Problems

There are 30 students in the class. Prove that the probability that some two students have the same birthday is more than 50%.

Prove that in any infinite decimal fraction you can rearrange the numbers so that the resulting fraction becomes a rational number.

Prove that $\sqrt{\frac{a^2+b^2}{2}}\ge \frac{a+b}{2}$.

We are given rational positive numbers $p,q$ where $1/p+1/q=1$. Prove that for positive $a$ and $b$, the following inequality holds: $ab\le \frac{a^p}{p}+\frac{b^q}{q}$.

Let $p$ and $q$ be positive numbers where $1/p+1/q=1$. Prove that $$a_1{b}_1+a_2{b}_2+\cdots +a_n{b}_n\le (a_1^p+\dots a_n^p{)}^{1/p}(b_1^q+\cdots +b_n^q{)}^{1/q}$$ The values of the variables are considered positive.

Find the largest value of the expression $a+b+c+d-ab-bc-cd-da$, if each of the numbers $a$, $b$, $c$ and $d$ belongs to the interval $[0,1]$.

Is it possible to:

a) load two coins so that the probability of “heads” and “tails” were different, and the probability of getting any of the combinations “tails, tails,” “heads, tails”, “heads, heads” be the same?

b) load two dice so that the probability of getting any amount from 2 to 12 would be the same?

On a calculator keypad, there are the numbers from 0 to 9 and signs of two actions (see the figure). First, the display shows the number 0. You can press any keys. The calculator performs the actions in the sequence of clicks. If the action sign is pressed several times, the calculator will only remember the last click.

a) The button with the multiplier sign breaks and does not work. The Scattered Scientist pressed several buttons in a random sequence. Which result of the resulting sequence of actions is more likely: an even number or an odd number?

b) Solve the previous problem if the multiplication symbol button is repaired.

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