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.
Prove that the polynomial \(P (x)\) is divisible by its derivative if and only if \(P (x)\) has the form \(P(x) = a_n(x - x_0)^n\).
Prove that for \(n > 0\) the polynomial \[P (x) = n^2x^{n + 2} - (2n^2 + 2n - 1) x^{n + 1} + (n + 1)^2x^n - x - 1\] is divisible by \((x - 1)^3\).
Prove that for \(n> 0\) the polynomial \(x^{2n + 1} - (2n + 1)x^{n + 1} + (2n + 1)x^n - 1\) is divisible by \((x - 1)^3\).
Prove that the polynomial \(P (x) = a_0 + a_1x + \dots + a_nx^n\) has a number \(-1\) which is a root of multiplicity \(m + 1\) if and only if the following conditions are satisfied: \[\begin{aligned} a_0 - a_1 + a_2 - a_3 + \dots + (-1)^{n}a_n &= 0,\\ - a_1 + 2a_2 - 3a_3 + \dots + (-1)^{n}na_n &= 0,\\ \dots \\ - a_1 + 2^{m}a_2 - 3^{m}a_3 + \dots + (-1)^{n}n^{m}a_n &= 0. \end{aligned}\]
In an attempt create diversity the government of the planet hired \(100\) truth tellers and \(100\) liars. Each of them has at least one friend. Once exactly \(100\) members said: “All my friends are honest” and exactly \(100\) members said: “All my friends are liars.” What is the smallest possible number of pairs of friends, one of whom is honest and the other a liar?
A class contains 33 pupils, who have a combined age of 430 years. Prove that if we picked the 20 oldest pupils they would have a combined age of no less than 260 years. The age of any given pupil is a whole number.
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).
Specify any solution of the puzzle: \(2014 + YES =BEAR\).
Of five coins, two are fake. One of the counterfeit coins is lighter than the real one, and the other is heavier than the real one by as much as the lighter one is lighter than the real coin.
Explain how in the three weighings, you can find both fake coins using scales without weights.