This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 2

2007 Gheorghe Vranceanu, 1
Tags: number theory , base 10 number theory , base 10
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.

2012 Grigore Moisil Intercounty, 2
Tags: number theory , inequalities , base 10 , base 10 number theory
[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 . $