Problems

At all rational points of the real line, integers are arranged. Prove that there is a segment such that the sum of the numbers at its ends does not exceed twice the number on its middle.

Prove that for any positive integer $n$ the inequality

is true.

The functions $f(x)-x$ and $f(x^2)-x^6$ are defined for all positive $x$ and increase. Prove that the function

also increases for all positive $x$.

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 $[0,1]$.

Find the largest natural number $n$ which satisfies $n^{200}<5^{300}$.

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

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.