Let $a$ be a positive number, and we show decimal part of the $a$ with $\left\{a\right\}$.For a positive number $x$ with $\sqrt 2< x <\sqrt 3$ such that, $\left\{\frac{1}{x}\right\}$=$\left\{x^2\right\}$.Find value of the $$x(x^7-21)$$

Let $a$ and $b$ is roots of the $x^2-6x+1$ equation. a) Show that, for all $n \in{\mathbb Z^+}$ , $a^n+b^n$ is a integer. b) Show that, for all $n \in{\mathbb Z^+}$ , $5$ isn't divide $a^n+b^n$

For all $x \geq 2$, $y \geq 2$ real numbers, prove that $$x(\frac{4x}{y-1}+\frac{1}{2y+x})+y(\frac{y}{6x-9}+\frac{1}{2x+y}) > \frac{26}{3}$$

Let an acute-angled triangle $ABC$'s circumcircle is $S$. $S$'s tangent from $B$ and $C$ intersects at point $M$. A line, lies $M$ and parallel to $[AB]$ intersects with $S$ at points $D$ and $E$, intersect with $[AC]$ at point $F$. Prove that $$[DF]=[FE]$$

For all $n \in {\mathbb Z^+}$ we define $$I_n=\{\frac{0}{n},\frac{1}{n},\frac{2}{n},\dotsm,\frac{n-1}{n},\frac{n}{n},\frac{n+1}{n},\dotsm\}$$infinite cluster. For whichever $x$ and $y$ real number, we say $\mid{x-y}\mid$ is between distance of the $x$ and $y$. a) For all $n$'s we find a number in $I_n$ such that, the between distance of the this number and $\sqrt 2$ is less than $\frac{1}{2n}$ b) We find a $n$ such that, between distance of the a number in $I_n$ and $\sqrt 2$ is less than $\frac{1}{2011n}$

Let $m,n$ positive integers and $p$ prime number with $p=3k+2$. If $p \mid {(m+n)^2-mn}$ , prove that $$p \mid m,n$$

Let $O$ is a point in a plane $P$ and let $[OX,[OY,[OZ$ is distinct ray in $P$. Prove that, if $A \in [OX$ , $B \in [OY$ and $C \in [OZ$ points such that $\triangle OAB$ , $\triangle OBC$ and $\triangle OCA$ 's perimeter is 2, there is only one $(A,B,C)$ triple

Let $a,b,c$ positive reals such that $a+b+c=3$. Show that following expression's minimum value is $2$. $$\frac{\sqrt a +\sqrt b +\sqrt c}{ab+bc+ca} + \frac{1}{1+2\sqrt {ab}} + \frac {1}{1+ 2\sqrt {bc}} + \frac{1}{1+ 2\sqrt {ca}}$$

$a_n$ sequence is a arithmetic sequence with all terms be positive integers. (for $a_n$ non-constant sequence) Let $p_n$ is greatest prime divisor of $a_n$. Prove that $$(\frac{a_n}{p_n})$$sequence is infinity. Click to reveal hidden textNote: If we find a $M>0$ constant such that $x_n \leq M$ for all $n \in {\mathbb N}$'s, $(x_n)$ sequence is non-infinite, but we can't find $M$, $(x_n)$ sequence is infinity

Let $ABC$ be an acute-angled triangle with $H$ orthocenter, $O$ circumcenter. $[AH]$'s perpendicular bisector intersects with $[AB]$ and $[AC]$ at $D$ and $E$ respectively. Prove that $$\angle ADE =\angle BDO$$