If $a,b\neq 0$ are real numbers such that $a^2b^2(a^2b^2+4)=2(a^6+b^6)$ , then show that $a,b$ can’t be both of them rational .

Find all real numbers $x$ which satisfied $|2x+1|+|x-1|=2-x$ .

Show that the sum $S=5+5^2+5^3+…+5^{2016}$ is divisible by $31$

In a planet $Papella$ year has $400$ days with days coundting from $1-400$ . A holiday would be that day which is divisible by $6$ . The new gonverment decide to reform a new calendar and split in $10$ months with $40$ day each month , and they decide that day of month which is divisible by $6$ to be holiday . Show that after reform the number of holidays after one year decreased less then $ 10 $ percent .

It is given rectangle $ABCD$ with length $|AB|=15cm$ and with length of altitude $|BE|=12cm$ where $BC$ is altitude of triangle $ABC$ . Find perimeter and area of rectangle $ABCD$ .

Find all triples $(x,y,z)$ of integers such that satisfied: $x^2+y^2+z^2+xy+yz+zx=6$

Show that the number $2017^{2016}-2016^{2017}$ is divisible by $5$ .

The distance from $A$ to $B$ is $408km$ . From $A$ in direction of $B$ move motorcyclist , and from $B$ in direction of $A$ move a bicyclist . If a motorcyclist start to move $2$ hours earlier then byciclist , then they will meet $7$ hours after bicyclist start to move . If a bicyclist start to move $2$ hours earlier then motorcyclist , then they will meet $8$ hours after after motorcyclist start to move . Find the velocity of motorcyclist and bicyclist if we now that the velocity of them was constant all the time .

Let be $f: (0,+\infty)\rightarrow \mathbb{R}$ monoton-decreasing . If $f(2a^2+a+1)<f(3a^2-4a+1)$ find interval of $a$ .

If $a,b,c$ are sides of right triangle with $c$ hypothenuse then show that for every positive integer $n>2$ we have $c^n>a^n+b^n$ .

Find all couples $(m,n)$ of positive integers such that satisfied $m^2+1=n^2+2016$ .

Evaluate the sum of all three digits number which are not divisible by $13$ .

If $\alpha $ is an acute angle and $a,b\geq 0$ then show that: $\left( a+\frac{b}{\sin \alpha}\right)\left(b+\frac{a}{\cos \alpha}\right)\geq a^2+b^2+3ab$

Solve equation in real numbers $\log_{2}(4^x+4)=x+\log_{2}(2^{x+1}-3)$

In angle $\angle AOB=60^{\circ}$ are two circle which circumscribed and tangjent to each other . If we write with $r$ and $R$ the radius of smaller and bigger circle respectively and if $r=1$ find $R$ .

Find all three digit numbers such that the square of that number is equal to the sum of their digits in power of $5$ .

Sum of all coefficients of polynomial $P(x)$ is equal with $2$ . Also the sum of coefficients which are at odd exponential in $x^k$ are equal to sum of coefficients which are at even exponential in $x^k$ . Find the residue of polynomial $P(x)$ when it is divide by $x^2-1$ .

Let be $a,b,c$ complex numbers such that $|a|=|b|=|c|=r$ then show that $\left | \frac{ab+bc+ca}{a+b+c}\right|=r$

In all rectangles with same diagonal $d$ find that one with bigger area .

In trapezoid $ABCD$ with $AB$ parallel to $CD$ show that : $\frac{|AB|^2-|BC|^2+|AC|^2}{|CD|^2-|AD|^2+|AC|^2}=\frac{|AB|}{|CD|}=\frac{|AB|^2-|AD|^2+|BD|^2}{|CD|^2-|BC|^2+|BD|^2}$