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) \(P_{k, l}(n) = P_{k, l} (kl - n)\).
Find the largest value of the expression \(a + b + c + d - ab - bc - cd - da\), if each of the numbers \(a\), \(b\), \(c\) and \(d\) belongs to the interval \([0, 1]\).
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?
In the Republic of mathematicians, the number \(\alpha > 2\) was chosen and coins were issued with denominations of 1 pound, as well as in \(\alpha^k\) pounds for every natural \(k\). In this case \(\alpha\) 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?
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 \([0,1]\).
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?
Solve the system of equations: \[\begin{aligned} \sin y - \sin x &= x-y; &&\text{and}\\ \sin y - \sin z &= z-y; && \text{and}\\ x-y+z &= \pi. \end{aligned}\]
At a round table, 2015 people are sitting down, each of them is either a knight or a liar. Knights always tell the truth, liars always lie. They were given one card each, and on each card a number is written; all the numbers on the cards are different. Looking at the cards of their neighbours, each of those sitting at the table said: “My number is greater than that of each of my two neighbors.” After that, \(k\) of the sitting people said: “My number is less than that of each of my two neighbors.” At what maximum \(k\) could this occur?
Author: I.I. Bogdanov
Peter wants to write down all of the possible sequences of 100 natural numbers, in each of which there is at least one 3, and any two neighbouring terms differ by no more than 1. How many sequences will he have to write out?
a) There are three identical large vessels. In one there are 3 litres of syrup, in the other – 20 litres of water, and the third is empty. You can pour all the liquid from one vessel into another or into a sink. You can choose two vessels and pour into one of them liquid from the third, until the liquid levels in the selected vessels are equal. How can you get 10 litres of diluted 30% syrup?
b) The same, but there is \(N\) l of water. At what integer values of \(N\) can you get 10 liters of diluted 30% syrup?