Let’s denote any two digits with the letters \(A\) and \(X\). Prove that the six-digit number \(XAXAXA\) is divisible by 7 without a remainder.
A continuous function \(f(x)\) is such that for all real \(x\) the following inequality holds: \(f(x^2) - (f (x))^2 \geq 1/4\). Is it true that the function \(f(x)\) necessarily has an extreme point?
A cinema contains 7 rows each with 10 seats. A group of 50 children went to see the morning screening of a film, and returned for the evening screening. Prove that there will be two children who sat in the same row for both the morning and the evening screening.
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.
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.
100 queens, that cannot capture each other, are placed on a \(100 \times 100\) chessboard. Prove that at least one queen is in each \(50 \times 50\) corner square.
The quadratic trinomials \(f (x)\) and \(g (x)\) are such that \(f' (x) g' (x) \geq | f (x) | + | g (x) |\) for all real \(x\). Prove that the product \(f (x) g (x)\) is equal to the square of some trinomial.
In 25 boxes there are spheres of different colours. It is known that for any \(k\) where \(1 \leq k \leq 25\) in any \(k\) of the boxes there are spheres of exactly \(k+1\) different colours. Prove that a sphere of one particular colour lies in every single box.
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.