MathDB

Problem 8 of First round - "Conjugate" numbers

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

8/29/2019

For a quadratic trinomial

f(x)

and the different numbers

a

and

b

it is known that

f(a)=b

and

f(b)=a

. We call such

a

and

b

conjugate for

f(x)

. Prove that

f(x)

has no other conjugate numbers.

algebrafunction

Problem 8 of Second round

Source: VII International Festival of Young Mathematicians Sozopol 2016, Theme for 10-12 grade

8/31/2019

Find all triples of natural numbers

(x,y,z)

for which:

xyz=x!+y^x+y^z+z!

.

number theory

Problem 8 of Third round - Sum as a power of a natural number

Source: VII International Festival of Young Mathematicians Sozopol 2016, Theme for 10-12 grade

9/1/2019

Let

a_i

,

i=1,2,…2016

, be fixed natural numbers. Prove that there exist infinitely many 2016-tuples

x_1,x_2…x_{2016}

of natural numbers, for which the sum

\sum_{i=1}^{2016}{a_i x_i^i}

is a 2017-th power of a natural number.

algebraSumpowers

Problem 8 of Fourth round - Function with conditions

Source: VII International Festival of Young Mathematicians Sozopol 2016, Theme for 10-12 grade

9/3/2019

Prove that there exist infinitely many natural numbers

n

, for which there

\exists \, f:\{0,1…n-1\}\rightarrow \{0,1…n-1\}

, satisfying the following conditions: 1)

f(x)\neq x

; 2)

f(f(x))=x

; 3)

f(f(f(x+1)+1)+1)=x

for

\forall x\in \{0,1…n-1\}

.

functionspecial conditionalgebra

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