Prove that the function \(\cos \sqrt {x}\) is not periodic.
Prove the formulae: \(\arcsin (- x) = - \arcsin x\), \(\arccos (- x) = \pi - \arccos x\).
Prove that for a monotonically increasing function \(f (x)\) the equations \(x = f (f (x))\) and \(x = f (x)\) are equivalent.
Prove that \(\sqrt{\frac{a^2 + b^2}{2}} \geq \frac{a+b}{2}\).
Prove that the equation \(\frac {x}{y} + \frac {y}{z} + \frac {z}{x} = 1\) is unsolvable using positive integers.
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 \((1, 1, 1, 1)\).
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.
Definition. The sequence of numbers \(a_0, a_1, \dots , a_n, \dots\), which, with the given \(p\) and \(q\), satisfies the relation \(a_{n + 2} = pa_{n + 1} + qa_n\) (\(n = 0,1,2, \dots\)) is called a linear recurrent sequence of the second order.
The equation \[x^2-px-q = 0\] is called a characteristic equation of the sequence \(\{a_n\}\).
Prove that, if the numbers \(a_0\), \(a_1\) are fixed, then all of the other terms of the sequence \(\{a_n\}\) are uniquely determined.
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?