1. 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
  2. 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 .
  3. 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
  4. 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
  5. Solve the equation y^3 + 3y^2 + 3y = x^3 + 5x^2 - 19x + 20  for positive integers x, y.
    SOLUTION: here
  6. 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.
    1. At which position in the resulting list does the number 1002 appear?
    2. What number occurs in the position 3600?
      SOLUTION: here