Found problems: 6
Given an arbitrary natural number $ n, $ is there a multiple of $ n $ whose base $ 10 $ representation can be written only with the digits $ 0,2,7? $ Explain.
Find all natural numbers $ n $ for which there exists two natural numbers $ a,b $ such that
$$ n=S(a)=S(b)=S(a+b) , $$
where $ S(k) $ denotes the sum of the digits of $ k $ in base $ 10, $ for any natural number $ k. $
[i]Vasile Zidaru[/i] and [i]Mircea Lascu[/i]
[b]a)[/b] Prove that
$$ k+\frac{1}{2}-\frac{1}{8k}<\sqrt{k^2+k}<k+\frac{1}{2}-\frac{1}{8k}+\frac{1}{16k^2} , $$
for any natural number $ k. $
[b]b)[/b] Prove that there exists four numbers $ \alpha,\beta,\gamma,\delta\in\{0,1,2,3,4,5,6,7,8,9\} $ such that
$$ \left\lfloor\sum_{k=1}^{2012} \sqrt{k(k+1)\left( k^2+k+1 \right)}\right\rfloor =\underbrace{\ldots\alpha \beta\gamma\delta}_{\text{decimal form}} $$
and $ \alpha +\delta =\gamma . $
Prove that for any natural number $ n, $ there exists a number having $ n+1 $ decimal digits, namely, $ k_0,k_1,k_2,\ldots ,k_n $, and a $ \text{(n+1)-tuple}, $ say $\left( \epsilon_0 ,\epsilon_1 ,\epsilon_2\ldots ,\epsilon_n \right)\in\{-1,1\}^{n+1} , $ that satisfies:
$$ 1\le \prod_{j=0}^n (2+j)^{k_j\cdot \epsilon_j}\le \sqrt[10^n-1]{2} $$
[i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]
Find all nonnegative numbers $ n $ which have the property that $ a_{2}\neq 9, $ where $ \sum_{i=1}^{\infty } a_i10^{-i} $ is the decimal representation of the fractional part of $ \sqrt{n(n+1)} . $
Let be two natural numbers $ m\ge n $ and a nonnegative integer $ r<2^n. $ How many numbers of $ m $ digits, each digit being either the number $ 1 $ or $ 2, $ are there whose residue modulo $ 2^n $ is $ r? $
[i]Dorel Miheț[/i]