For which values of the parameter  the expression , where and   are the distinct roots of the quadratic function given by the formula , is the smallest?

The distance from the center of the base of a regular quadrangular pyramid to its lateral edge is equal to , while the lateral edge makes an angle of with the plane of the base. Determine the tangent of the angle of inclination of the lateral face to the plane of the base. Calculate the lateral surface area of the pyramid.

Find all real values of the parameter for which the function  has exactly two zeros, with one zero being twice the other.

Determine the equations of the circles with radius that pass through the points and .

A line parallel to the base of a triangle divides the triangle into two figures of equal area. In what ratio does this line divide the sides of the triangle?

Determine the equation of the line containing a chord of the circle given by the equation , knowing that the point is the midpoint of this chord.

Let and . Solve the inequality

.

Consider the function . For any , , the following equation holds

The expression 

Prove that for every real number and every positive real number the inequality holds.