- Find the minimum value of \(\displaystyle { \frac{ ( x + \frac{1}{x} )^6 – ( x^6 + \frac{1}{x^6}) – 2}{(x+\frac{1}{x})^3 + (x^3 + \frac{1}{x^3} )} } \) and \(x \in \mathbb{R} \) and \(x > 0 \)
**SOLUTION:****here** - Given that P and Q are points on the sides AB and AC respectively of \(\Delta ABC \). The perpendiculars to the sides AB and AC at P and Q respectively meet at D, an interior point of \(\Delta ABC \). If M is the midpoint of BC, prove that PM = QM if and only if \(\angle BDP = \angle CDQ \).
- Let \(N = 2^5 + 2^{5^2} + 2^{5^3} + … + 2^{5^{2013}} \). Written in the usual decimal form, find the last two digits of the number N.
**SOLUTION:here** - Two circles \(\Sigma_1 \) and \(\Sigma_2 \) having centers at \(C_1 \) and \(C_2 \) intersect at A and B. Let P be a point on the segment AB and let \(AP \neq PB \). The line through P perpendicular to \(C_1 P \) meets \(\Sigma_1 \) at C and D. The line through P perpendicular to \(C_2P \) meets \(\Sigma_2 \) at E and F. prove that C,D, E and F form a rectangle.
**SOLUTION: here** - Solve the equation \(y^3 + 3y^2 + 3y = x^3 + 5x^2 – 19x + 20 \) for positive integers x, y.
**SOLUTION: here** - From the list of natural numbers 1, 2, 3, … suppose we remove all multiples of 7, all multiples of 11 and all multiples of 13.
- At which position in the resulting list does the number 1002 appear?
- What number occurs in the position 3600?
**SOLUTION: here**

sir solution for 5th problem

can you give solution for other problems too?waiting in anticipation.