Problems

Is it possible for the mean of some 35 whole numbers to equal $6.35$?

One of $n$ prizes is embedded in each chewing gum pack, where each prize has probability $1/n$ of being found.

How many packets of gum, on average, should I buy to collect the full collection prizes?

We call the geometric-harmonic mean of numbers $a$ and $b$ the general limit of the sequences $\{a_n\}$ and $\{b_n\}$ constructed according to the rule $a_0=a$, $b_0=b$, $a_{n+1}=\frac{2a_n{b}_n}{a_n+b_n}$, $b_{n+1}=\sqrt{a_n{b}_n}$ ($n\ge 0$).

We denote it by $\nu (a,b)$. Prove that $\nu (a,b)$ is related to $\mu (a,b)$ (see problem number 61322) by $\nu (a,b)\times \mu (1/a,1/b)=1$.

Problem number 61322 says that both of these sequences have the same limit.

This limit is called the arithmetic-geometric mean of the numbers $a,b$ and is denoted by $\mu (a,b)$.

A class contains 33 pupils, who have a combined age of 430 years. Prove that if we picked the 20 oldest pupils they would have a combined age of no less than 260 years. The age of any given pupil is a whole number.

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?

Gabby and Joe cut rectangles out of checkered paper. Joe is lazy; He throws a die once and cuts out a square whose side is equal to the number of points that are on the upwards facing side of the die. Gabby throws the die twice and cuts out a rectangle with the length and width equal to the numbers which come out from the die. Who has the mathematical expectation of the rectangle of a greater area?

An exam is made up of three trigonometry problems, two algebra problems and five geometry problems. Martin is able to solves trigonometry problems with probability $p_1=0.2$, geometry problems with probability $p_2=0.4$, and algebra problems with probability $p_3=0.5$. To get a $B$, Martin needs to solve at least five problems, where the grades are as follows $(A+,A,B,C,D)$.

a) With what probability does Martin solve at least five problems?

Martin decided to work hard on the problems of any one section. Over a week, he can increase the probability of solving the problems of this section by 0.2.

b) What section should Martin complete, so that the probability of solving at least five problems becomes the greatest?

c) Which section should Martin deal with, so that the mathematical expectation of the number of solved problems becomes the greatest?

According to the rules of a chess match, the winner is declared to be the one who has beaten their opponent by two defeats. Draws do not count. The probability of winning for both rivals is the same. The number of successful games played in such a match is random. Find its mathematical expectation.

$N$ people lined up behind each other. The taller people obstruct the shorter ones, and they cannot be seen.

What is the mathematical expectation of the number of people that can be seen?

In the centre of a rectangular billiard table that is 3 m long and 1 m wide, there is a billiard ball. It is hit by a cue in a random direction. After the impact the ball stops passing exactly 2 m. Find the expected number of reflections from the sides of the table.