## Problem 4

Find $x>1$ for which $\frac{1}{[x]}+\frac{1}{\{x\}}=2014 x$, where $[x]$ denotes the integer part of $x$, and $\{x\}$ represents the fractional part of $x$.

## Mathematical Gazette

## Note.

All problems are mandatory.

Each problem is worth 7 points.

Working time 3 hours.

Proposers: Prof. Hecser Enikö, Pescărus Teodor

## MURES COUNTY SCHOOL INSPECTION  S.S.M.R - MURES BRANCH  Mathematics Olympiad  Local phase  14.02.2014  9th Grade  Grading Rubric
## Problem 4

Find $x>1$ for which $\frac{1}{[x]}+\frac{1}{\{x\}}=2014 x$, where $[x]$ denotes the integer part of $x$, and $\{x\}$ represents the fractional part of $x$.

## Solution

The equation is equivalent to $\frac{\{x\}+[x]}{[x] \cdot\{x\}}=2014 x$.

Since $\{x\}+[x]=x$ and $x>1$, we get $[x] \cdot\{x\}=\frac{1}{2014}$.

If $[x]=n$, then $\{x\}=\frac{1}{2014 n}\frac{1}{2014}$, which means $n \geq 1$. (2p)

In conclusion, any $x=n+\frac{1}{2014 n}$, with $n \in \mathbf{N}^{*}$ is a solution to the equation. (1p)