11.3

Part of V Soros Olympiad 1998 - 99 (Russia)

Problems(3)

a cos x + b cos 2x <= 0 (V Soros Olympiad 1998-99 Round 1 11.3)

Source:

For what a from the interval

[0,\pi]

a

b

that are not simultaneously equal to zero, for which the inequality

a \cos x + b \cos 2x \le 0

is satisfied for all

x

belonging to the segment

[a, \pi]

algebrainequalitiestrigonometry

x - \sqrt{x^2-a^2} = (x-a)^2}/2(x+a) (V Soros Olympiad 1998-99 Round 2 11.3)

Source:

For each value of parameter

a

, solve the the equation

x - \sqrt{x^2-a^2} = \frac{(x-a)^2}{2(x+a)}

y =\sqrt{x^3+1},y=-\sqrt[3]{x^2+ 2x} (V Soros Olympiad 1998-99 Round 3 11.3)

Source:

Find the area of the figure on the coordinate plane bounded by the straight lines

x = 0

x = 2

and the graphs of the functions

y =\sqrt{x^3+ 1}

y = - \sqrt[3]{x^2+ 2x}

analytic geometrygeometryareas