Problems

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 marketing company decided to carry out a sociological survey to find out which part of the urban population learns news mostly from radio programs, which part – from TV programs, which part – from the press, and which – from the Internet. For the study, it was decided to use a sample of 2,000 randomly chosen owners of email addresses. Can this sample be considered representative?

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 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?

A numerical set $x_1,\dots ,x_n$ is given. Consider the function $d(t)=\frac{mi{n}_{i=1,\dots ,n}|x_i-t|+ma{x}_{i=1,\dots ,n}|x_i-t|}{2}$.

a) Is it true that the function $d(t)$ takes the smallest value at a single point, for any set of numbers $x_1,\dots ,x_n$?

b) Compare the values of $d(c)$ and $d(m)$ where $c=\frac{mi{n}_{i=1,\dots ,n}{x}_i+ma{x}_{i=1,\dots ,n}{x}_i}{2}$ and $m$ is the median of the specified set.

In a numerical set there is 100 numbers. If you remove one number, the median of the remaining numbers will be 78. If you remove a different number, the median of the remaining numbers is 66. Find the median of the entire set.

In a numerical set of $n$ numbers, one of the numbers is 0 and another is 1.

a) What is the smallest possible variance of such a set of numbers?

b) What should be the set of numbers for this?

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) $2{\pi }^2/9$.

In the set $-5$, $-4$, $-3$, $-2$, $-1$, $0$, $1$, $2$, $3$, $4$, $5$, replace one number with two other integers so that the set variance and its mean remain unchanged.

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?