The polynomial \(P (x)\) of degree \(n\) has \(n\) distinct real roots.
What is the largest number of its coefficients that can be equal to zero?
Solve the equation \((x + 1)^3 = x^3\).
The sum of the positive numbers \(a, b, c\) is \(\pi / 2\). Prove that \(\cos a + \cos b + \cos c > \sin a + \sin b + \sin c\).
Prove that for each \(x\) such that \(\sin x \neq 0\), there is a positive integer \(n\) such that \(|\sin nx| \geq \sqrt{3}/2\).
For what natural numbers \(n\) are there positive rational but not whole numbers \(a\) and \(b\), such that both \(a + b\) and \(a^n + b^n\) are integers?
The base of the pyramid is a square. The height of the pyramid crosses the diagonal of the base. Find the largest volume of such a pyramid if the perimeter of the diagonal section containing the height of the pyramid is 5.
The volume of the regular quadrangular pyramid \(SABCD\) is equal to \(V\). The height \(SP\) of the pyramid is the edge of the regular tetrahedron \(SPQR\), the plane of the face \(PQR\) which is perpendicular to the edge \(SC\). Find the volume of the common part of these pyramids.
The height \(SO\) of a regular quadrilateral pyramid \(SABCD\) forms an angle \(\alpha\) with a side edge and the volume of this pyramid is equal to \(V\). The vertex of the second regular quadrangular pyramid is at the point \(S\), the centre of the base is at the point \(C\), and one of the vertices of the base lies on the line \(SO\). Find the volume of the common part of these pyramids.
The sequence \((a_n)\) is given by the conditions \(a_1 = 1000000\), \(a_{n + 1} = n \lfloor a_n/n\rfloor + n\). Prove that an infinite subsequence can be found within it, which is an arithmetic progression.
In the infinite sequence \((x_n)\), the first term \(x_1\) is a rational number greater than 1, and \(x_{n + 1} = x_n + \frac{1}{\lfloor x_n\rfloor }\) for all positive integers \(n\).
Prove that there is an integer in this sequence.
Note that in this problem, square brackets represent integers and curly brackets represent non-integer values or 0.