Olympiad problems

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Found problems: 3

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2011 Romania National Olympiad, 1

Let be a natural number $ n $ and $ n $ real numbers $ a_1,a_2,\ldots ,a_n $ such that $$ a_m+a_{m+1} +\cdots +a_n\ge \frac{(m+n)(n-m+1)}{2} ,\quad\forall m\in\{ 1,2,\ldots ,n \} . $$ Prove that $ a_1^2+a_2^2+\cdots +a_n^2\ge\frac{n(n+1)(2n+1)}{6} . $

2014 German National Olympiad, 2

Tags: combinatorics , number theory , divisibility , sequence , formula

For a positive integer $n$, let $y_n$ be the number of $n$-digit positive integers containing only the digits $2,3,5, 7$ and which do not have a $5$ directly to the right of a $2.$ If $r\geq 1$ and $m\geq 2$ are integers, prove that $y_{m-1}$ divides $y_{rm-1}.$

2012 Centers of Excellency of Suceava, 1

Tags: algebra , formula

Let be three nonzero rational numbers $ a,b,c $ under the relation $ (a+b)(b+c)(c+a)=a^2b^2c^2. $ Show that the expression $ \sqrt[3]{3+1/a^3+1/b^3+1/c^3} $ is rational. [i]Ion Bursuc[/i]

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