Prove that: $\sqrt{10+\sqrt{24}+\sqrt{40}+\sqrt{60}}=\sqrt{2}+\sqrt3+\sqrt5$

Find all integer $n$ such that $n-5$ divide $n^2+n-27$.

For all real numbers $a$ prove that $3(a^4+a^2+1)\geq (a^2+a+1)^2$

Find all value of parameter $a$ such that equations $x^2-ax+1=0$ and $x^2-x+a=0$ have at least one same solution. For this value $a$ find same solution of this equations(real or imaginary).

Let $ABC$ be an equilateral triangle, with sidelength equal to $a$. Let $P$ be a point in the interior of triangle $ABC$, and let $D,E$ and $F$ be the feet of the altitudes from $P$ on $AB, BC$ and $CA$, respectively. Prove that $\frac{|PD|+|PE|+|PF|}{3a}=\frac{\sqrt{3}}{6}$

Let be $a,b$ real numbers such that $|a|\neq |b|$ and $\frac{a+b}{a-b}+\frac{a-b}{a+b}=6$ . Calculate: $\frac{a^3+b^3}{a^3-b^3}+\frac{a^3-b^3}{a^3+b^3}$

Three numbers have sum $k$ (where $k\in \mathbb{R}$) such that the numbers are arethmetic progression.If First of two numbers remain the same and to the third number we add $\frac{k}{6}$ than we have geometry progression. Find those numbers?

How many positive integers which are less or equal with $2013$ such that $3$ or $5$ divide the number.

Let be $n$ positive integer than calculate: $1\cdot 1!+2\cdot2!+...+n\cdot n!$

Let $P$ be a point inside or outside (but not on) of a triangle $ABC$. Prove that $PA +PB +PC$ is greater than half of the perimeter of the triangle

Let be $z_1$ and $z_2$ two complex numbers such that $|z_1+2z_2|=|2z_1+z_2|$.Prove that for all real numbers $a$ is true $|z_1+az_2|=|az_1+z_2|$

Solve equation $27\cdot3^{3\sin x}=9^{\cos^2x}$ where $x\in [0,2\pi )$

Prove that solution of equation $y=x^2+ax+b$ and $x=y^2+cy+d$ it belong a circle.

Let be $a,b,c$ three positive integer.Prove that $4$ divide $a^2+b^2+c^2$ only and only if $a,b,c$ are even.

Let $ABCD$ be a convex quadrilateral with perpendicular diagonals. . Assume that $ABCD$ has been inscribed in the circle with center $O$. Prove that $AOC$ separates $ABCD$ into two quadrilaterals of equal area

Which number is bigger $\sqrt[2012]{2012!}$ or $\sqrt[2013]{2013!}$.

Math teacher wrote in a table a polynomial $P(x)$ with integer coefficients and he said: "Today my daughter have a birthday.If in polynomial $P(x)$ we have $x=a$ where $a$ is the age of my daughter we have $P(a)=a$ and $P(0)=p$ where $p$ is a prime number such that $p>a$." How old is the daughter of math teacher?

Find all numbers $x$ such that: $1+2\cdot2^x+3\cdot3^x<6^x$

Calculate: $\sqrt{3\sqrt{5\sqrt{3\sqrt{5...}}}}$

A trapezium has parallel sides of length equal to $a$ and $b$ ($a <b$), and the distance between the parallel sides is the altitude $h$. The extensions of the non-parallel lines intersect at a point that is a vertex of two triangles that have as sides the parallel sides of the trapezium. Express the areas of the triangles as functions of $a,b$ and $h$.