Problems

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?

Author: L.N. Vaserstein

For any natural numbers $a_1,a_2,\dots ,a_m$, no two of which are equal to each other and none of which is divisible by the square of a natural number greater than one, and also for any integers and non-zero integers $b_1,b_2,\dots ,b_m$ the sum is not zero. Prove this.

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 $n=1,2,3,\dots$

Out of the given numbers 1, 2, 3, ..., 1000, find the largest number $m$ that has this property: no matter which $m$ of these numbers you delete, among the remaining $1000-m$ numbers there are two, of which one is divisible by the other.

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: A.A. Egorov

Calculate the square root of the number $0.111\dots 111$ (100 ones) to within a) 100; b) 101; c)* 200 decimal places.

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?