Inside a square with side 1 there are several circles, the sum of the radii of which is 0.51. Prove that there is a line that is parallel to one side of the square and that intersects at least 2 circles.
The base of the pyramid is a square. The height of the pyramid crosses the diagonal of the base. Find the largest volume of such a pyramid if the perimeter of the diagonal section containing the height of the pyramid is 5.
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?
Let’s denote any two digits with the letters \(A\) and \(X\). Prove that the six-digit number \(XAXAXA\) is divisible by 7 without a remainder.
A continuous function \(f(x)\) is such that for all real \(x\) the following inequality holds: \(f(x^2) - (f (x))^2 \geq 1/4\). Is it true that the function \(f(x)\) necessarily has an extreme point?
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 cinema contains 7 rows each with 10 seats. A group of 50 children went to see the morning screening of a film, and returned for the evening screening. Prove that there will be two children who sat in the same row for both the morning and the evening screening.
The numbers \(p\) and \(q\) are such that the parabolas \(y = - 2x^2\) and \(y = x^2 + px + q\) intersect at two points, bounding a certain figure.
Find the equation of the vertical line dividing the area of this figure in half.
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?