Does there exist a function \(f (x)\) defined for all real numbers such that \(f(\sin x) + f (\cos x) = \sin x\)?
Are there functions \(p (x)\) and \(q (x)\) such that \(p (x)\) is an even function and \(p (q (x))\) is an odd function (different from identically zero)?
The function \(f (x)\) is defined for all real numbers, and for any \(x\) the equalities \(f (x + 2) = f (2 - x)\) and \(f (x + 7) = f (7 - x)\) are satisfied. Prove that \(f (x)\) is a periodic function.
For all real \(x\) and \(y\), the equality \(f (x^2 + y) = f (x) + f (y^2)\) holds. Find \(f(-1)\).
Author: V.A. Popov
On the interval \([0; 1]\) a function \(f\) is given. This function is non-negative at all points, \(f (1) = 1\) and, finally, for any two non-negative numbers \(x_1\) and \(x_2\) whose sum does not exceed 1, the quantity \(f (x_1 + x_2)\) does not exceed the sum of \(f (x_1)\) and \(f (x_2)\).
a) Prove that for any number \(x\) on the interval \([0; 1]\), the inequality \(f (x_2) \leq 2x\) holds.
b) Prove that for any number \(x\) on the interval \([0; 1]\), the \(f (x_2) \leq 1.9x\) must be true?
What has a greater value: \(300!\) or \(100^{300}\)?
We consider a function \(y = f (x)\) defined on the whole set of real numbers and satisfying \(f (x + k) \times (1 - f (x)) = 1 + f (x)\) for some number \(k \ne 0\). Prove that \(f (x)\) is a periodic function.
\(f(x)\) is an increasing function defined on the interval \([0, 1]\). It is known that the range of its values belongs to the interval \([0, 1]\). Prove that, for any natural \(N\), the graph of the function can be covered by \(N\) rectangles whose sides are parallel to the coordinate axes so that the area of each is \(1/N^2\). (In a rectangle we include its interior points and the points of its boundary).
A numerical sequence is defined by the following conditions: \[a_1 = 1, \quad a_{n+1} = a_n + \lfloor \sqrt{a_n}\rfloor .\]
Prove that among the terms of this sequence there are an infinite number of complete squares.
The function \(f(x)\) on the interval \([a, b]\) is equal to the maximum of several functions of the form \(y = C \times 10^{- | x-d |}\) (where \(d\) and \(C\) are different, and all \(C\) are positive). It is given that \(f (a) = f (b)\). Prove that the sum of the lengths of the sections on which the function increases is equal to the sum of the lengths of the sections on which the function decreases.