MathDB

Problem 6 of First round

Source: VI International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

1/13/2020

The points

A_1

,

B_1

,

C_1

are middle points of the arcs

\widehat{BC}, \widehat{CA}, \widehat{AB}

of the circumscribed circle of

\Delta ABC

, respectively. The points

I_a,I_b,I_c

are the reflections in the middle points of

BC,CA,AB

of the center

I

of the inscribed circle in the triangle. Prove that

I_a A_1,I_b B_1

, and

I_c C_1

are concurrent.

geometryconcurrency

Problem 6 of Fourth round

Source: VI International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

12/31/2019

Find all functions

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

such that for

\forall

x,y\in \mathbb{R}

:

f(x+f(x+y))+xy=yf(x)+f(x)+f(y)+x

.

algebrafunctional equation

Problem 6 of Third round

Source: VI International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

12/24/2019

The natural number

n>1

is called “heavy”, if it is coprime with the sum of its divisors. What’s the maximal number of consecutive “heavy” numbers?

number theorycoprime

Problem 6 of Second round - 12 consecutive numbers with exactly 2 prime factors

Source: VI International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

12/19/2019

A natural number is called “sozopolian”, if it has exactly two prime divisors. Does there exist 12 consecutive “sozopolian” numbers?

number theory

Problem 6 of Finals

Source: VI International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

1/11/2020

In

\Delta ABC

points

A_1

,

B_1

, and

C_1

are the tangential points of the excircles of

ABC

with its sides. a) Prove that

AA_1

,

BB_1

, and

CC_1

intersect in one point

N

. b) If

AC+BC=3AB

, prove that the center of the inscribed circle of

ABC

, its tangential point with

AB

, and the point

N

are collinear.

geometryInscribed circleexcirclecollinearconcurrent

6

Problems(5)