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).
A castle is surrounded by a circular wall with nine towers, at which there are knights on duty. At the end of each hour, they all move to the neighbouring towers, each knight moving either clockwise or counter-clockwise. During the night, each knight stands for some time at each tower. It is known that there was an hour when at least two knights were on duty at each tower, and there was an hour when there was precisely one knight on duty on each of exactly five towers. Prove that there was an hour when there were no knights on duty on one of the towers.
Find the largest value of the expression \(a + b + c + d - ab - bc - cd - da\), if each of the numbers \(a\), \(b\), \(c\) and \(d\) belongs to the interval \([0, 1]\).
A polynomial of degree \(n > 1\) has \(n\) distinct roots \(x_1, x_2, \dots , x_n\). Its derivative has the roots \(y_1, y_2, \dots , y_{n-1}\). Prove the inequality \[\frac{x_1^2 + \dots + x_n^2}{n}> \frac{y_1^2 + \dots + y_n^2}{n}.\]
We are given 111 different natural numbers that do not exceed 500. Could it be that for each of these numbers, its last digit coincides with the last digit of the sum of all of the remaining numbers?
The number \(x\) is such a number that exactly one of the four numbers \(a = x - \sqrt{2}\), \(b = x-1/x\), \(c = x + 1/x\), \(d = x^2 + 2\sqrt{2}\) is not an integer. Find all such \(x\).
The numbers \(x\), \(y\) and \(z\) are such that all three numbers \(x + yz\), \(y + zx\) and \(z + xy\) are rational, and \(x^2 + y^2 = 1\). Prove that the number \(xyz^2\) is also rational.
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?
A function \(f\) is given, defined on the set of real numbers and taking real values. It is known that for any \(x\) and \(y\) such that \(x > y\), the inequality \((f (x)) ^2 \leq f (y)\) is true. Prove that the set of values generated by the function is contained in the interval \([0,1]\).
Three players are playing knockout table tennis – that is, the player who loses a game swaps places with the player who did not take part in that game and the winner stays on. In total Andrew played 10 games, Ben played 15, and Charlotte played 17. Which player lost the second game played?