Problems

According to one implausible legend, Cauchy and Bunyakovsky were very fond of playing darts in the evenings. But the target was unusual – the sectors on it were unequal, so the probability of getting into different sectors was not the same. Once Cauchy throws a dart and hits the target. Bunyakovsky throws the next one. Which is more likely: that Bunyakovsky will hit the same sector that Cauchy’s dart went into, or that his dart will land on the next sector clockwise?

Three cyclists travel in one direction along a circular track that is 300 meters long. Each of them moves with a constant speed, with all of their speeds being different. A photographer will be able to make a successful photograph of the cyclists, if all of them are on some part of the track which has a length of $d$ meters. What is the smallest value of $d$ for which the photographer will be able to make a successful photograph sooner or later?

We took several positive numbers and constructed the following sequence: $a_1$ is the sum of the initial numbers, $a_2$ is the sum of the squares of the original numbers, $a_3$ is the sum of the cubes of the original numbers, and so on.

a) Could it happen that up to $a_5$ the sequence decreases ($a_1>a_2>a_3>a_4>a_5$), and starting with $a_5$ – it increases ($a_5<a_6<a_7<\dots$)?

b) Could it be the other way around: before $a_5$ the sequence increases, and starting with $a_5$ – decreases?

A grasshopper can make jumps of 8, 9 and 10 cells in any direction on a strip of $n$ cells. We will call the natural number $n$ jumpable if the grasshopper can, starting from some cell, bypass the entire strip, having visited each cell exactly once. Find at least one $n>50$ that is not jumpable.

One hundred gnomes weighing each 1, 2, 3, ..., 100 pounds, gathered on the left bank of a river. They cannot swim, but on the same shore is a rowing boat with a carrying capacity of 100 pounds. Because of the current, it’s hard to swim back, so each gnome has enough power to row from the right bank to the left one no more than once (it’s enough for any one of the gnomes to row in the boat, the rower does not change during one voyage). Will all gnomes cross to the right bank?

The number $x$ is such that both the sums $S=\sin 64x+\sin 65x$ and $C=\cos 64x+\cos 65x$ are rational numbers.

Prove that in both of these sums, both terms are rational.

For all real $x$ and $y$, the equality $f(x^2+y)=f(x)+f(y^2)$ holds. Find $f(-1)$.

Prove that for any odd natural number, $a$, there exists a natural number, $b$, such that $2^b-1$ is divisible by $a$.

On a lottery ticket, it is necessary for Mary to mark 8 cells from 64. What is the probability that after the draw, in which 8 cells from 64 will also be selected (all such possibilities are equally probable), it turns out that Mary guessed

a) exactly 4 cells? b) exactly 5 cells? c) all 8 cells?

Prove that for any positive integer $n$, it is always possible to find a number, consisting of the digits $1$ and $2,$ that is divisible by $2^n$. (For example, $2$ is divisible by $2$, $12$ is divisible by $4,$ $112$ is divisible by $8,$ $2112$ is divisible by $16$ and so on...).