MathDB

Existence in number theory

Source:

11/26/2021

Prove that there are infinitely many integers can't be written as

\frac{p^a-p^b}{p^c-p^d}

, with a,b,c,d are arbitrary integers and p is an arbitrary prime such that the fraction is an integer too.

number theory

Geometry with line passing through circumcenter

Source: Russian TST 2014, Day 8 P3 (Groups A & B)

1/8/2024

On the sides

AB{}

and

AC{}

of the acute-angled triangle

ABC{}

the points

M{}

and

N{}

are chosen such that

MN

passes through the circumcenter of

ABC.

Let

P{}

and

Q{}

be the midpoints of the segments

CM{}

and

BN{}.

Prove that

\angle POQ=\angle BAC.

geometryangles

Minimum value of cosine sum (easy variant)

Source: Russian TST 2014, Day 10 P3 (Groups A & B)

1/8/2024

Let

x,y,z

be real numbers. Find the minimum value of the sum \begin{align*}|\cos(x)|+|\cos(y)|+|\cos(z)|+|\cos(x-y)|+|\cos(y-z)|+|\cos(z-x)|.\end{align*}

algebratrigonometryminimum value

Minimum of sum of cosines of differences

Source: Russian TST 2014, Day 10 P3 (Group NG)

1/8/2024

Let

n>1

be an integer and

x_1,x_2,\ldots,x_n

be

n{}

arbitrary real numbers. Determine the minimum value of

\sum_{i<j}|\cos(x_i-x_j)|.

algebraminimum valuetrigonometry

Very hard FE

Source: Russian TST 2014, Day 11 P3 (Group NG), P4 (Groups A & B)

1/8/2024

Find all functions

f : \mathbb{R}\to\mathbb{R}

such that

f(0) = 0

and for any real numbers

x, y

the following equality holds

f(x^2+yf(x))+f(y^2+xf(y))=f(x+y)^2.

algebrafunctional equation

P3