There are 13 weights. It is known that any 12 of them could be placed in 2 scale cups with 6 weights in each cup in such a way that balance will be held.
Prove the mass of all the weights is the same, if it is known that:
a) the mass of each weight in grams is an integer;
b) the mass of each weight in grams is a rational number;
c) the mass of each weight could be any real (not negative) number.
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: \(ab \leq \frac{a^p}{p} + \frac{b^q}{q}\).
Prove that the equation \(\frac {x}{y} + \frac {y}{z} + \frac {z}{x} = 1\) is unsolvable using positive integers.
Prove the inequality: \[\frac{(b_1 + \dots b_n)^{b_1 + \dots b_n}}{(a_1 + \dots a_n)^{b_1 + \dots + b_n}}\leq \left(\frac{b_1}{a_1}\right)^{b_1}\dots \left( \frac{b_n}{a_n}\right)^{b_n}\] where all variables are considered positive.
Prove that if the function \(f (x)\) is convex upwards on the line \([a, b]\), then for any distinct points \(x_1, x_2\) in \([a; b]\) and for any positive \(\alpha_{1}, \alpha_{2}\) such that \(\alpha_{1} + \alpha_ {2} = 1\) the following inequality holds: \(f(\alpha_1 x_1 + \alpha_2 x_2 ) > \alpha_1 f (x_1) + \alpha_2 f(x_2)\).
Inequality of Jensen. Prove that if the function \(f (x)\) is convex upward on \([a, b]\), then for any distinct points \(x_1, x_2, \dots , x_n\) (\(n \geq 2\)) from \([a; b]\) and any positive \(\alpha_{1}, \alpha_{2}, \dots , \alpha_{n}\) such that \(\alpha_ {1} + \alpha_{2} + \dots + \alpha_{n} = 1\), the following inequality holds: \(f (\alpha_{1} x_1 + \dots + \alpha_{n} x_n) > \alpha_{1} f (x_1) + \dots + \alpha_{n} f (x_n)\).
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.
Let the sequences of numbers \(\{a_n\}\) and \(\{b_n\}\), that are associated with the relation \(\Delta b_n = a_n\) (\(n = 1, 2, \dots\)), be given. How are the partial sums \(S_n\) of the sequence \(\{a_n\}\) \(S_n = a_1 + a_2 + \dots + a_n\) linked to the sequence \(\{b_n\}\)?
Definition. Let the function \(f (x, y)\) be valid at all points of a plane with integer coordinates. We call a function \(f (x, y)\) harmonic if its value at each point is equal to the arithmetic mean of the values of the function at four neighbouring points, that is: \[f (x, y) = 1/4 (f (x + 1, y) + f (x-1, y) + f(x, y + 1) + f (x, y-1)).\] Let \(f(x, y)\) and \(g (x, y)\) be harmonic functions. Prove that for any \(a\) and \(b\) the function \(af (x, y) + bg (x, y)\) is also harmonic.
Let \(f (x, y)\) be a harmonic function. Prove that the functions \(\Delta_{x} f (x, y) = f (x + 1, y) - f (x, y)\) and \(\Delta_{y}f(x , y) = f(x, y + 1) - f(x, y)\) will also be harmonic.