Hello , help please I need to solve those equations :
1) $\displaystyle [x]+[2x]+[4x]+[8x]+[16x]+[32x]=12345$
2) $\displaystyle [x^2]=[x]^2$
3) $\displaystyle [\frac{x^2-2x}{3}]=\frac{x}{2}$
Ah, right. Are you familiar with the following property, for the first problem:
$\displaystyle \lfloor kx \rfloor=\left\lfloor x\right\rfloor + \left\lfloor x+\frac{1}{k}\right\rfloor +\dots+\left\lfloor x+\frac{k-1}{k}\right\rfloor$
In other words:
$\displaystyle \lfloor 2x \rfloor=\left\lfloor x\right\rfloor + \left\lfloor x+\frac{1}{2}\right\rfloor $
$\displaystyle \lfloor 4x \rfloor=\left\lfloor x\right\rfloor + \left\lfloor x+\frac{1}{4}\right\rfloor+ \left\lfloor x+\frac{2}{4}\right\rfloor + \left\lfloor x+\frac{3}{4}\right\rfloor$
Followed by the rule that:
$\displaystyle \lfloor x + n \rfloor = \lfloor x \rfloor + n $
Hence:
$\displaystyle \lfloor 2x \rfloor=\left\lfloor x\right\rfloor + \left\lfloor x\right\rfloor +\frac{1}{2}$ etc...
For the 2nd one, think of it like this:
$\displaystyle x = k + p $ where k is an integer, and p is a non-integer between 0 and 1.
In mathematical terms: $\displaystyle k \in Z \text{, } p \notin Z \text{ and } 0 < p<1 $
Hence:
$\displaystyle \lfloor (k + p)^2 \rfloor = \lfloor k + p \rfloor^2 $
$\displaystyle \lfloor k^2+ 2pk+ p^2 \rfloor = \lfloor k + p \rfloor^2 $
On the RHS, we know that k is the integer part of the number so:
$\displaystyle \lfloor k^2 +2pk+ p^2 \rfloor = k^2 $
In other words, the NON-integer part of the LHS must be $\displaystyle 2pk+p^2 $
So now you need to come up with values of k and p such that:
$\displaystyle k \in Z \text{, } p \notin Z \text{, } 0 < p < 1\text{ and } 0 < 2pk+p^2 < 1 $
For example, p = 0.1, k = 1 is a solution. (x = 1.1, in other words).
i'm not sure about 1) but i guess there is no solution. my solutions to 2) and 3):
clearly any integer is a solution. so we'll assume that $\displaystyle x \notin \mathbb{Z}.$ let $\displaystyle [x]=n.$ then $\displaystyle n < x < n+1 $ and $\displaystyle [x^2 - n^2]=0,$ which is equivalent to $\displaystyle 0 < x^2 - n^2 < 1. \ \ \ \ (1)$
case 1. $\displaystyle x < 0$: in this case, since $\displaystyle x > n,$ we'll have: $\displaystyle x^2 < n^2.$ but from (1) we have $\displaystyle x^2 > n^2.$ this contradiction means there's no solution in this case.
case 2. $\displaystyle x > 0$: then the conditions $\displaystyle n < x < n+1$ and (1) are equivalent to $\displaystyle n < x < \sqrt{n^2 + 1}.$ conversely if $\displaystyle n < x < \sqrt{n^2 + 1},$ for some positive integer $\displaystyle n,$ then
$\displaystyle n < x < n+1$ and $\displaystyle n^2 < x^2 < n^2 + 1.$ thus $\displaystyle [x]=n$ and $\displaystyle [x^2]=n^2=[x]^2.$
so the complete set of solutions of the equation is: $\displaystyle \bigcup_{n \in \mathbb{N} \cup \{0 \}} (n , \sqrt{n^2 + 1}) \cup \mathbb{Z}.$
the equation is equivalent to these two conditions together: $\displaystyle \frac{x}{2} \in \mathbb{Z}$ and $\displaystyle \frac{x^2 - 2x}{3} - 1 < \frac{x}{2} \leq \frac{x^2 - 2x}{3}. \ \ \ \ (2)$
3) $\displaystyle [\frac{x^2-2x}{3}]=\frac{x}{2}$
let $\displaystyle x = 2n,$ where $\displaystyle n \in \mathbb{Z}.$ solving the inequalities in (2) for $\displaystyle n$ gives you: $\displaystyle \left[\frac{7-\sqrt{97}}{8}, 0 \right] \cup \left[\frac{7}{4}, \frac{7 + \sqrt{97}}{8} \right],$ which gives us $\displaystyle n = 0,2.$ thus $\displaystyle x=2n=0,4.$
There is no solution
Using the following two properties:
$\displaystyle \lfloor kx \rfloor=\left\lfloor x\right\rfloor + \left\lfloor x+\frac{1}{k}\right\rfloor +\dots+\left\lfloor x+\frac{k-1}{k}\right\rfloor$
$\displaystyle \lfloor x + n \rfloor = \lfloor x \rfloor + n $
It can be deduced that:
$\displaystyle \displaystyle \lfloor kx \rfloor = k \lfloor x \rfloor + \sum_{n = 1}^{k-1} \frac{n}{k} $
Hence:
$\displaystyle \displaystyle \lfloor 2x \rfloor = 2\lfloor x \rfloor + \sum_{n = 1}^{1} \frac{n}{2} =2\lfloor x \rfloor + \frac{1}{2} $
$\displaystyle \displaystyle \lfloor 4x \rfloor = 4 \lfloor x \rfloor + \sum_{n = 1}^{3} \frac{n}{4} = 4\lfloor x \rfloor + \frac{3}{2} $
And in general $\displaystyle \displaystyle \lfloor kx \rfloor = k\lfloor x \rfloor + \frac{k-1}{2} $
Continuing like this, the equation can be reduced to:
$\displaystyle 63 \lfloor x \rfloor + \frac{1+3+7+15+31}{2} = 12345 $
$\displaystyle 63\lfloor x \rfloor + \frac{57}{2} = 12345 $
$\displaystyle 63\lfloor x \rfloor = \frac{24633}{2} $
$\displaystyle \lfloor x \rfloor = \frac{391}{2} $
It is non-integer, and so there are no solutions.
this is not true! why? see below for example:
$\displaystyle \lfloor 2x \rfloor - 2\lfloor x \rfloor$ is an integer but $\displaystyle \frac{1}{2}$ is not. so they can never be equal. the function $\displaystyle \lfloor 2x \rfloor - 2\lfloor x \rfloor$ is always either 0 or 1.
Hence:
$\displaystyle \displaystyle \lfloor 2x \rfloor = 2\lfloor x \rfloor + \sum_{n = 1}^{1} \frac{n}{2} =2\lfloor x \rfloor + \frac{1}{2} $
look this
When he plugs $\displaystyle x = 196$, the result is $\displaystyle 12348$.We're going to need to take off three. Now imagine we decrease that $\displaystyle 196$ slightly. Every term on the lefthand side will decrease by AT LEAST one (for example, $\displaystyle [32\cdot 196] = 6272, [32\cdot 195.99] = 6271$), therefore the sum will decrease by AT LEAST six, and that's too much.
some one can more explain this way