Let \(p\) and \(q\) be positive numbers where \(1 / p + 1 / q = 1\). Prove that \[a_1b_1 + a_2b_2 + \dots + a_nb_n \leq (a_1^p + \dots a_n^p)^{1/p}(b_1^q +\dots + b_n^q)^{1/q}\] The values of the variables are considered positive.
Draw all of the stairs made from four bricks in descending order, starting with the steepest \((4, 0, 0, 0)\) and ending with the shallowest \((1, 1, 1, 1)\).
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.
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?
Let \((1 + \sqrt {2} + \sqrt {3})^n = p_n + q_n \sqrt {2} + r_n \sqrt {3} + s_n \sqrt {6}\) for \(n \geq 0\). Find:
a) \(\lim \limits_ {n \to \infty} {\frac {p_n} {q_n}}\); b) \(\lim \limits_ {n \to \infty} {\frac {p_n} {r_n}}\); c) \(\lim \limits_ {n \to \infty} {\frac {p_n} {s_n}}\);
Find the generating functions of the sequences of Chebyshev polynomials of the first and second kind: \[F_T(x,z) = \sum_{n=0}^{\infty}T_n(x)z^n;\quad F_U(x,z) = \sum_{n=0}^{\infty}U_n(X)z^n.\]
Definitions of Chebyshev polynomials can be found in the handbook.
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) \(P_{k, l}(n) = P_{k, l} (kl - n)\).
Author: D.E. Shnol
On the island of Truthland, all of the inhabitants may be mistaken, but the younger ones never contradict the elders, and when the older ones contradict the younger ones, they (the elders) are not mistaken. Between the residents A, B and C there was such a conversation:
A: B is the tallest.
B: A is the tallest.
C: I’m taller than B.
Does it follow from this conversation that the younger the person, the taller he or she is (for the three people having this conversation)?
Author: I.V. Izmestyev
Postman Pat did not want to give away the parcel. So, Matt suggested that he play the following game: every move, Pat writes in a line from left to right the letters M and P, randomly alternating them, until he has a line made up of 11 letters. Matt, after each of Pat’s moves, if he wants, swaps any two letters. If in the end it turns out that the recorded word is a palindrome (that is, it is the same if read from left to right and right to left), then Pat gives Matt the parcel. Can Matt play in such a way as to get the parcel?
Authors: Folklore
There are 13 pupils in the school of witchcraft and wizardry. Before the Clairvoyance exam, the teacher put them at a round table and asked to guess who would receive the clairvoyant’s diploma. The students said nothing about themselves and two of their neighbours, but they wrote the following about all of the others: “None of these ten will get the diploma!" Of course, all of those who passed the exam guessed correctly, and all of the other students were mistaken. How many wizards received a diploma?