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\(\color{red}{\text{Unity}} \ \color{brown}{\text{Is}} \ \color{blue}{\text{Strength}} \ \color{green}{\text{Contest}}\)

\(\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad\)


Hello every \(-e^{i\pi}\) ,

I am holding a contest named "Unity Is Strength".The aim of this contest is to motivate the community to create beautiful mathematical expressions that equal 1 i.e unity.


What you have to do?

You have to generate an expression that equals \(1\) and post it on this note in the following format:

"Here's my entry: <expression> = 1"


Rules And Regulations:

  • The expression must be original and it must equal one.

  • Every member is allowed only 1 entry. You can update your entry by editing your comment, or deleting it and adding a new one.

  • Submission of famous identities is prohibited. For example : Proposing Euler's identity in this form: \(-e^{i\pi}=1\) is not allowed.

  • Just modifying the known identities/equations to make them equal 1 is prohibited. For example: We all know in \(\Delta ABC , \sum \tan A = \prod \tan A\) , so submitting it in this form: \(\dfrac{\sum \tan A}{\prod \tan A}=1\) is not allowed.

  • Trivial proposals like \(2-1=1\) or \(\dfrac{a\times 1}{a}=1\) or putting some numbers to force the expression to be equal to \(1\) are not expected.

  • Irrelevant/abusive comments would be deleted.

  • Please follow the rules and regulations.


Details about the contest:

  • This contest starts from the time this note is posted and ends on 18 October 2015 at 3:30 pm GMT. The entries after this time won't be accepted.

  • The entry with the maximum value of \(upvotes\) will be declared the winning entry. The member who posted that entry will be the Winner of the Unity Is Strength Contest. So please Do Upvote all the entries you like, and even if you downvote any entries ,it won't affect the score of that/those individuals.

  • The results will be announced at any day in the next week of the last date of submission.


The contest starts now. Please like,reshare this note so that it will reach most of the brilliantians!

Best Of Luck!

Note by Nihar Mahajan
1 year, 3 months ago

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Here's my entry :

\[\Re\left(\frac{2}{1-e^{i\theta}}\right)=1~~\forall~\theta\in\Bbb R\setminus\{x\mid x=2\pi n~,~n\in\Bbb Z\}\]

where \(\Re(z)\) denotes the real part of \(z\in \Bbb C\).

This is something that I noticed a few months ago while solving a few trigonometry problems on Brilliant. I'm not sure whether this complies with Rule #5 or not but I'm posting it anyway. Prasun Biswas · 1 year, 3 months ago

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@Prasun Biswas This is great! Julian Poon · 1 year, 3 months ago

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Here's my submission.

\[\lim _{ n\rightarrow \infty }{ \frac { \sin \left( 2\times 10^{ -n }° \right) -\sin \left( 2\times 10^{ -(n+1) }° \right) }{ \pi \times 10^{ -(n+2) } } } =1\]

This was not found by me, it was found by my friend about a year ago. Julian Poon · 1 year, 3 months ago

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@Julian Poon Can you tell your friend's name? Is he/she on brilliant? Nihar Mahajan · 1 year, 3 months ago

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@Nihar Mahajan He is called Isaac Lee. Not very active though. Julian Poon · 1 year, 3 months ago

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@Julian Poon It's very easy though it seems to be tough. It can be achieved by using L'Hospital Rule. Surya Prakash · 1 year, 3 months ago

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@Surya Prakash There's an even easier method, and that is using the approximation

\[\sin { \theta = } \theta \]

When \(\theta\) is very small. Julian Poon · 1 year, 3 months ago

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@Julian Poon yes!! it's very easy. You are right. But we should convert degrees to radians. Surya Prakash · 1 year, 3 months ago

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Here is my entry

\[\dfrac{\left(\displaystyle\int_0^1\ln\left(\frac{1}{x}\right)^{49} \dfrac{x^3}{1-2x^4} \text{dx} \right)\times{2^{101}}}{49! \text{Li}_{50}}=1\]

\(\text{Li}_{x} \) is the polylogarithm function. Kritarth Lohomi · 1 year, 3 months ago

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@Kritarth Lohomi Invite Parth too :)

  • Kritarth Lohomi : In response to Nihar Mahajan: He is busy..

  • Aditya Kumar In response to Kritarth Lohomi: Nice:) Well, this doesn't deserve downvotes.

  • Julian Poon In response to Aditya Kumar: Maybe downvotes should not be counted

  • Aditya Kumar In response to Julian Poon: Yes but still... it doesn't look good.

  • Nihar Mahajan In response to Julian Poon: They are not counted.

  • Anik Mandal In response to Kritarth Lohomi: Cool!

  • Siddharth Bhatt In response to Kritarth Lohomi: Nice!
Nihar Mahajan · 1 year, 3 months ago

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\[\int_{-1}^{1}\frac{d}{dx}\left(\frac{1.5}{2^{\frac{1}{x}} + 1}\right)dx\] Andrew Ellinor · 1 year, 3 months ago

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@Andrew Ellinor Isn't it an improper integral? It is discontinuous at \(x=0\) . Aditya Agarwal · 1 year, 3 months ago

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@Aditya Agarwal Yes, it is an improper integral. But that doesn't mean it can't be evaluated. Andrew Ellinor · 1 year, 3 months ago

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@Andrew Ellinor \(\color{white}{\text{=1 That's a wise one :P}}\) Nihar Mahajan · 1 year, 3 months ago

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Here's my entry:

\[1=\lim_{x\to0}\sum_{n=1}^{\infty}\sum_{r=0}^{\infty}\frac{(-1)^r}{n+r}\dbinom{n+r}{r}x^{n+r-1}\] Kenny Lau · 1 year, 3 months ago

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@Kenny Lau Nice! Julian Poon · 1 year, 3 months ago

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My entry: \[\left( \frac { \int _{ 0 }^{ \infty }{ \frac { a^{ 3 } }{ { e }^{ a }-1 } } da }{ \int _{ 0 }^{ \infty }{ { b }^{ 3 }{ e }^{ -b } } db } \right) .\frac { 90 }{ { \pi }^{ 4 } } =1\] Aditya Kumar · 1 year, 3 months ago

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@Aditya Kumar Lol!! Numerator is Riemann zeta function and denominator is Gamma function. Surya Prakash · 1 year, 3 months ago

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@Surya Prakash Yes. U r right! Aditya Kumar · 1 year, 3 months ago

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My submission: \[\Large\int^1_0\frac{\ln(\sin^{-1}[\int^3_e\Gamma(x)dx]+e)}{e^{\delta(\int^\pi_0[\cot x]dx)}}dx\]
where \([.]\) denotes the floor function/ greatest integer function, \(\delta(.)\) denotes the Delta Function and \(\Gamma(.)\) denotes the Gamma Function. All the best! Aditya Agarwal · 1 year, 3 months ago

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Here's my entry, \[\displaystyle{ \lim_{x \to 0}} \log_{tan^{2} x} \left(tan^{2} 2x \right) = 1\] Athiyaman Nallathambi · 1 year, 3 months ago

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@Athiyaman Nallathambi \(\quad\displaystyle\lim_{x\to0}\log_{\tan^2x}(\tan^22x)\)

\(=\displaystyle\lim_{x\to0}\frac{\log(\tan^22x)}{\log{\tan^2x}}\)

\(=\displaystyle\lim_{t\to0}\frac{\log\left(\frac{2t}{1-t^2}\right)^2}{\log{t^2}}\)

\(=\displaystyle\lim_{t\to0}\frac{\log\left(\frac{2t}{1-t^2}\right)}{\log{t}}\)

\(=\displaystyle\lim_{t\to0}\frac{\frac{2(1-t^2)-2t(-2t)}{(1-t^2)^2}/\left(\frac{2t}{1-t^2}\right)}{1/t}\)

\(=\displaystyle\lim_{t\to0}t\frac{2+2t^2}{(1-t^2)^2}/\left(\frac{2t}{1-t^2}\right)\)

\(=\displaystyle\lim_{t\to0}t\frac{2+2t^2}{1-t^2}/(2t)\)

\(=\displaystyle\lim_{t\to0}\frac{2+2t^2}{2(1-t^2)}\)

\(=\displaystyle\lim_{t\to0}\frac{1+t^2}{1-t^2}\)

\(=\displaystyle\frac11\)

\(=1\) Kenny Lau · 1 year, 3 months ago

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@Kenny Lau Well yeah you could do that. But wouldnt it be easier to apply L'Hopital rule right after applying change of base theorem? Athiyaman Nallathambi · 1 year, 3 months ago

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@Athiyaman Nallathambi Right.

\(\quad\displaystyle\lim_{x\to0}\log_{\tan^2x}(\tan^22x)\)

\(=\displaystyle\lim_{x\to0}\frac{\log(\tan^22x)}{\log{\tan^2x}}\)

\(=\displaystyle\lim_{x\to0}\frac{\log(\tan2x)}{\log{\tan x}}\)

\(=\displaystyle\lim_{x\to0}\frac{2\sec^2x/(\tan2x)}{\sec^2x/(\tan x)}\)

\(=\displaystyle\lim_{x\to0}\frac{2/(\tan2x)}{1/(\tan x)}\)

\(=\displaystyle\lim_{x\to0}\frac{2\tan x}{\tan2x}\)

\(=\displaystyle\lim_{x\to0}\frac{2\sec^2x}{2\sec^2x}\)

\(=\displaystyle1\) Kenny Lau · 1 year, 3 months ago

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@Kenny Lau Yeah that was my solution too.Good job. Athiyaman Nallathambi · 1 year, 3 months ago

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@Athiyaman Nallathambi Thanks! Kenny Lau · 1 year, 3 months ago

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my late entry @Nihar Mahajan \[\prod_{k=2}^\infty \left(\zeta(k)\sum_{n=1}^\infty \dfrac{\mu(n)}{n^k}\right) =1\] Aareyan Manzoor · 12 months ago

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@Aareyan Manzoor No problem. Though I am unable to actually evaluate the LHS , the equation looks interesting! Nihar Mahajan · 12 months ago

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@Sravanth Chebrolu Sorry to say , but I have confirmed that your proposal is not original with some people. Your proposal is not accepted. I am deleting it. Don't worry , you can post another , you have whole week in hand! :) Nihar Mahajan · 1 year, 3 months ago

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Here's my entry

\[\large{\frac { 7 }{ 26 } \left[ \frac { d }{ dx } { \left[ \tan ^{ -1 }{ \left( \frac { 2{ x }^{ 4 } }{ 3{ x }^{ 2 }-7x+1 } \right) } +\sin ^{ -1 }{ \left( \frac { 26x+4 }{ 7 } \right) } \right] }_{ x=0 } \right] =1}\] Lakshya Sinha · 1 year, 3 months ago

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@Lakshya Sinha The contest is already over and I am in process of posting the results. Don't delete this , I would mention this as a "late entry". Nihar Mahajan · 1 year, 3 months ago

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I just think that this would be fun if it implies the rule (but I don't think it is): \[\displaystyle \ce{^16_8A} \ce{^20_10B} = 1\] Jimmy PrevailLone · 1 year, 3 months ago

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@Jimmy PrevailLone Is the equation in your entry somehow related to radioactivity? Because I see a nice correlation between A and B (according to your notation and how it's written). B reduces to A after occurrence of one \(\alpha\)-decay (followed by a \(\gamma\)-decay, as usual to stabilize energy levels).

Then again, looking at the numbers, this might be something else because B doesn't satisfy the "n/p ratio > 1.5" criteria for radioactivity to occur. Prasun Biswas · 1 year, 3 months ago

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@Jimmy PrevailLone Even I don't understand your notation.Wait , are you using the notation for atomic mass , atomic number of an element? If so , then it is not allowed. Nihar Mahajan · 1 year, 3 months ago

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@Jimmy PrevailLone I don't really understand your notation. Mind if you elaborate on the notation used? Thanks a lot! Julian Poon · 1 year, 3 months ago

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@Jimmy PrevailLone This was just a silly equation which won't make any sense here. So A is Oxygen, denoted by O and B is Neon, denoted by Ne. When we connect both, we get ONe = 1 Jimmy PrevailLone · 1 year, 3 months ago

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@Jimmy PrevailLone Lol... I like that Nehemiah Osei · 1 year, 3 months ago

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@Jimmy PrevailLone Ah.. ok lol Julian Poon · 1 year, 3 months ago

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Here is my entry \[-(\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{\ldots}}}})(\large \lim_{n\to\infty}(\frac {n^2(n sin(\frac {1}{n})-2)}{1+3n^2}))=1\] Atul Shivam · 1 year, 3 months ago

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@Atul Shivam You mean \(n\to\infty\) right? And use "\left(...\right)" and also "\displaystyle" to make things neater :) Kenny Lau · 1 year, 3 months ago

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@Atul Shivam Sorry , but your submission includes the famous Ramanujan's nested radicals which is not allowed as stated in rule number 3. Don't delete this , let it stay :) Nihar Mahajan · 1 year, 3 months ago

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@Nihar Mahajan I have sent you some others as message on fb Atul Shivam · 1 year, 3 months ago

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@Atul Shivam This is actually a bit forced and not really original. The nested radical is actually a problem by Ramanujan which is well known to equal \(3\). So, you just multiply it with \(\frac {-1}3\) to get \(-1\), the \(\frac {-1}3\) coming very trivially from the limit. The negative sign at the very left results in getting \((-1)\times (-1)=1\).

Seeing that your profile picture is a picture of Ramanujan, this post looks quite apt, haha. Prasun Biswas · 1 year, 3 months ago

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@Prasun Biswas Lol I have not copied exactly the same I have made edits in it :-p Atul Shivam · 1 year, 3 months ago

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