Factorise $$(a+2b-3c)^3+(b+2c-3a)^3+(c+2a-3b)^3$$

At the round table there are $10$ students. Every of the students thinks of a number and says that number to its immediate neighbors (left and right) such that others do not hear him. So every student knows three numbers. After that every student publicly says arithmetic mean of two numbers he found out from his neghbors. If those arithmetic means were $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$ and $10$, respectively, which number thought student who told publicly number $6$

Triangle $AOB$ is rotated in plane around point $O$ for $90^{\circ}$ and it maps in triangle $A_1OB_1$ ($A$ maps to $A_1$, $B$ maps to $B_1$). Prove that median of triangle $OAB_1$ of side $AB_1$ is orthogonal to $A_1B$

For positive integer $n$, prove that at least one of the numbers $$A=2n-1 , B=5n-1, C=13n-1$$is not perfect square

Find the real number coefficient $c$ of polynomial $x^2+x+c$, if his roots $x_1$ and $x_2$ satisfy following: $$\frac{2x_1^3}{2+x_2}+\frac{2x_2^3}{2+x_1}=-1$$

If $p>2$ is prime number and $m$ and $n$ are positive integers such that $$\frac{m}{n}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{p-1}$$Prove that $p$ divides $m$

Let $I$ be the incircle and $O$ a circumcenter of triangle $ABC$ such that $\angle ACB=30^{\circ}$. On sides $AC$ and $BC$ there are points $E$ and $D$, respectively, such that $EA=AB=BD$. Prove that $DE=IO$ and $DE \perp IO$

Let $n$ be a positive integer and set $S=\{n,n+1,n+2,...,5n\}$ $a)$ If set $S$ is divided into two disjoint sets , prove that there exist three numbers $x$, $y$ and $z$(possibly equal) which belong to same subset of $S$ and $x+y=z$ $b)$ Does $a)$ hold for set $S=\{n,n+1,n+2,...,5n-1\}$

Determine value of real parameter $\lambda$ such that equation $$\frac{1}{\sin{x}} + \frac{1}{\cos{x}} = \lambda $$has root in interval $\left(0,\frac{\pi}{2}\right)$

For positive integers $a$ and $b$ holds $a^3+4a=b^2$. Prove that $a=2t^2$ for some positive integer $t$

Let $AD$ and $BE$ be angle bisectors in triangle $ABC$. Let $x$, $y$ and $z$ be distances from point $M$, which lies on segment $DE$, from sides $BC$, $CA$ and $AB$, respectively. Prove that $z=x+y$

Prove that among any $6$ irrational numbers you can pick three numbers $a$, $b$ and $c$ such that numbers $a+b$, $b+c$ and $c+a$ are irrational

If for real numbers $x$ and $y$ holds $\left(x+\sqrt{1+y^2}\right)\left(y+\sqrt{1+x^2}\right)=1$ prove that $$\left(x+\sqrt{1+x^2}\right)\left(y+\sqrt{1+y^2}\right)=1$$

If $n$ is a positive integer and $n+1$ is divisible with $24$, prove that sum of all positive divisors of $n$ is divisible with $24$