A class has 25 pupils. It is known that for any two girls in the class, the number of male friends they have in the class is different. What is the maximum number of girls that it is possible for there to be in the class?
In the king’s prison, there are five cells numbered from 1 to 5. In each cell, there is one prisoner. Kristen persuaded the king to conduct an experiment: on the wall of each cell she writes at one point a number and at midnight, each prisoner will go to the cell with the indicated number (if the number on the wall coincides with the cell number, the prisoner does not go anywhere). On the following night at midnight, the prisoners again must move from their cell to another cell according to the instructions on the wall, and they do this for five nights. If the location of prisoners in the cells for all six days (including the first) is never repeated, then Kristen will be given the title of Wisdom, and the prisoners will be released. Help Kristen write numbers in the cells.
The \(KUB\) is a cube. Prove that the ball, \(CIR\), is not a cube. (\(KUB\) and \(CIR\) are three-digit numbers, where different letters denote different numbers).
Prove that if the numbers \(x, y, z\) satisfy the following system of equations for some values of \(p\) and \(q\): \[\begin{aligned} y &= x^2 + px + q,\\ z &= y^2 + py + q,\\ x &= z^2 + pz + q, \end{aligned}\] then the inequality \(x^2y + y^2z + z^2x \geq x^2z + y^2x + z^2y\) is satisfied.
Can I replace the letters with numbers in the puzzle \(RE \times CTS + 1 = CE \times MS\) so that the correct equality is obtained (different letters need to be replaced by different numbers, and the same letters must correspond to the same digits)?
Find \(x^3 +y^3\) if \(x+y=5\) and \(x+y+x^2 y +xy^2 =24\).
On the \(xy\)-plane shown below is the graph of the function \(y=ax^2 +c\). At which points does the graph of the function \(y=cx+a\) intersect the \(x\) and \(y\) axes?
Find the largest natural number \(n\) which satisfies \(n^{200} <5^{300}\).
Does there exist a natural number which, when divided by the sum of its digits, gives a quotient and remainder both equal to the number 2011?
Three circles are constructed on a triangle, with the medians of the triangle forming the diameters of the circles. It is known that each pair of circles intersects. Let \(C_{1}\) be the point of intersection, further from the vertex \(C\), of the circles constructed from the medians \(AM_{1}\) and \(BM_{2}\). Points \(A_{1}\) and \(B_{1}\) are defined similarly. Prove that the lines \(AA_{1}\), \(BB_{1}\) and \(CC_{1}\) intersect at the same point.