A fly moves from the origin only to the right or upwards along the lines of the integer grid (a monotonic wander). In each node of the net, the fly randomly selects the direction of further movement: upwards or to the right.
a) Prove that sooner or later the fly will reach the point with abscissa 2011.
b) Find the mathematical expectation of the ordinate of the fly at the moment when the fly reached the abscissa 2011.
Hercules meets the three-headed snake Hydra of Lerna. Every minute, Hercules chops off one head of the snake. Let \(x\) be the survivability of the snake (\(x > 0\)). The probability \(p_s\) of the fact that in the place of the severed head will grow s new heads \((s = 0, 1, 2)\) is equal to \(\frac{x^s}{1 + x + x^2}\).
During the first 10 minutes of the battle, Hercules recorded how many heads grew in place of each chopped off one. The following vector was obtained: \(K = (1, 2, 2, 1, 0, 2, 1, 0, 1, 2)\). Find the value of the survivability of the snake, under which the probability of the vector \(K\) is greatest.
The functions \(f\) and \(g\) are defined on the entire number line and are reciprocal. It is known that \(f\) is represented as a sum of a linear and a periodic function: \(f (x) = kx + h (x)\), where \(k\) is a number, and \(h\) is a periodic function. Prove that \(g\) is also represented in this form.
It is known that \(a > 1\). Is it always true that \(\lfloor \sqrt{\lfloor \sqrt{a}\rfloor }\rfloor = \lfloor \sqrt{4}{a}\rfloor\)?
Does there exist a function \(f (x)\) defined for all real numbers such that \(f(\sin x) + f (\cos x) = \sin x\)?
A firm recorded its expenses in pounds for 100 items, creating a list of 100 numbers (with each number having no more than two decimal places). Each accountant took a copy of the list and found an approximate amount of expenses, acting as follows. At first, he arbitrarily chose two numbers from the list, added them, discarded the sum after the decimal point (if there was anything) and recorded the result instead of the selected two numbers. With the resulting list of 99 numbers, he did the same, and so on, until there was one whole number left in the list. It turned out that in the end all the accountants ended up with different results. What is the largest number of accountants that could work in the company?
Are there functions \(p (x)\) and \(q (x)\) such that \(p (x)\) is an even function and \(p (q (x))\) is an odd function (different from identically zero)?
The function \(f (x)\) is defined for all real numbers, and for any \(x\) the equalities \(f (x + 2) = f (2 - x)\) and \(f (x + 7) = f (7 - x)\) are satisfied. Prove that \(f (x)\) is a periodic function.
We took several positive numbers and constructed the following sequence: \(a_1\) is the sum of the initial numbers, \(a_2\) is the sum of the squares of the original numbers, \(a_3\) is the sum of the cubes of the original numbers, and so on.
a) Could it happen that up to \(a_5\) the sequence decreases (\(a_1> a_2> a_3> a_4> a_5\)), and starting with \(a_5\) – it increases (\(a_5 < a_6 < a_7 <\dots\))?
b) Could it be the other way around: before \(a_5\) the sequence increases, and starting with \(a_5\) – decreases?
The number \(x\) is such that both the sums \(S = \sin 64x + \sin 65x\) and \(C = \cos 64x + \cos 65x\) are rational numbers.
Prove that in both of these sums, both terms are rational.