Find the graph of the function $y=x-|x+x^2|$

If $x_1$ and $x_2$ are the solutions of the equation $x^2-(m+3)x+m+2=0$ Find all real values of $m$ such that the following inequations are valid $\frac {1}{x_1}+\frac {1}{x_2}>\frac{1}{2}$ and $x_1^2+x_2^2<5$

Prove that $\sqrt 2$ is irrational.

Prove that if in the product of four consequtive natural numbers we add $1$, we get a perfect square.

In a circle four distinct points are fixed and each of them is assigned with a real number. Let those numbers be $x_1,x_2,x_3,x_4$ such that $x_1+x_2+x_3+x_4>0$. Now we define a game with these numbers: If one of them, i.e. $x_i$, is a negative number, the player makes a move by adding the number $x_i$ to his neighbors and changes the sign of the chosen number. The game ends when all the numbers are negative. Prove that this game ends in a finite number of steps.

Find the graph of the function $y=1-|1-sin x|$.

Solve the equation: $x^2+2xcos(x-y)+1=0$

Let $n\geq2$ be an integer. $n$ is a prime if it is only divisible by $1$ and $n$. Prove that there are infinitely many prime numbers.

Prove that $n^{11}-n$ is divisible by $11$.

Find the graph of the function $y=x+|1-x^3|$.

Let $p$ be a prime number and $n$ a natural one. How many natural numbers are between $1$ and $p^n$ that are relatively prime with $p^n$?

Let $a,b$ and $c$ be the sides of a triangle, prove that $\frac {a}{b+c}+\frac {b}{c+a}+\frac {c}{a+b}<2$.

$(a)$ Let $a_1,a_2,a_3$ be three real numbers. Prove that $(a_1-a_2)(a_1-a_3)+(a_2-a_1)(a_2-a_3)+(a_3-a_1)(a_2-a_2)\geq 0$. $(b)$ Prove that the inequality above doesn't hold if we use four number instead of three.