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: 27

2018 VJIMC, 2
Tags: inequalities , floor function , integer part , algebra
Let $n$ be a positive integer and let $a_1\le a_2 \le \dots \le a_n$ be real numbers such that \[a_1+2a_2+\dots+na_n=0.\] Prove that \[a_1[x]+a_2[2x]+\dots+a_n[nx] \ge 0\] for every real number $x$. (Here $[t]$ denotes the integer satisfying $[t] \le t<[t]+1$.)

2019 District Olympiad, 4
Tags: algebra , floor function , integer part
Solve the equation in the set of real numbers: $$\left[ x+\frac{1}{x} \right] = \left[ x^2+\frac{1}{x^2} \right]$$ where $[a]$, represents the integer part of the real number $a$.