A frog jumps over the vertices of the triangle \(ABC\), moving each time to one of the neighbouring vertices.
How many ways can it get from \(A\) to \(A\) in \(n\) jumps?
In the Republic of mathematicians, the number \(\alpha > 2\) was chosen and coins were issued with denominations of 1 pound, as well as in \(\alpha^k\) pounds for every natural \(k\). In this case \(\alpha\) was chosen so that the value of all the coins, except for the smallest, was irrational. Could it be that any amount of a natural number of pounds can be made with these coins, using coins of each denomination no more than 6 times?
Author: A.V. Shapovalov
We call a triangle rational if all of its angles are measured by a rational number of degrees. We call a point inside the triangle rational if, when we join it by segments with vertices, we get three rational triangles. Prove that within any acute-angled rational triangle there are at least three distinct rational points.
Is there a positive integer \(n\) such that \[\sqrt{n}{17\sqrt{5} + 38} + \sqrt{n}{17\sqrt{5} - 38} = 2\sqrt{5}\,?\]
Author: A. Khrabrov
Do there exist integers \(a\) and \(b\) such that
a) the equation \(x^2 + ax + b = 0\) does not have roots, and the equation \(\lfloor x^2\rfloor + ax + b = 0\) does have roots?
b) the equation \(x^2 + 2ax + b = 0\) does not have roots, and the equation \(\lfloor x^2\rfloor + 2ax + b = 0\) does have roots?
Note that here, square brackets represent integers and curly brackets represent non-integer values or 0.
Author: A.K. Tolpygo
12 grasshoppers sit on a circle at various points. These points divide the circle into 12 arcs. Let’s mark the 12 mid-points of the arcs. At the signal the grasshoppers jump simultaneously, each to the nearest clockwise marked point. 12 arcs are formed again, and jumps to the middle of the arcs are repeated, etc. Can at least one grasshopper return to his starting point after he has made a) 12 jumps; b) 13 jumps?
Author: A.K. Tolpygo
An irrational number \(\alpha\), where \(0 <\alpha <\frac 12\), is given. It defines a new number \(\alpha_1\) as the smaller of the two numbers \(2\alpha\) and \(1 - 2\alpha\). For this number, \(\alpha_2\) is determined similarly, and so on.
a) Prove that for some \(n\) the inequality \(\alpha_n <3/16\) holds.
b) Can it be that \(\alpha_n> 7/40\) for all positive integers \(n\)?
Author: L.N. Vaserstein
For any natural numbers \(a_1, a_2, \dots , a_m\), no two of which are equal to each other and none of which is divisible by the square of a natural number greater than one, and also for any integers and non-zero integers \(b_1, b_2, \dots , b_m\) the sum is not zero. Prove this.
The segment \(OA\) is given. From the end of the segment \(A\) there are 5 segments \(AB_1, AB_2, AB_3, AB_4, AB_5\). From each point \(B_i\) there can be five more new segments or not a single new segment, etc. Can the number of free ends of the constructed segments be 1001? By the free end of a segment we mean a point belonging to only one segment (except point \(O\)).
Solve the equation \(x^3 - \lfloor x\rfloor = 3\).