It is known that \(\cos \alpha^{\circ} = 1/3\). Is \(\alpha\) a rational number?
Let \(f (x)\) be a polynomial of degree \(n\) with roots \(\alpha_1, \dots , \alpha_n\). We define the polygon \(M\) as the convex hull of the points \(\alpha_1, \dots , \alpha_n\) on the complex plane. Prove that the roots of the derivative of this polynomial lie inside the polygon \(M\).
a) Using geometric considerations, prove that the base and the side of an isosceles triangle with an angle of \(36^{\circ}\) at the vertex are incommensurable.
b) Invent a geometric proof of the irrationality of \(\sqrt{2}\).
The Newton method (see Problem 61328) does not always allow us to approach the root of the equation \(f(x) = 0\). Find the initial condition \(x_0\) for the polynomial \(f(x) = x (x - 1)(x + 1)\) such that \(f(x_0) \neq x_0\) and \(x_2 = x_0\).
Find the largest number of colours in which you can paint the edges of a cube (each edge with one colour) so that for each pair of colours there are two adjacent edges coloured in these colours. Edges are considered to be adjacent if they have a common vertex.
In a one-on-one tournament 10 chess players participate. What is the least number of rounds after which the single winner could have already been determined? (In each round, the participants are broken up into pairs. Win – 1 point, draw – 0.5 points, defeat – 0).
Hannah placed 101 counters in a row which had values of 1, 2 and 3 points. It turned out that there was at least one counter between every two one point counters, at least two counters lie between every two two point counters, and at least three counters lie between every two three point counters. How many three point counters could Hannah have?
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?
The numbers \(1, 2, 3,\dots , 10\) are written around a circle in a particular order. Peter calculated the sum of each of the 10 possible groups of three adjacent numbers around the circle and wrote down the smallest value he had calculated. What is the largest possible value he could have written down?
There are two symmetrical cubes. Is it possible to write some numbers on their faces so that the sum of the points when throwing these cubes on the upwards facing face on landing takes the values 1, 2, ..., 36 with equal probabilities?