We solve problems

Problems Calculus Functions of several variables

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 $\forall (x, y) \in Z^{2}$ $| f (x, y) | \leq M$ . Prove that the function $f (x, y)$ is equal to a constant.

Problem #PRU-61458

Problems Algebra Sequences Recurrent relations Linear recurrent relations Methods Mathematical induction

13–16 2.0

Definition. The sequence of numbers $a_{0}, a_{1}, \dots, a_{n}, \dots$ , which, with the given $p$ and $q$ , satisfies the relation $a_{n + 2} = p a_{n + 1} + q a_{n}$ ( $n = 0, 1, 2, \dots$ ) is called a linear recurrent sequence of the second order.

The equation $x^{2} - p x - q = 0$ 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.

Problem #PRU-61474

Problems Probability and statistics Probability theory Discrete distribution Algebra Sequences Recurrent relations Linear recurrent relations

15–16 4.0

The frog jumps over the vertices of the hexagon $A B C D E F$ , each time moving to one of the neighbouring vertices.

a) How many ways can it get from $A$ to $C$ in $n$ jumps?

b) The same question, but on condition that it cannot jump to $D$ ?

c) Let the frog’s path begin at the vertex $A$ , and at the vertex $D$ there is a mine. Every second it makes another jump. What is the probability that it will still be alive in $n$ seconds?

d)* What is the average life expectancy of such frogs?

Problem #PRU-61535

Problems Number Theory Divisibility Division with remainders. Arithmetic of remainders Division with remainder Calculus Functions of one variable. Continuity Periodicity and aperiodicity Probability and statistics Probability theory Probability theory (other) Methods Algebraic methods Proof by exhaustion

13–16 4.0

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.

Problem #PRU-61538

Problems Fun Problems Joke Problems Algebra Polynomials Polynomials(other)

12–16 2.0

Find the coefficient of $x$ for the polynomial $(x - a) (x - b) (x - c) \dots (x - z)$ .

Problem #PRU-64409

Problems Calculus Derivative Increasing and decreasing. Researching functions Algebra Surds. Rational powers Irrational equations.

15–16 3.0

Prove that the polynomial $P (x)$ is divisible by its derivative if and only if $P (x)$ has the form $P (x) = a_{n} (x - x_{0})^{n}$ .

Problem #PRU-64410

Problems Calculus Derivative Derivative and multiple roots

15–16 3.0

Prove that for $n > 0$ the polynomial $P (x) = n^{2} x^{n + 2} - (2 n^{2} + 2 n - 1) x^{n + 1} + (n + 1)^{2} x^{n} - x - 1$ is divisible by $(x - 1)^{3}$ .

Problem #PRU-64411

Problems Calculus Derivative Derivative and multiple roots

15–16 3.0

Prove that for $n > 0$ the polynomial $x^{2 n + 1} - (2 n + 1) x^{n + 1} + (2 n + 1) x^{n} - 1$ is divisible by $(x - 1)^{3}$ .

Problem #PRU-64412

Problems Calculus Derivative Derivative and multiple roots

15–16 4.0

Prove that the polynomial $P (x) = a_{0} + a_{1} x + \dots + a_{n} x^{n}$ has a number $- 1$ which is a root of multiplicity $m + 1$ if and only if the following conditions are satisfied: $\begin{aligned} a_{0} - a_{1} + a_{2} - a_{3} + \dots + (- 1)^{n} a_{n} & = 0, \\ - a_{1} + 2 a_{2} - 3 a_{3} + \dots + (- 1)^{n} n a_{n} & = 0, \\ \dots \\ - a_{1} + 2^{m} a_{2} - 3^{m} a_{3} + \dots + (- 1)^{n} n^{m} a_{n} & = 0. \end{aligned}$

Problem #PRU-64433

Problems Algebra Mean values Methods Pigeonhole principle Pigeonhole principle (other)

13–16 3.0

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.