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?
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?
In Anchuria, presidential elections are being prepared, in which President Miraflores wants to win. Exactly half of the voters support Miraflores, and the other half support Dick Maloney. Miraflores is also a voter. According to the law, he has the right to divide all of the voters into two constituencies at his own discretion. In each of the districts, the voting is conducted as follows: each voter marks the name of their candidate on the ballot; all ballots are placed in the ballot box. Then one random ballot is chosen from the ballot box, and the one whose name is marked on it will win in this district. The candidate wins the election only if he wins in both districts. If the winner does not appear, the next round of voting is appointed according to the same rules. How should Miraflores divide the electorate in order to maximize the probability of his victory on the first round?
In a terrible thunderstorm, along a rope ladder, \(n\) dwarfs ascend in a chain. If suddenly there is a thunderbolt, then from fear, every gnome, regardless of others, can fall with probability \(p\) (\(0 < p < 1\)). If the dwarf falls, then he also hits all of the dwarfs that are below him. Find:
a) The probability that exactly \(k\) dwarfs will fall.
b) The mathematical expectation of the number of fallen dwarfs.
What is the minimum number of \(1\times 1\) squares that need to be drawn in order to get an image of a \(25\times 25\) square divided into 625 smaller 1x1 squares?
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?
A box contains 111 red, blue, green, and white marbles. It is known that if we remove 100 marbles from the box, without looking, we will always have removed at least one marble of each colour. What is the minimum number of marbles we need to remove to guarantee that we have removed marbles of 3 different colours?
A box contains 100 red, blue, and white marbles. It is known that if we remove 26 marbles from the box, without looking, we will always have removed at least 10 marbles of one colour. What is the minimum number of marbles we need to remove to guarantee that we have removed 30 marbles of the same colour?
At a contest named “Ah well, monsters!”, 15 dragons stand in a row. Between neighbouring dragons the number of heads differs by 1. If the dragon has more heads than both of his two neighbors, he is considered cunning, if he has less than both of his neighbors – strong, the rest (including those standing at the edges) are considered ordinary. In the row there are exactly four cunning dragons – with 4, 6, 7 and 7 heads and exactly three strong ones – with 3, 3 and 6 heads. The first and last dragons have the same number of heads.
a) Give an example of how this could occur.
b) Prove that the number of heads of the first dragon in all potential examples is the same.