Find all functions \(f (x)\) such that \(f (2x + 1) = 4x^2 + 14x + 7\).
The function \(f\) is such that for any positive \(x\) and \(y\) the equality \(f (xy) = f (x) + f (y)\) holds. Find \(f (2007)\) if \(f (1/2007) = 1\).
The function \(f (x)\) is defined and satisfies the relationship \((x-1) f((x=1)/(x-1)) - f (x) = x\) for all \(x \neq 1\). Find all such functions.
Find all the functions \(f\colon \mathbb {R} \rightarrow \mathbb {R}\) which satisfy the inequality \(f (x + y) + f (y + z) + f (z + x) \geq 3f (x + 2y + 3z)\) for all \(x, y, z\).
Is there a bounded function \(f\colon \mathbb{R} \rightarrow \mathbb{R}\) such that \(f (1)> 0\) and \(f (x)\) satisfies the inequality \(f^2 (x + y) \geq f^2 (x) + 2f (xy) + f^2 (y)\) for all \(x, y \in \mathbb{R}\)?
On a function \(f (x)\), defined on the entire real line, it is known that for any \(a>1\) the function \(f (x) + f (ax)\) is continuous on the whole line. Prove that \(f (x)\) is also continuous on the whole line.
Find all functions \(f (x)\) defined for all real values of \(x\) and satisfying the equation \(2f (x) + f (1 - x) = x^2\).
Suppose that there are 15 prime numbers forming an arithmetic progression with a difference of \(d\). Prove that \(d >30,000\).