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Regional Math Olympiad (India) Algebra Problems

(This is a work in progress. More problems will be added soon).

  1. Three positive real numbers a, b, c are such that \( a^2 + 5b^2 + 4c^2 − 4ab − 4bc = 0 \). Can a, b, c be the lengths of the sides of a triangle? Justify your answer. (RMO 2014, Mumbai Region)
  2. The roots of the equation \(x^3 − 3ax^2 + bx + 18c = 0 \) form a non-constant arithmetic progression and the roots of the equation \( x^3 + bx^2 + x − c 3 = 0 \) form a non constant geometric progression. Given that a, b, c are real numbers, find all positive integral values of a and b. (RMO 2014, Mumbai Region)
  3. Let a, b, c be positive numbers such that 1 1 + a + 1 1 + b + 1 1 + c ≤ 1. Prove that \( (1 + a^2 )(1 + b^2 )(1 + c^2 ) ≥ 125 \). When does the equality hold? (RMO 2014, Mumbai Region)
  4. Let \( P(x) = x^3 + ax^2 + b \) and \( Q(x) = x^3 + bx + a \), where a, b are non-zero real numbers. Suppose that the roots of the equation P(x) = 0 are the reciprocals of the roots of the equation Q(x) = 0. Prove that a and b are integers. Find the greatest common divisor of P(2013! + 1) and Q(2013! + 1). (RMO 2013, Mumbai Region)
  5. Let a, b, c be positive integers such that a divides \( b^3 \) , b divides \(c^3\) and c divides \(a^3 \) . Prove that abc divides \( (a + b + c)^{13} \) . (RMO 2012)
  6. Let a and b be positive real numbers such that a + b = 1. Prove that \( a^a b^b + a^b b^a ≤ 1 \). (RMO 2012)
  7. Let a and b be real numbers such that a 6= 0. Prove that not all the roots of \(ax^4 + bx^3 + x^2 + x + 1 = 0\) can be real. (RMO 2012)
  8. Find all pairs (x, y) of real numbers such that \( 16x^2+y + 16x+y^2 = 1\) (RMO 2011)
December 14, 2016

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