The sum of two numbers is equal to , and their difference is equal to . Prove that the product of these numbers is an integer.

Solve the equation

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The sides opposite the angles , and of a triangle have lengths , and , respectively.  The radius of the circle circumscribed about this triangle has length

We draw three vertices of a cube. Calculate the probability of the event that the triangle defined by these vertices is equilateral. In the answer, give the first three digits after the decimal point of the decimal expansion of the obtained result.

The limit is equal to for equal to

The sequence is an arithmetic sequence with . Find for which the expression is the smallest.

In a regular hexagonal pyramid, the lateral edge has length and the dihedral angle between adjacent side walls is equal to . Determine the volume of this pyramid depending on and . Make a drawing.

Consider a rectangular trapezium circumscribed around a circle built by tangents to the circle inclined at an angle to the axis and a tangent to the circle parallel to   axis. Determine the ratio of the area of the trapezium to its perimeter , given that . Make a drawing.

Consider the circles and with equations 

a) Find the intersection points and of the circles.

b) Prove that the area of the quadrilateral with vertices at points , and the centers of the circles and is equal to half of the square of the distance between the centers of those circles.

There are boys and   girls in the room. We draw two children. Then, we draw another child from the remaining children. Calculate the probability that the drawn child is a girl.