In the first term of the year Daniel received five grades in mathematics with each of them being on a scale of 1 to 5, and the most common grade among them was a 5 . In this case it turned out that the median of all his grades was 4, and the arithmetic mean was 3.8. What grades could Daniel have?
A sailor can only serve on a submarine if their height does not exceed 168 cm. There are four teams \(A\), \(B\), \(C\) and \(D\). All sailors in these teams want to serve on a submarine and have been rigorously selected. There remains the last selection round – for height.
In team \(A\), the average height of sailors is 166 cm.
In team \(B\), the median height of the sailors is 167 cm.
In team \(C\), the tallest sailor has a height of 169 cm.
In team \(D\), the mode of the height of the sailors is 167 cm.
In which team, can at least half of the sailors definitely serve on the submarine?
The point \(O\) is randomly chosen on piece of square paper. Then the square is folded in such a way that each vertex is overlaid on the point \(O\). The figure shows one of the possible folding schemes. Find the mathematical expectation of the number of sides of the polygon that appears.
The point \(O\), lying inside the triangle \(ABC\), is connected by segments with the vertices of the triangle. Prove that the variance of the set of angles \(AOB\), \(AOC\) and \(BOC\) is less than a) \(10\pi ^2/27\); b) \(2\pi ^2/9\).
Valerie wrote the number 1 on the board, and then several more numbers. As soon as Valerie writes the next number, Mike calculates the median of the already available set of numbers and writes it in his notebook. At some point, in Mike’s notebook, the numbers: 1; 2; 3; 2.5; 3; 2.5; 2; 2; 2; 2.5 are written.
a) What is the fourth number written on the board?
b) What is the eighth number written on the board?
On a laundry drying line \(n\) socks hang in a random order (the order in which they got out of the washing machine). Among them there are the two favourite socks of the Scattered Scientist. The socks are covered by a drying sheet, so the Scientist does not see them, and takes out one sock by touch. Find the mathematical expectation of the number of socks taken out by the Scientist by the time he has both of his favourite socks.
An ant goes out of the origin along a line and makes \(a\) steps of one unit to the right, \(b\) steps of one unit to the left in some order, where \(a > b\). The wandering span of the ant is the difference between the largest and smallest coordinates of the ant for the entire length of its journey.
a) Find the largest possible wandering range.
b) Find the smallest possible range.
c) How many different sequences of motion of the ant are there, where the wandering range is the greatest possible?
What is the smallest number of cells that can be chosen on a \(15\times15\) board so that a mouse positioned on any cell on the board touches at least two marked cells? (The mouse also touches the cell on which it stands.)
There are 40 identical cords. If you set any cord on fire on one side, it burns, and if you set it alight on the other side, it will not burn. Ahmed arranges the cords in the form of a square (see the figure below, each cord makes up a side of a cell). Then, Helen arranges 12 fuses. Will Ahmed be able to lay out the cords in such a way that Helen will not be able to burn all of them?
What is the largest number of horses that can be placed on an \(8\times8\) chessboard so that no horse touches more than seven of the others?