Problems

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...).

A sequence of natural numbers $a_1<a_2<a_3<\cdots <a_n<\dots$ is such that each natural number is either a term in the sequence, can be expressed as the sum of two terms in the sequence, or perhaps the same term twice. Prove that $a_n\le n^2$ for any $n=1,2,3,\dots$

Let’s call a natural number good if in its decimal record we have the numbers 1, 9, 7, 3 in succession, and bad if otherwise. (For example, the number 197,639,917 is bad and the number 116,519,732 is good.) Prove that there exists a positive integer $n$ such that among all $n$-digit numbers (from ${10}^{n-1}$ to ${10}^{n-1}$) there are more good than bad numbers.

Try to find the smallest possible $n$.

An infinite sequence of digits is given. One may consider a finite set of consecutive digits and view it as a number in decimal expression, whose digits shall be read from left to right, as usual. Prove that, for any natural number $n$ which is relatively prime with 10, you can choose a finite set of consecutive digits which gives you a multiple of $n$.

Author: V.A. Popov

On the interval $[0;1]$ a function $f$ is given. This function is non-negative at all points, $f(1)=1$ and, finally, for any two non-negative numbers $x_1$ and $x_2$ whose sum does not exceed 1, the quantity $f(x_1+x_2)$ does not exceed the sum of $f(x_1)$ and $f(x_2)$.

a) Prove that for any number $x$ on the interval $[0;1]$, the inequality $f(x_2)\le 2x$ holds.

b) Prove that for any number $x$ on the interval $[0;1]$, the $f(x_2)\le 1.9x$ must be true?

The triangle $C_1{C}_{2}O$ is given. Within it the bisector $C_2{C}_3$ is drawn, then in the triangle $C_2{C}_{3}O$ – bisector $C_3{C}_4$ and so on. Prove that the sequence of angles ${\gamma }_n=C_{n+1}{C}_{n}O$ tends to a limit, and find this limit if $C_{1}O{C}_2=\alpha$.

A rectangular chocolate bar size $5\times 10$ is divided by vertical and horizontal division lines into 50 square pieces. Two players are playing the following game. The one who starts breaks the chocolate bar along some division line into two rectangular pieces and puts the resulting pieces on the table. Then players take turns doing the same operation: each time the player whose turn it is at the moment breaks one of the parts into two parts. The one who is the first to break off a square slice $1\times 1$ (without division lines) a) loses; b) wins. Which of the players can secure a win: the one who starts or the other one?

What has a greater value: $300!$ or ${100}^{300}$?

Prove that, for any integer $n$, among the numbers $n,n+1,n+2,\dots ,n+9$ there is at least one number that is mutually prime with the other nine numbers.

Numbers $1,2,3,\dots ,101$ are written out in a row in some order. Prove that one can cross out 90 of them so that the remaining 11 will be arranged in their magnitude (either increasing or decreasing).