Solve the equation \(2x^x = \sqrt {2}\) for positive numbers.
One and a half diggers dig for a half hour and end up having dug half a pit. How many pits will two diggers dig in two hours?
Find the first 99 decimal places in the number expansion of \((\sqrt{26} + 5)^{99}\).
Let \(M\) be a finite set of numbers. It is known that among any three of its elements there are two, the sum of which belongs to \(M\).
What is the largest number of elements in \(M\)?
Someone arranged a 10-volume collection of works in an arbitrary order. We call a “disturbance” a situation where there are two volumes for which a volume with a large number is located to the left. For this volume arrangement, we call the number \(S\) the number of all of the disturbances. What values can \(S\) take?
Each of the three cutlets should be fried in a pan on both sides for five minutes each side. Only two cutlets can fit onto the frying pan. Is it possible to fry all three cutlets more quickly than in 20 minutes (if the time to turn over and transfer the cutlets is neglected)?
An area of airspace contains clouds. It turns out that the area can be divided by 10 aeroplanes into regions such that each region contains no more than one cloud. What is the largest number of clouds an aircraft can fly through whilst holding a straight line course.
A standard chessboard has more than a quarter of its squares filled with chess pieces. Prove that at least two adjacent squares, either horizontally, vertically, or diagonally, are occupied somewhere on the board.
Let \(f\) be a continuous function defined on the interval \([0; 1]\) such that \(f (0) = f (1) = 0\). Prove that on the segment \([0; 1]\) there are 2 points at a distance of 0.1 at which the function \(f 4(x)\) takes equal values.
A convex figure and point \(A\) inside it are given. Prove that there is a chord (that is, a segment joining two boundary points of a convex figure) passing through point \(A\) and dividing it in half at point \(A\).