^2+n}}<\frac{19}{10}

.

algebraInequalitySum

no of subsets of {1, 2,... , 2100} with sum of elements 3 mod 7

Source: IFYM - XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade, 4th round p5

11/12/2022

Find the number of subsets of

\{1, 2,... , 2100\}

such that each has sum of the elements giving a remainder of

3

when divided by

7

.

remaindernumber theory

f(p) divides f(n)^p- n

Source: IFYM - XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade,finals p5

11/13/2022

Find all functions

f : N \to N

such that

f(p)

divides

f(n)^p -n

by any natural number

n

and prime number

p

.

algebradivides

5

Problems(5)