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Prove that the trace of a matrix A i.e TR(A)=sum of the diagonal elements of A.
45 people came to the meeting. It turned out that any two of them having the same number of friends among the visitors are not familiar with each other. What is the greatest number of pairs of friends could be among the participants of the meeting?
The parts of a compass are located at the nodes of an infinite sheet of checkered paper, the cells of which are squares with the side 1. You may, without changing the radius of a compass, turn it around one of its parts to move the second part to the other node on the worksheet. Is it possible in several of such steps to swap the parts of a compass?
Three brothers visited their sick friend in one day, and in the same day their wives also visited him. No one of the visitors came more than once. Every brother has met both of his brothers’ wives in the house of their sick friend. Prove that at least one of the brothers have also met his wife it that house.
In what case the vectors AB and BA are equal?
It is known that the sequences {xn} and {yn} have a limit. If we form the sequence {x1; y1; x2; y2; … ; xn; yn; … } would it have a limit?
2/3 (x+9) = 1/3 (12-x)
The sequence {xn} tends to zero.
a) Could there be members bigger than 1000000 in this sequence?
b) Could all the members of this sequence be negative?
c) Could all the members of this sequence be bigger than 0.000001?
With what accuracy should the length of the equator be measured to figure out the volume of the Earth with the accuracy of 1 km³ (If consider the Earth as the ideal sphere with the radius of 64000 km)?
Sides of a rectangle are measured with an accuracy of 1 cm. With what accuracy it is possible to measure the perimeter and area of the rectangle?
