Definition. Let the function \(f (x, y)\) be valid at all points of a plane with integer coordinates. We call a function \(f (x, y)\) harmonic if its value at each point is equal to the arithmetic mean of the values of the function at four neighbouring points, that is: \[f (x, y) = 1/4 (f (x + 1, y) + f (x-1, y) + f(x, y + 1) + f (x, y-1)).\] Let \(f(x, y)\) and \(g (x, y)\) be harmonic functions. Prove that for any \(a\) and \(b\) the function \(af (x, y) + bg (x, y)\) is also harmonic.
Let \(f (x, y)\) be a harmonic function. Prove that the functions \(\Delta_{x} f (x, y) = f (x + 1, y) - f (x, y)\) and \(\Delta_{y}f(x , y) = f(x, y + 1) - f(x, y)\) will also be harmonic.
Liouville’s discrete theorem. Let \(f (x, y)\) be a bounded harmonic function (see the definition in problem number 11.28), that is, there exists a positive constant \(M\) such that \(\forall (x, y) \in \mathbb {Z}^2\) \(| f (x, y) | \leq M\). Prove that the function \(f (x, y)\) is equal to a constant.
Definition. The sequence of numbers \(a_0, a_1, \dots , a_n, \dots\), which, with the given \(p\) and \(q\), satisfies the relation \(a_{n + 2} = pa_{n + 1} + qa_n\) (\(n = 0,1,2, \dots\)) is called a linear recurrent sequence of the second order.
The equation \[x^2-px-q = 0\] is called a characteristic equation of the sequence \(\{a_n\}\).
Prove that, if the numbers \(a_0\), \(a_1\) are fixed, then all of the other terms of the sequence \(\{a_n\}\) are uniquely determined.
A frog jumps over the vertices of the triangle \(ABC\), moving each time to one of the neighbouring vertices.
How many ways can it get from \(A\) to \(A\) in \(n\) jumps?
Prove that the 13th day of the month is more likely to occur on a Friday than on other days of the week. It is assumed that we live in the Gregorian style calendar.
Find the coefficient of \(x\) for the polynomial \((x - a) (x - b) (x - c) \dots (x - z)\).
The following words/sounds are given: look, yar, yell, lean, lease. Determine what will happen if the sounds that make up these words are pronounced in reverse order.
A class contains 33 pupils, who have a combined age of 430 years. Prove that if we picked the 20 oldest pupils they would have a combined age of no less than 260 years. The age of any given pupil is a whole number.
In a one-on-one tournament 10 chess players participate. What is the least number of rounds after which the single winner could have already been determined? (In each round, the participants are broken up into pairs. Win – 1 point, draw – 0.5 points, defeat – 0).