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Problem Writing Party: June 6th to 26th

Problem Writing Party 8 was a great success, and we had so many comments that the page started loading slowly for some people. Thanks for being part of the conversation!

I'm working on creating the quizzes, and will update the following list ASAP. Some of you would also have received B-notifications that your problems were added to the quizzes over the course of the previous week. Keep it up! We value your submissions, and would love to feature more of them.

Here are the quizzes that the Brilliant community helped create:

31 New Brilliant Challenge Quizzes

Arithmetic Progressions: Level 1, Level 2, Level 3
GCD/LCM: Level 1, Level 2, Level 3, Level 4, Level 5
Conditional Probability: Level 2, Level 3, Level 4, Level 5
Euler's Theorem: Level 2, Level 3, Level 4, Level 5
Distribution into Bins: Level 2, Level 3, Level 4, Level 5
Limits of Functions: Level 1, Level 2, Level 3, Level 4
Chess Tactical: Level 2, Level 3, Level 4
Chess Abstract: Level 2, Level 3
Pattern Recognition:

Let's kick off our 9th Problem Writing Party!

How it Works

The party starts right now (June 6th, 2016) and will last for the next two weeks. Throughout the two weeks, we will be focusing on writing awesome problems for the topics listed in quizzes that need your help on the publish page. The topics are:

All levels:Chess TacticalRectangular Grid WalkPower Mean inequalityRoots of Unity
Low levels:FunctionsClassification of TrianglesNewton's Law of Gravity
High levels:Triangle CentersMarkov Chains

To join, submit as many problems as you want to these listed topics. At the end of the party, Brilliant staff will be picking the best 5-10 problems for each topic. These problems will then be immortalized and formed into a challenge quiz. If we pick your problem, then you can brag to your friends because it will be displayed on Brilliant forever! Your problem has a better chance of being selected if you include a graphic (when appropriate) and a solution.

Happy writing and keep the party alive!

Use this note to

  1. Ask questions about the party or brainstorming ideas from Brilliant staff.

  2. Share links to great relevant problems.

  3. Bounce your ideas off each other to help formulate the best problem you can.

  4. If you're posting your problems, please keep it to one rooted comment (and I will be merging such comments from the same person). This helps us keep the page more orderly.
    You can link to your own problems by using the markdown syntax of [text](url link).

Note by Calvin Lin
3 months, 3 weeks ago

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I had a really good one once but I forgot it. I hate that. Whitney Clark · 3 months, 3 weeks ago

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@Whitney Clark Is it like a fabulous proof for which this comment is too small to contain? I hate that too. Calvin Lin Staff · 3 months, 3 weeks ago

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@Calvin Lin :-) I have a bad memory for some things. Some of my best ideas come when I don't have a pen and paper on hand, it seems. Whitney Clark · 3 months, 3 weeks ago

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This is my submission for the Power Mean Inequality... And my first submission for the problem writing parties... Manuel Kahayon · 3 months, 3 weeks ago

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@Manuel Kahayon This is an interesting geometric application of the Power Mean Inequality! It is well presented, and the diagram helps too. I am looking forward to more of your problems :) Pranshu Gaba · 3 months, 2 weeks ago

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Hey guys, check out my new problems for the PWP. Hope you like it :) .

Roots of Unity - Let's get United.

Classical Inequalities - Classic ! , Classical ! .

Law of Gravity - Who are A and B ? . Anish Harsha · 3 months, 3 weeks ago

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@Anish Harsha Ah yes, that's a really important​ property of roots if unity :)

Note that it's the "Power Mean inequalities" chapter instead of all classical inequalities. Yes, we're developing out this section and splitting up the chapter! Finally! So, the IMO question doesn't quite count, but the other one is great!

The phrasing of the gravitational question could be improved on. Right now, it sounds quite forced / unnatural. Can you tweak it slightly? Calvin Lin Staff · 3 months, 3 weeks ago

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@Calvin Lin Ok, sir. I'll work on the question on Law of gravity. Anyways, thanks. Anish Harsha · 3 months, 3 weeks ago

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Thanks for inviting me to this party but I am currently facing crisis in my school that my free time is totally cut off. I will see to that I can add something good if I can! Akhash Raja Raam · 3 months, 1 week ago

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@Akhash Raja Raam Same here, thats why I cant contribute nowadays :( Ashish Siva · 3 months, 1 week ago

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A problem for Rectangular Grid Walk:

Heist of the Diamond Tarmo Taipale · 3 months, 2 weeks ago

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@Tarmo Taipale Woah! Very nice question! +1 for storytelling! Reshared Pi Han Goh · 3 months, 1 week ago

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@Pi Han Goh Thanks! Tarmo Taipale · 3 months, 1 week ago

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Here is mine for Roots of unity Akeel Howell · 3 months, 3 weeks ago

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@Akeel Howell I think asking for the minimum number of sides was a good question if the point wasn't already of the form \( ( \cos \frac{2\pi}{n}, \sin \frac{2\pi}{n} \). Giving the 3rd vertex of a 7-gon would have made it more interesting Calvin Lin Staff · 3 months, 1 week ago

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Here is another rectangular grid walk program. It has to do with a random walk in four dimension, where time is the fourth dimension... Enjoy! :) Geoff Pilling · 3 months, 3 weeks ago

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@Geoff Pilling Haha, I'm a Doctor Who fan!

Your question brings up a loophole in the Doctor Who universe, where events are "time locked". They could have just travelled to \(T-1\) and then wait it out, or travel to a nearby city and then walk over :) Calvin Lin Staff · 3 months, 1 week ago

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@Geoff Pilling TARDISSSSSS! =D =D Pi Han Goh · 3 months, 1 week ago

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@Pi Han Goh Hahahaha... Yup, an inside joke for Dr. Who fans! :) (Apropos for a time traveling problem!) Geoff Pilling · 3 months, 1 week ago

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Here is my question for Rectangular grid walk. I edited the previous one. Arkodipto Dutta · 3 months, 3 weeks ago

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@Arkodipto Dutta Thanks, the final answer is in a much simpler form now. It would be great if you could add a diagram to your problem; it would make the problem easier to visualize. Pranshu Gaba · 3 months, 2 weeks ago

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@Pranshu Gaba Ok, I will add one soon. Arkodipto Dutta · 3 months, 2 weeks ago

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@Arkodipto Dutta Thanks for the image! I've edited it to make it clearer, and to also show an example of a possible path. Calvin Lin Staff · 3 months, 1 week ago

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Here is a question on functions ( inverse ) . Log inverse Prince Loomba · 3 months, 3 weeks ago

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@Prince Loomba The final answer format of \( \ln(\log_a e)^{6}\) is very contrived. Can we simplify that further? It might be better to ask what the constant \(a\) is, and provide options like \( e^e, e^{ \frac{1}{e}, e^ 1 , e^2 \). Thoughts? Calvin Lin Staff · 3 months, 3 weeks ago

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@Calvin Lin Sir I cannot change the options but you can. Surely it will be better then. But then the question can be solved by hit and trial. Thats why I did this. Anyways as you wish. Prince Loomba · 3 months, 3 weeks ago

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@Prince Loomba I've edited the problem. Let me know what you think of it. Calvin Lin Staff · 3 months, 1 week ago

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@Calvin Lin This is an excellent question that was fun to solve. However, I wonder if replacing the two choices that you can immediately eliminate because they are <1, violating the first assumption in the question, would partially address Prince Loomba's concern about trial and error. Perhaps \(e^{e-1}\) and \(e^{e^{-2}}\) can replace \(e^{-1}\) and \(e^{-e}\)? Aaron Miller Staff · 3 months, 1 week ago

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Welcome to my dazzling problem on Triangle Centers: Tangling Circles. Worranat Pakornrat · 3 months, 1 week ago

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@Worranat Pakornrat That's interesting (and colorful)! Calvin Lin Staff · 3 months, 1 week ago

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Another one on Roots of Unity: Prime Power Roots. Worranat Pakornrat · 3 months, 1 week ago

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Here's one for Roots of Unity: Odd Powers Only. Worranat Pakornrat · 3 months, 1 week ago

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Triangle problem: It's a whole number!. \[\] Triangle centers: It's a Triangle!.\[\] Powers: Powers of 2.\[\] Roots of Unity: What value does the area approach?. Akeel Howell · 3 months, 1 week ago

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@Akeel Howell Short and simple problems. These problems test basic understanding of the concepts. Thanks for sharing these problems, Akeel :) Pranshu Gaba · 3 months ago

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This is my new question on Power-Mean Inequality: Yummy Jelly.

Should be pretty easy though. ;) Worranat Pakornrat · 3 months, 2 weeks ago

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Two easy (I think) problems for functions:

Not many solutions? Part 1

Not many solutions? Part 2 Tarmo Taipale · 3 months, 2 weeks ago

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Here's the applied question for Classical Inequality: Rectangular Inequality.

Enjoy! ;) Worranat Pakornrat · 3 months, 2 weeks ago

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Another Function Problem

Click Here Jason Chrysoprase · 3 months, 2 weeks ago

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An easy one on Classification of Triangles: Tri-Max. Worranat Pakornrat · 3 months, 2 weeks ago

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function problem Margaret Zheng · 3 months, 2 weeks ago

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@Margaret Zheng Cute question! I thought sinusoidal = periodic only. +1 Pi Han Goh · 3 months, 2 weeks ago

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Here are my entries for the topics: Functions

Maps to itself... eventually and Use a telescope. Pi Han Goh · 3 months, 2 weeks ago

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coupling a few 1008 roots!!

problem on roots of unity i just made !!! :) Rohith M.Athreya · 3 months, 2 weeks ago

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@Rohith M.Athreya Doesn't your expression of \( \omega\omega^{2n} + \omega^2 \omega^{2n-1} + \cdots + \omega^{2n} \omega \) simplifies to just \((2n+1)\omega^{2n+1} \)? Or am I interpreting your question wrongly

Interesting question nonetheless! +1 Pi Han Goh · 3 months, 2 weeks ago

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@Pi Han Goh according to me, it adds upto \(2n\omega^{2n+1}\) which is nothing but \(2n\omega\) which can be expressed as

\(2ne^{\frac{2\pi i}{2n}}\) which finally yields an answer of 2

but a moderator(perhaps) editted the question and the answer has been changed to 1 which i cant account for Rohith M.Athreya · 3 months, 2 weeks ago

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Can i post Similar Triangle Problem for Classification Triangle ? Jason Chrysoprase · 3 months, 2 weeks ago

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@Jason Chrysoprase Hmmmm, I don't think so. They are two different stuffs. Pi Han Goh · 3 months, 2 weeks ago

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@Pi Han Goh And if you are not busy, try to write solution in this problem. Yes i know the answer but i don't know how to prove it. Jason Chrysoprase · 3 months, 2 weeks ago

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@Pi Han Goh ;(, that's mean i only have one posted problem Jason Chrysoprase · 3 months, 2 weeks ago

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Here's my problem for Function( Sorry i was a little bit late ;) ) Jason Chrysoprase · 3 months, 2 weeks ago

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Here is my fourth submission for triangle centers section. Ashish Siva · 3 months, 3 weeks ago

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Here is mine for functions Abc Xyz · 3 months, 3 weeks ago

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@Abc Xyz Well u learnt hoe to link!!! (+1) Ashish Siva · 3 months, 3 weeks ago

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@Ashish Siva How do you link? Finn C · 3 months, 2 weeks ago

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@Finn C Use the Syntax:- [Text you want to link](url of the page you want to link) Ashish Siva · 3 months, 2 weeks ago

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@Ashish Siva Ya I refered to the latex help page :> Abc Xyz · 3 months, 3 weeks ago

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@Abc Xyz Actually it is mentioned in the note too Prince Loomba · 3 months, 3 weeks ago

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Here is another rectangular grid walk problem. Geoff Pilling · 3 months, 3 weeks ago

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A set on Markov Chains :

Christopher Boo · 3 months, 3 weeks ago

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@Christopher Boo Winnie the Poo!!!!!!!!!! Akeel Howell · 3 months, 1 week ago

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I have posted two more problems, both relating to classification of triangles. I hope that these two problems are more suitable than my last two:

Stubborn triangles

Believe in yourself, Tommy!

I am also planning to create some better problems for roots of unity, keeping in mind how Euler's formula can be used as a means of simplifying trigonometric functions. Jonas Katona · 3 months, 3 weeks ago

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@Jonas Katona Thank you! I enjoyed solving the problems :) They are easy to understand, and slightly tricky to solve.

I am looking forward to your problems in roots of unity! Pranshu Gaba · 3 months, 2 weeks ago

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@Pranshu Gaba Thank you for the compliment, Pranshu! :D Jonas Katona · 3 months, 1 week ago

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Here is my question about Triangle Centers. :> Nowras Otmen · 3 months, 3 weeks ago

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@Nowras Otmen Thanks for making your first problem contribution to Brilliant! I look forward to seeing more :) Calvin Lin Staff · 3 months, 3 weeks ago

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Not sure if this one counts as a Markov chain? Geoff Pilling · 3 months, 3 weeks ago

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@Geoff Pilling Very tangentially. It is more of linearity of expectation, then it has to do with understanding a Markov chain. Great question BTW. Calvin Lin Staff · 3 months, 3 weeks ago

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@Calvin Lin Yeah, that's kinda what I figured... Still trying to think of a good Markov Chain one... Oh, and thanks! :) Geoff Pilling · 3 months, 3 weeks ago

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Here is my question for Rectangle grid walk

Here is my question for Rectangular grid walk. I edited the previous one. Arkodipto Dutta · 3 months, 3 weeks ago

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@Arkodipto Dutta Thank you! Can you add a solution to the problem? To me, it's more natural to ask for the total number of ways, in part because the \( Z! ^3 \) in the denominator seems out of place. Calvin Lin Staff · 3 months, 3 weeks ago

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Than you, sir, for the invitation, I just posted my problems under the community page. I do not know how to add them here. Hana Nakkache · 3 months, 3 weeks ago

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@Hana Nakkache Well write the syntax:- [word you want to link](the url of the webpage) is my submission for xxx section. Ashish Siva · 3 months, 3 weeks ago

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Here is my first entry for classification of triangles section.

Here is my entry for triangle centers section.

Here is my third submission for triangle centers section.

Thanks. Here is that \({91}^°\) question. Ashish Siva · 3 months, 3 weeks ago

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@Ashish Siva FYI I combined your posts into 1, to keep this note cleaner.

Great question about triangle classifications :) A follow up question could be "Which non-degenerate triangle cannot have \( 91^\circ \) as one of its angles.

I like the triangle centers question too. It's kept simple, and interesting to play around with.

The third problem seems like tedious computation, and I'm not excited to get started with it. You can also tell that from the low attempt rate of the community.

Overall, great improvement​ in the problems that you're posting, especially in comparison to what you started out from. Keep it up! Calvin Lin Staff · 3 months, 3 weeks ago

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Not sure if it fits, but this is my question for Rectangular Grid Walk: Dark Room Hunt.
And for function: Fibonacci Reversed. Worranat Pakornrat · 3 months, 3 weeks ago

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@Worranat Pakornrat The first problem is really interesting! It doesn't fit under Rectangular grid walks, but I've placed it into the Grid puzzles quiz.

The second problem is pretty fun too. I see that it's been placed in the Functions chapter, and will review​ it later :) Calvin Lin Staff · 3 months, 3 weeks ago

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@Calvin Lin Thanks. The first one is one of my favorite detective question. :) Worranat Pakornrat · 3 months, 3 weeks ago

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Here is one on rectangular grid walk: Paths

My problem on Classification of Triangles: Get It Right

Problems on functions: Fitting Functions, Flipping Functions Lee Care Gene · 3 months, 3 weeks ago

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@Lee Care Gene Ah yes, that's pretty interesting. At first glance, it seems almost impossible to proceed and we have to resort to the old method of listing out paths to every single node. But as it turns out, there is a nice way to interpret the problem and play with it. Thanks for sharing! Could you add a solution to the problem?

The second problem is good too. Ideally, we want to avoid trigonometry in this basic chapter. I have moved the problem into trigonometry instead. Calvin Lin Staff · 3 months, 3 weeks ago

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@Calvin Lin I've just posted a solution. Lee Care Gene · 3 months, 2 weeks ago

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For Classification of Triangles: That's all?

For Roots of Unity: Unification Jonas Katona · 3 months, 3 weeks ago

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@Jonas Katona The first question seems computational / tedious, as it's just memorization of an angle. It also doesn't help explain how/why we care about classifying triangles.

Similarly for the second question, which doesn't yet showcase the beauty of roots of unity. It would have been nicer if final form was \( \cos \frac{pi}{n} \times e^{ i \theta } \), which starts to get at the usefulness of the identities.

Here are some suggestions for writing a great problem. I look forward to seeing your improvements over time :) Calvin Lin Staff · 3 months, 3 weeks ago

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@Calvin Lin Thank you for the advice! I will keep this in mind for next time. Jonas Katona · 3 months, 3 weeks ago

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Roots of unity!. will contribute to it Prakhar Bindal · 3 months, 3 weeks ago

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@Prakhar Bindal Looking forward to it! Calvin Lin Staff · 3 months, 3 weeks ago

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@Calvin Lin Sir like gravitation requires low level problems but i have some level 4-5 problems can i still contribute? Prakhar Bindal · 3 months, 3 weeks ago

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@Prakhar Bindal Sure of course! Calvin Lin Staff · 3 months, 3 weeks ago

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@Christopher Boo Great! I've added both of these to the Markov Chains chapter. I really like Otto's problem, and the discussion. I was pleasantly surprised at the result. Calvin Lin Staff · 3 months, 3 weeks ago

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Here is one for rectangular grids: Avoiding the accident sites.

And a simple one on Newton's Law of Universal Gravitation: Rock. Arjen Vreugdenhil · 3 months, 3 weeks ago

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@Arjen Vreugdenhil Thanks! I combined your posts into 1, to keep this note cleaner.

The first problem could definitely benefit from an image, which will remove the need to define the streets and avenues. Calvin Lin Staff · 3 months, 3 weeks ago

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Here is a first problem for Classification of Triangles Hjalmar Orellana Soto · 3 months, 3 weeks ago

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@Hjalmar Orellana Soto Great. I've improved the phrasing of the problem. Old version

If \(ABC\) it a triangle such that its orthocenter lies on its circumscribed circumference. Find the greatest internal angle of \(ABC\) (in degrees).

into

If \(ABC\) is a triangle such that the orthocenter lies on the circumcircle, then find the greatest internal angle of \(ABC\) (in degrees).

Can you add a solution to it? Calvin Lin Staff · 3 months, 3 weeks ago

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Here are my entries for Triangle Centres :

Deeparaj Bhat · 3 months, 3 weeks ago

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@Deeparaj Bhat Thanks! These are great problems that the community likes and will be in the list that I review. In fact, the first problem was already added to a challenge quiz :) Calvin Lin Staff · 3 months, 3 weeks ago

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@Calvin Lin Oh! I didn't get a notification about it though. Thanks for selecting it. :) Deeparaj Bhat · 3 months, 3 weeks ago

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@Deeparaj Bhat Yea, that happened back in the day before we started sending out notifications lol. I'm so glad that we do it now. Calvin Lin Staff · 3 months, 3 weeks ago

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I would like to submit these problems for Newton's Law of Gravity: Entry 1, Entry 2, and Entry 3. Pranshu Gaba · 3 months, 3 weeks ago

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@Pranshu Gaba Oh, I really like question 1. That same thought occurred to me as I was reading the book and thinking about the environmental differences.

For question 2, I'm not quite sure that 5N of gravitational force between 2 stones that we're holding is a reasonable estimate. It might be better to just say "N newtons" and express the options in terms of that.

For question 3, one potential thought experiment is to ask what happens as \( r \rightarrow 0 \). If the gravitational force goes to infinity, why am I allowed to touch something and release it? Calvin Lin Staff · 3 months, 3 weeks ago

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I have one more question to add to this party! I just created it today, for functions: Inseparable Jonas Katona · 3 months, 1 week ago

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Final entries for this party::::

Triangle centers:
Centroid?
Some other center?

Grid walk::::
Sudoku
Tron
Limit Pi Han Goh · 3 months, 1 week ago

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Here's my entry for triangle centers Pi Han Goh · 3 months, 2 weeks ago

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Hope I'm not late to the party. Here's my entries for Power mean inequality::

Do I need to expand it all?

Maximum value is not as big as you think Pi Han Goh · 3 months, 3 weeks ago

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@Pi Han Goh Man you come up with the craziest questions, Pi! ;-) Geoff Pilling · 3 months, 3 weeks ago

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Here is my rectangular grid walk question... Enjoy! :^) Geoff Pilling · 3 months, 3 weeks ago

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Here's a link to a problem I wrote for Newton's Law of Gravity: Splitting the Sun in Two Aaron Miller Staff · 3 months, 3 weeks ago

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@Aaron Miller Here's another Gravity problem: A Gloopelhopper Problem Aaron Miller Staff · 3 months, 1 week ago

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@Aaron Miller Interesting problems, Aaron! The scenarios are described in great detail, and the solutions are excellently written. Awesome! Pranshu Gaba · 3 months, 1 week ago

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@Aaron Miller Here's another problem I wrote for Newton's Law of Gravity: Voyage from the Sun Aaron Miller Staff · 3 months, 1 week ago

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@Aaron Miller Thank you, Aaron! It is a lovely problem with a nice diagram.

I have a suggestion; the problem statement was long and slightly difficult to follow:

Express your answer as a ratio \(\dfrac{F_2}{F_1}\), where \(F_1\) is the gravitational force applied by the (actual) Sun on the Earth at an orbital distance of \(1 \text{AU}\).

It can be simplified to:

Suppose the gravitational force applied by the Sun on the Earth at an orbital distance of \(1 \text{ AU}\) is \(F_1\).

What is \(\dfrac{F_2}{F_1}\)?

By using shorter sentences, we can make the problem more easy to understand. Pranshu Gaba · 3 months, 2 weeks ago

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@Pranshu Gaba Hi Pranshu,

Thank you for your helpful comment! I totally agree with your rephrasing. I will change the problem to incorporate your suggestion. Aaron Miller Staff · 3 months, 2 weeks ago

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My entry for roots of unity Joao Miguel Coelho · 3 months, 3 weeks ago

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@Joao Miguel Coelho Cool problem, Joao! It is simply stated, and clear and concise. Keep it up! :) Pranshu Gaba · 3 months, 2 weeks ago

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@Pranshu Gaba Thanks! Joao Miguel Coelho · 3 months, 2 weeks ago

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This is my question on Markov's Chain: Sun & Beaches. Worranat Pakornrat · 3 months, 3 weeks ago

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@Worranat Pakornrat Nice problem, Dr. Warm. I have added a diagram so it's easier to see what is happening. Pranshu Gaba · 3 months, 2 weeks ago

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@Pranshu Gaba Thanks. I was so busy with my work that I didn't have time to draw the diagram. Worranat Pakornrat · 3 months, 2 weeks ago

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Ahh i'm late, need to work fast ;) Jason Chrysoprase · 3 months, 3 weeks ago

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@Jason Chrysoprase There's still lots of time, don't worry.

And lots of parties too! Calvin Lin Staff · 3 months, 3 weeks ago

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Do it by algebra Aditya Sharma · 3 months, 3 weeks ago

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@Aditya Sharma Hm, after our discussion, I don't see what makes this problem interesting / special.

The problem is not in any of the chapters that I listed out. These chapters will get special attention over the next 2 weeks, and thus contributing into those chapters now would increase the likelihood that it makes it into the quizzes. Calvin Lin Staff · 3 months, 3 weeks ago

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@Calvin Lin sir what about this? Will it work for the AP quiz: Just, AP Rishabh Sood · 3 months, 3 weeks ago

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@Rishabh Sood I did review the question as I was putting together the AP set, and it's interesting in it's own way. However, the community didn't find it interesting and wasn't excited to work on it, which is why I didn't place it into the Level 4 quiz.

Here are some guidelines to help you improve the quality of your problems, and I would love to feature them :) Calvin Lin Staff · 3 months, 3 weeks ago

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