We are given rational positive numbers \(p, q\) where \(1/p + 1/q = 1\). Prove that for positive \(a\) and \(b\), the following inequality holds: \(ab \leq \frac{a^p}{p} + \frac{b^q}{q}\).
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) \(P_{k, l}(n) = P_{k, l} (kl - n)\).
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?