We solve problems

Problems Geometry Solid geometry Visual geometry in space

Find the generating functions of the sequences of Chebyshev polynomials of the first and second kind: $F_{T} (x, z) = \sum_{n = 0}^{\infty} T_{n} (x) z^{n}; F_{U} (x, z) = \sum_{n = 0}^{\infty} U_{n} (X) z^{n} .$

Definitions of Chebyshev polynomials can be found in the handbook.

Problem #PRU-61525

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

We denote by $P_{k, l} (n)$ the number of partitions of the number $n$ into at most $k$ terms, each of which does not exceed $l$ . Prove the equalities:

a) $P_{k, l} (n) - P_{k, l - 1} (n) = P_{k - 1, l} (n - l)$ ;

b) $P_{k, l} (n) - P_{k - 1, l} (n) = P_{k, l - 1} (n - k)$ ;

c) $P_{k, l} (n) = P_{l, k} (n)$ ;

d) $P_{k, l} (n) = P_{k, l} (k l - n)$ .

Problem #PRU-64409

Problems Calculus Derivative Increasing and decreasing. Researching functions Algebra and arithmetic 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-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-64548

Problems Calculus Functions of one variable. Continuity Monotonicity, boundedness Algebra and arithmetic Algebraic inequalities and systems of inequalities Quadratic inequalites (several variables)

3.0

Find the largest value of the expression $a + b + c + d - a b - b c - c d - d a$ , if each of the numbers $a$ , $b$ , $c$ and $d$ belongs to the interval $[0, 1]$ .

Problem #PRU-64613

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

15–16 3.0

A polynomial of degree $n > 1$ has $n$ distinct roots $x_{1}, x_{2}, \dots, x_{n}$ . Its derivative has the roots $y_{1}, y_{2}, \dots, y_{n - 1}$ . Prove the inequality $\frac{x_{1}^{2} + \dots + x_{n}^{2}}{n} > \frac{y_{1}^{2} + \dots + y_{n}^{2}}{n} .$

Problem #PRU-64683

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

A group of several friends was in correspondence in such a way that each letter was received by everyone except for the sender. Each person wrote the same number of letters, as a result of which all together the friends received 440 letters. How many people could be in this group of friends?

Problem #PRU-64767

Problems Methods Examples and counterexamples. Constructive proofs Calculus Real numbers Rational and irrational numbers

13–15 4.0

In the Republic of mathematicians, the number $α > 2$ was chosen and coins were issued with denominations of 1 pound, as well as in $α^{k}$ pounds for every natural $k$ . In this case $α$ was chosen so that the value of all the coins, except for the smallest, was irrational. Could it be that any amount of a natural number of pounds can be made with these coins, using coins of each denomination no more than 6 times?

Problem #PRU-64770

Problems Methods Mathematical induction Mathematical induction (other) Calculus Functions of one variable. Continuity Monotonicity, boundedness Proof by contradiction

14–16 4.0

A function $f$ is given, defined on the set of real numbers and taking real values. It is known that for any $x$ and $y$ such that $x > y$ , the inequality $(f (x))^{2} \leq f (y)$ is true. Prove that the set of values generated by the function is contained in the interval $[0, 1]$ .

Problem #PRU-64883

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

In a line 40 signs are written out: 20 crosses and 20 zeros. In one move, you can swap any two adjacent signs. What is the least number of moves in which it is guaranteed that you can ensure that some 20 consecutive signs are crosses?