Definite Integral : Problem with a twist!

Sum of squares of 5 consecutive natural numbers!

Here's a difficult problem of proving that a given expression is not a perfect square. A direct approach will be a nightmare ( even not sure it can be proved) but use of a simple property of perfect squares will ease the problem. We know that any perfect squares leaves a remainder either 1 or 0 when being divided by 3 or 4. This simple result would be used to solve the problem.

Number of solutions of the given equation

Here the equation to be solved is very simple |2x-[x]|=4. But unfortunately the traditional methods of solving will not apply! You have solved equations by methods like factorisation, vanishing method, hit and trial.... But in many equations these methods don't work or requires huge computation and guess work to solve it. Here is an equation which can be solved from the simple fact that if one side of an equation is an integer the we look to the other side only and see for what values its also an integer. This simple but powerful method can be used in many problems