Found problems: 12
Let $n$ be a positive integer. Prove that there exists a finite sequence $S$ consisting of only zeros and ones, satisfying the following property: for any positive integer $d \geq 2$, when $S$ is interpreted in base $d$, the resulting number is non-zero and divisible by $n$.
[i]Remark: The sequence $S=s_ks_{k-1} \cdots s_1s_0$ interpreted in base $d$ is the number $\sum_{i=0}^{k}s_id^i$[/i]
Let the base $2$ representation of $x\in[0;1)$ be $x=\sum_{i=0}^\infty \frac{x_i}{2^{i+1}}$. (If $x$ is dyadically rational, i.e. $x\in\left\{\frac{k}{2^n}\,:\, k,n\in\mathbb{Z}\right\}$, then we choose the finite representation.) Define function $f_n:[0;1)\to\mathbb{Z}$ by
$$f_n(x)=\sum_{j=0}^{n-1}(-1)^{\sum_{i=0}^j x_i}.$$Does there exist a function $\varphi:[0;\infty)\to[0;\infty)$ such that $\lim_{x\to\infty} \varphi(x)=\infty$ and
$$\sup_{n\in\mathbb{N}}\int_0^1 \varphi(|f_n(x)|)\mathrm{d}x<\infty\, ?$$
Let a sequence $(a_i)_{i=10}^{\infty}$ be defined as follows:
[list=a]
[*] $a_{10}$ is some positive integer, which can of course be written in base 10.
[*] For $i \geq 10$ if $a_i > 0$, let $b_i$ be the positive integer whose base-$(i + 1)$ representation is the same as $a_i$'s base-$i$ representation. Then let $a_{i + 1} = b_i - 1$. If $a_i = 0$, $a_{i + 1} = 0$.
[/list]
For example, if $a_{10} = 11$, then $b_{10} = 11_{11} (= 12_{10})$; $a_{11} = 11_{11} - 1 = 10_{11} (= 11_{10})$; $b_{11} = 10_{12} (= 12_{10})$; $a_{12} = 11$.
Does there exist $a_{10}$ such that $a_i$ is strictly positive for all $i \geq 10$?
The sequence $a_n$ is defined for any $n\geq 10$ by the following inductive rule:
[list]
[*] $a_{10}=5778$
[*] If $a_n=0$ then $a_{n+1}=0$.
[*] If $a_n\neq0$ then $a_{n+1}$ is the number whose base-$(n+1)$ representation equals the base $n$ representation of the number $a_n -1$.
[/list]
For example,
$a_{11}=5\cdot11^3+7\cdot11^2+7\cdot11^1+7\cdot11^0=7586$
$a_{12}=5\cdot12^3+7\cdot12^2+7\cdot12^1+6\cdot12^0=9738$
[list=a]
[*] Does there exist $n\geq10$ for which $a_n=0$?
[*] Is $a_{1,000,000}=0$?
[*] Is $a_{100^{100^{100}}}=0$?
[/list]
Determine all integers $n \geq 2$ for which the number $11111$ in base $n$ is a perfect square.
Suppose that $y$ is a positive integer written only with digit $1$, in base $9$ system. Prove that $y$ is a triangular number, that is, exists positive integer $n$ such that the number $y$ is the sum of the $n$ natural numbers from $1$ to $n$.
Determine all integers $n \geq 2$ for which the number $11111$ in base $n$ is a perfect square.
Given an integer $m > 1$, we call the number $x{}$ dangerous if $x{}$ divides the number $y{}$, which is obtained by writing the digits of $x{}$ in base $m{}$ in reverse order, with $x\neq y$. Prove that if there exists a three-digit (in base $m$) dangerous number for a given $m$, then there exists a two-digit (in base $m$) dangerous number.
We consider the following operation applied to a positive integer: The integer is represented in an arbitrary base $b \ge 2$, in which it has exactly two digits and in which both digits are different from $0$. Then the two digits are swapped and the result in base $b$ is the new number.
Is it possible to transform every number $> 10$ to a number $\le 10$ with a series of such operations?
(Theresia Eisenkölbl)
Determine (with proof) whether there is a subset $X$ of the integers with the following property: for any integer $n$ there is exactly one solution of $a + 2b = n$ with $a,b \in X$.
Show that, for each integer $z \ge 3$, there exist two two-digit numbers $A$ and $B$ in base $z$, one equal to the other one read in reverse order, such that the equation $x^2 -Ax+B$ has one double root. Prove that this pair is unique for a given $z$. For instance, in base $10$ these numbers are $A = 18, B = 81$.
Let positive integers $K$ and $d$ be given. Prove that there exists a positive integer $n$ and a sequence of $K$ positive integers $b_1,b_2,..., b_K$ such that the number $n$ is a $d$-digit palindrome in all number bases $b_1,b_2,..., b_K$.