MathDB

Problem 6 of Second round

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

9/22/2018

Prove that there exist infinitely many positive integers

n

, for which at least one of the numbers

2^{2^n}+1

and

2018^{2^n}+1

is composite.

number theory

Problem 6 of First round

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

9/21/2018

Find all functions

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

, such that

f(x+y) = f(y) f(x f(y))

for every two real numbers

x

and

y

.

functional equationfunctionalgebra

Problem 6 of third round

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

9/22/2018

Let

S

be a real number. It is known that however we choose several numbers from the interval

(0, 1]

with sum equal to

S

, these numbers can be separated into two subsets with the following property: The sum of the numbers in one of the subsets doesn’t exceed 1 and the sum of the numbers in the other subset doesn’t exceed 5.
Find the greatest possible value of

S

.

algebra

Problem 6 of Fourth round

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

9/22/2018

Find all sets

(a, b, c)

of different positive integers

a

,

b

,

c

, for which:
*

2a - 1

is a multiple of

b

;
*

2b - 1

is a multiple of

c

;
*

2c - 1

is a multiple of

a

.

number theory

Problem 6 of Finals

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

9/22/2018

There are

a

straight lines in a plane, no two of which are parallel to each other and no three intersect in one point.
a) Prove that there exist a straight line for which each of the two Half-Planes defined by it contains at least

\lfloor \frac{(a-1)(a-2)}{10} \rfloor

intersection points.
b) Find all

a

for which the evaluation in a) is the best possible.

floor functionstraight linesIntersection

6

Problems(5)