Let $m \in {\mathbb R}$ and $$x^2+(m-4)x+(m^2-3m+3)=0$$equations roots are $x_1$ and $x_2$ and $x_1^2+x_2^2=6$. Find all $m$ values.

If $x$ and $y$ are positive reals, prove that $$x^2\sqrt{\frac{x}{y}}+y^2\sqrt{\frac{y}{x}} \geq x^2+y^2$$

Let for all $k \in {\mathbb N}$ $k$'s sum of the digits is $T(k)$. If a natural number $n$ such that $T(n)=T(1997n)$, prove that $$9\mid n$$

A plane dividing like a chessboard and write a real number each square such that, for a squares' number equal to its up, down ,left and right squares' numbers arithmetic mean. Prove that all number are equal.

A $ABC$ triangle divide by a $d$ line such that, new two pieces' areas are equal. $d$ line intersects with $[AB]$ at $D$, $[AC]$ at $E$. Prove that $$\frac{AD+AE}{BD+DE+EC+CB} > \frac{1}{4}$$

Prove that, $$15x^2-7y^2=9$$equation has any solutions in integers.

Let $x,y,z,t$ be real numbers such that, $1 \leq x \leq y \leq z \leq t \leq 100$. Find minimum value of $$\frac{x}{y}+\frac{z}{t}$$

$(x_n)$ be a sequence with $x_1=0$, $$x_{n+1}=5x_n + \sqrt{24x_n^2+1}$$. Prove that for $k \geq 2$ $x_k$ is a natural number.

A polygon with $1997$ vertices is given. Write a positive real number each vertex such that, each number equal to its right and left numbers' arithmetic or geometric mean. Prove that all numbers are equal.

An $ABC$ triangle divide by a $d$ line such that, new two pieces' areas and perimeters are equal. Prove that $ABC$'s incenter lies $d$