Regional Math Olympiad 2015 Chennai Region

  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

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