Problems
Prove that for all \(x \in (0;\pi /2)\) for \(n > m\), where \(n, m\) are natural, we have the inequality \(2 | \sin^n x-\cos^n x | \leq 3 | \sin^m x-\cos^m x |\);
A regular hexagon with sides of length \(5\) is divided by straight lines, that are parallel to its sides, to form regular triangles with sides of length 1. We call the vertices of all such triangles nodes. It is known that more than half of the nodes are marked. Prove that there are five marked nodes lying on one circle.
Members of the State parliament formed factions in such a way that for any two factions \(A\) and \(B\) (not necessarily different)
– also a faction (through
the set of all parliament members not included in \(C\) is denoted). Prove that for any two factions \(A\) and \(B\), \(A \cup % \includegraphics{https://problems-static.s3.amazonaws.com/production/task_images/700/109909-3.png} B\) is also a faction.
Peter has some coins in his pocket. If Peter pulls \(3\) coins from his pocket, without looking, there will always be a £1 coin among them. If Peter pulls \(4\) coins from his pocket, without looking, there will always be a £2 coin among them. Peter pulls \(5\) coins from his pocket. Identify these coins.
One day all the truth tellers on the planet decided to carry a clearly visible mark of truth in order to be distinguished from liars. Two truth tellers and two liars met and looked at each other. Which of them could say the phrase:
“All of us are truth tellers.”
“Only one of you is a truth teller.”
“Exactly two of you are truth tellers.”
A New Year’s garland, hanging along the school corridor, consists of red and blue light bulbs. Next to each red light bulb there must necessarily be a blue one. What is the largest number of red light bulbs in this garland, if it consists of only 50 light bulbs?
Looking back at her diary, Natasha noticed that in the date 17/02/2008 the sum of the first four numbers are equal to the sum of the last four. When will this coincidence happen for the last time in 2008?
A magician with a blindfold gives a spectator five cards with the numbers from 1 to 5 written on them. The spectator hides two cards, and gives the other three to the assistant magician. The assistant indicates to the spectator two of them, and the spectator then calls out the numbers of these cards to the magician (in the order in which he wants). After that, the magician guesses the numbers of the cards hidden by the spectator. How can the magician and the assistant make sure that the trick always works?
In 10 boxes there are pencils (there are no empty boxes). It is known that in different boxes there is a different number of pencils, and in each box, all pencils are of different colors. Prove that from each box you can choose a pencil so that they will all be of different colors.
A block of cheese comes in packaging with parallel lines of different colours printed on it. If you cut along the red lines then you will get 5 slices of cheese, if you cut along the yellow lines then there will be 7 slices, and along the green lines you will get 11 slices. How many slices will you get if you cut along the lines of all three colours?