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 \(f (1/2007) = 1\).
Suppose that there are 15 prime numbers forming an arithmetic progression with a difference of \(d\). Prove that \(d >30,000\).
Prove that for a monotonically increasing function \(f (x)\) the equations \(x = f (f (x))\) and \(x = f (x)\) are equivalent.
Prove that the 13th day of the month is more likely to occur on a Friday than on other days of the week. It is assumed that we live in the Gregorian style calendar.
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 \([0, 1]\).
Does there exist a function \(f (x)\) defined for all real numbers such that \(f(\sin x) + f (\cos x) = \sin x\)?
What has a greater value: \(300!\) or \(100^{300}\)?
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.
In a row there are 2023 numbers. The first number is 1. It is known that each number, except the first and the last, is equal to the sum of two neighboring ones. Find the last number.