Solve the equation \((x + 1)^3 = x^3\).
The numbers \(p\) and \(q\) are such that the parabolas \(y = - 2x^2\) and \(y = x^2 + px + q\) intersect at two points, bounding a certain figure.
Find the equation of the vertical line dividing the area of this figure in half.
Of the four inequalities \(2x > 70\), \(x < 100\), \(4x > 25\) and \(x > 5\), two are true and two are false. Find the value of \(x\) if it is known that it is an integer.
The function \(f (x)\) is defined on the positive real \(x\) and takes only positive values. It is known that \(f (1) + f (2) = 10\) and \(f(a+b) = f(a) + f(b) + 2\sqrt{f(a)f(b)}\) for any \(a\) and \(b\). Find \(f (2^{2011})\).
Does there exist a real number \({\alpha}\) such that the number \(\cos {\alpha}\) is irrational, and all the numbers \(\cos 2{\alpha}\), \(\cos 3{\alpha}\), \(\cos 4{\alpha}\), \(\cos 5{\alpha}\) are rational?
The sequence of numbers \(a_1, a_2, \dots\) is given by the conditions \(a_1 = 1\), \(a_2 = 143\) and
for all \(n \geq 2\).
Prove that all members of the sequence are integers.
Solve the inequality: \(\lfloor x\rfloor \times \{x\} < x - 1\).
Can 100 weights of masses 1, 2, 3, ..., 99, 100 be arranged into 10 piles of different masses so that the following condition is fulfilled: the heavier the pile, the fewer weights in it?
How many integers are there from 0 to 999999, in the decimal notation of which there are no two identical numbers next to each other?
Prove that \(\frac {1}{2} (x^2 + y^2) \geq xy\) for any \(x\) and \(y\).