MathDB

1

Part of 1984 Vietnam National Olympiad

Problems(2)

Find polynomial with irrational root and solve an equation.

Source: Vietnam MO 1984 P1

3/20/2011
(a)(a) Find a polynomial with integer coefficients of the smallest degree having 2+33\sqrt{2} + \sqrt[3]{3} as a root. (b)(b) Solve 1 +\sqrt{1 + x^2}(\sqrt{(1 + x)^3}-\sqrt{(1- x)^3}) = 2\sqrt{1 - x^2}.
algebrapolynomialcalculusintegrationalgebra unsolved
Finding minimum value of expression and solving an equation.

Source: Vietnam MO 1984 P4

3/20/2011
(a)(a) Let x,yx, y be integers, not both zero. Find the minimum possible value of ∣5x2+11xy−5y2∣|5x^2 + 11xy - 5y^2|. (b)(b) Find all positive real numbers tt such that 9t10=[t]t−[t]\frac{9t}{10}=\frac{[t]}{t - [t]}.
calculusintegrationquadraticsnumber theoryalgebra unsolvedalgebra