Find the sum \(1/3 + 2/3 + 2^2/3 + 2^3/3 + \dots + 2^{1000}/3\).
The function \(f (x)\) is defined on the positive real \(x\) and takes only positive values. It is known that \(f (1) + f (2) = 10\) and \(f(a+b) = f(a) + f(b) + 2\sqrt{f(a)f(b)}\) for any \(a\) and \(b\). Find \(f (2^{2011})\).
How many integers are there from 0 to 999999, in the decimal notation of which there are no two identical numbers next to each other?
\(A\) and \(B\) shoot in a shooting gallery, but they only have one six-shot revolver with one cartridge. Therefore, they agreed in turn to randomly rotate the drum and shoot. \(A\) goes first. Find the probability that the shot will occur when \(A\) has the revolver.
At the Antarctic station, there are \(n\) polar explorers, all of different ages. With the probability \(p\) between each two polar explorers, friendly relations are established, regardless of other sympathies or antipathies. When the winter season ends and it’s time to go home, in each pair of friends the senior gives the younger friend some advice. Find the mathematical expectation of the number of those who did not receive any advice.