Found problems: 5
Prove the equality
$$\frac{a_1}{a_2(a_1+a_2)}+\frac{a_2}{a_3(a_2+a_3)}+...+\frac{a_n}{a_1(a_n+a_1)}=\frac{a_2}{a_1(a_1+a_2)}+\frac{a_3}{a_2(a_2+a_3)}+...+\frac{a_1}{a_n(a_n+a_1)} $$
(R. Zhenodarov)
$n \ge 4$ real numbers are arranged in a circle. It turned out that for any four consecutive numbers $a, b, c, d$, that lie on the circle in this order, holds $a+d = b+c$. For which $n$ does it follow that all numbers on the circle are equal?
[i]Proposed by Oleksiy Masalitin[/i]
Let $1<k\leq n$ be positive integers and $x_1 , x_2 , \ldots , x_k$ be positive real numbers such that $x_1 \cdot x_2 \cdot \ldots \cdot x_k = x_1 + x_2 + \ldots +x_k.$
a) Show that $x_{1}^{n-1} +x_{2}^{n-1} + \ldots +x_{k}^{n-1} \geq kn.$
b) Find all numbers $k,n$ and $x_1, x_2 ,\ldots , x_k$ for which equality holds.
Let $a$, $b$, $c$, $d$ and $e$ be distinct positive real numbers such that $a^2+b^2+c^2+d^2+e^2=ab+ac+ad+ae+bc+bd+be+cd+ce+de$
$a)$ Prove that among these $5$ numbers there exists triplet such that they cannot be sides of a triangle
$b)$ Prove that, for $a)$, there exists at least $6$ different triplets
Do there exist $10$ real numbers, not all of which are equal, each of which is equal to the square of the sum of the remaining $9$ numbers?
[i]Proposed by Bogdan Rublov[/i]