Like the title said:

Is it true that if \(\lceil x\lfloor x\rfloor\rceil+\lfloor x\lceil x\rceil\rfloor=A\) has real solutions of \(x\) with positive range of \(\alpha\le x\le\beta\), then \(A=\alpha\left\lceil\ \sqrt{\frac A2}\ \right\rceil+\beta\left\lfloor\ \sqrt{\frac A2}\ \right\rfloor\) must be true?

Inspired by Raghav Vaidyanathan which was inspired by Pi Han Goh.

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestHmm , nice inspirations !

@Raghav Vaidyanathan , @Pi Han Goh

Log in to reply

@Pi Han Goh Sir , does the equality hold on the upper limit side?

Log in to reply

I don't understand your question at all. The claim clearly states that we have to first find \(\alpha\) and \(\beta\).

Log in to reply

C++ code... Tested for a lot of numbers.. seems that it isn't always true... especially in the vicinity of numbers of form \(2n^2\)

Log in to reply

So you've found a counterexample? Can you show me which values of \(A\) shows that is not true?

Log in to reply

I think \(A=129\) doesn't satisfy. And also, numbers like \(A=128\) have \(\alpha=\beta\).

Log in to reply

Log in to reply

Log in to reply

So my statement still holds.

If this is a cop out explanation, then you've successfully disproven my statement! (and I can't have that, haha!)

Log in to reply

Log in to reply

That means to say that we can't take \(A=137\) as an example because the range of solution of positive \(x\) is an open interval: \( \left(8, \frac{73}9\right)\).

Log in to reply

Okay this is disproven! Thank you for your cooperation. I knew this is too good to be true.

Log in to reply