a) The vertices (corners) in a regular polygon with 10 sides are colored black and white in an alternating fashion (i.e. one vertex is black, the next is white, etc). Two people play the following game. Each player in turn draws a line connecting two vertices of the same color. These lines must not have common vertices (i.e. must not begin or end on the same dot as another line) with the lines already drawn. The winner of the game is the player who made the final move. Which player, the first or the second, would win if the right strategy is used?
b) The same problem, but for a regular polygon with 12 sides.
The edges of a cube are assigned with integer values. For each vertex we look at the numbers corresponding to the three edges coming from this vertex and add them up. In case we get 8 equal results we call such cube “cute”. Are there any “cute” cubes with the following numbers corresponding to the edges:
(a) \(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\);
(b) \(-6, -5, -4, -3, -2, -1, 1, 2, 3, 4, 5, 6\)?
The area of a rectangle is 1 cm\(^2\). Can its perimeter be greater than 1 km?
A unit square contains 51 points. Prove that it is always possible to cover three of them with a circle of radius \(\frac{1}{7}\).
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 length of the hypotenuse of a right-angled triangle is 3.
a) The Scattered Scientist calculated the dispersion of the lengths of the sides of this triangle and found that it equals 2. Was he wrong in the calculations?
b) What is the smallest standard deviation of the sides that a rectangular triangle can have? What are the lengths of its sides, adjacent to the right angle?
The upper side of a piece of square paper is white, and the lower one is red. In the square, a point F is randomly chosen. Then the square is bent so that one randomly selected vertex overlaps the point F. Find the mathematical expectation of the number of sides of the red polygon that appears.
40% of adherents of some political party are women. 70% of the adherents of this party are townspeople. At the same time, 60% of the townspeople who support the party are men. Are the events “the adherent of the party is a townsperson” and “the adherent of party is a woman” independent?
On weekdays, the Scattered Scientist goes to work along the circle line on the London Underground from Cannon Street station to Edgware Road station, and in the evening he goes back (see the diagram).
Entering the station, the Scientist sits down on the first train that arrives. It is known that in both directions the trains run at approximately equal intervals, and along the northern route (via Farringdon) the train goes from Cannon Street to Edgware Road or back in 17 minutes, and along the southern route (via St James Park) – 11 minutes. According to an old habit, the scientist always calculates everything. Once he calculated that, from many years of observation:
– the train going counter-clockwise, comes to Edgware Road on average 1 minute 15 seconds after the train going clockwise arrives. The same is true for Cannon Street.
– on a trip from home to work the Scientist spends an average of 1 minute less time than a trip home from work.
Find the mathematical expectation of the interval between trains going in one direction.
Out of the given numbers 1, 2, 3, ..., 1000, find the largest number \(m\) that has this property: no matter which \(m\) of these numbers you delete, among the remaining \(1000 - m\) numbers there are two, of which one is divisible by the other.