Try the solution of problem from RMO (Regional Mathematical Olympiad) 2015 Problem 1 based on Triangle. Problem: Triangle Problem Two circles $latex \Gamma $ and $latex \Sigma $, with centers O and O', respectively, are such that O' lies on $latex \Gamma $. Let A be a point on $latex \Sigma $, and let M […]
Try this problem from RMO 2015 from Chennai Region based on Integer Solution of Polynomial. Problem: Integer Solution of Polynomial Solve the equation $latex y^3 + 3y^2 + 3y = x^3 + 5x^2 - 19x + 20 &s=2 $ for positive integers x, y. Discussion: $latex y^3 + 3y^2 + 3y = x^3 + 5x^2 […]
This is a problem from RMO 2015 Mumbai Region based on inequality. Problem: RMO 2015 Mumbai Region Let x, y, z be real numbers such that $ x^2 + y^2 + z^2 - 2xyz = 1 $ and $ s=2$ . Prove that $ (1+x)(1+y)(1+z) \le 4 + 4xyz $ and $ s=2$ Discussion Note […]
This is a problem from RMO 2015 from Mumbai Region based on Cyclic Quadrilaterals and Incenters. Problem: RMO 2015 Mumbai Region Let ABC be a right angled triangle with $ \angle B = 90^0 $ and $ s=2 $ and let BD be the altitude from B on to AC. Draw $ DE \perp AB […]
This is a problem from Regional Mathematics Olympiad, RMO 2015 Mumbai Region based on Polynomial with positive integers. Try to solve it. Site title Title Primary category Separator Problem: Let P(x) be a polynomial whose coefficients are positive integers. If P(n) divides P(P(n) -2015) for every natural number n, prove that P(-2015) = 0. Discussion: Let […]
This is a problem from the Regional Mathematics Olympiad, RMO 2015 Mumbai Region based on the Number of Three-digit numbers. Try to solve it. Problem: Number of Three-digit numbers Determine the number of 3 digit numbers in base 10 having at least one 5 and at most one 3. Discussion: (Suggested by Shuborno Das in […]
This is a problem from the Regional Mathematics Olympiad, RMO 2015 Mumbai Region based on Diagonal of a Quadrilateral. Try to solve it. Problem: Diagonal of a Quadrilateral Let ABCD be a convex quadrilateral with AB = a, BC = b, CD = c and DA = d. Suppose $ a^2 + b^2 + c^2 […]
This is a problem from Regional Mathematics Olympiad, RMO 2015 Chennai Region based on the Minimal value problem. Try to solve it. Problem: Minimal value problem 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 […]
This is a problem from the Regional Mathematics Olympiad, RMO 2015 Chennai Region based on a List of numbers. Problem: 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 […]