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Ok-guys-We-all-LOVE-mathematics-So-I-have-came-up-with-a-simple-and-possibly-effect-way-to-not-only-improve-our-abilities-as-mathematicians-but-also-have-some-fun-too-We-should-hold-a-competiti

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Question Number 2649 by Filup last updated on 24/Nov/15
Ok guys. We all LOVE mathematics. So, I have came up with a simple and possibly effect way to not only improve our abilities as mathematicians, but also have some fun, too. We should hold a competition every now and again. We can, for example have questions such as the following: (v^→ , k^→ )∈R^n , v^→ =⟨v_1 , ..., v_n ⟩, k^→ =⟨1, 1, ..., 1⟩. Prove that if v•k=0, v and k are orthoganal. Do so in the most creative way(s) or most complex. Let this post be a trial competition. If this goes well, we should continue doing more in the future! Good Luck Edit: Let me know what you all think!
$$\mathrm{Ok}\:\mathrm{guys}.\:\mathrm{We}\:\mathrm{all}\:\mathscr{LOVE}\:\mathrm{mathematics}. \\ $$$$\mathrm{So},\:\mathrm{I}\:\mathrm{have}\:\mathrm{came}\:\mathrm{up}\:\mathrm{with}\:\mathrm{a}\:\mathrm{simple}\:\mathrm{and} \\ $$$$\mathrm{possibly}\:\mathrm{effect}\:\mathrm{way}\:\mathrm{to}\:\mathrm{not}\:\mathrm{only}\:\mathrm{improve} \\ $$$$\mathrm{our}\:\mathrm{abilities}\:\mathrm{as}\:\mathrm{mathematicians},\:\mathrm{but}\:\mathrm{also} \\ $$$$\mathrm{have}\:\mathrm{some}\:\mathrm{fun},\:\mathrm{too}. \\ $$$$ \\ $$$$\mathrm{We}\:\mathrm{should}\:\mathrm{hold}\:\mathrm{a}\:\mathrm{competition}\:\mathrm{every} \\ $$$$\mathrm{now}\:\mathrm{and}\:\mathrm{again}.\:\mathrm{We}\:\mathrm{can},\:\mathrm{for}\:\mathrm{example} \\ $$$$\mathrm{have}\:\mathrm{questions}\:\mathrm{such}\:\mathrm{as}\:\mathrm{the}\:\mathrm{following}: \\ $$$$ \\ $$$$\left(\overset{\rightarrow} {\boldsymbol{{v}}},\:\overset{\rightarrow} {\boldsymbol{{k}}}\right)\in\mathbb{R}^{{n}} ,\:\overset{\rightarrow} {\boldsymbol{{v}}}=\langle{v}_{\mathrm{1}} ,\:…,\:{v}_{{n}} \rangle,\:\overset{\rightarrow} {\boldsymbol{{k}}}=\langle\mathrm{1},\:\mathrm{1},\:…,\:\mathrm{1}\rangle. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{if}\:\boldsymbol{{v}}\bullet\boldsymbol{{k}}=\mathrm{0},\:\boldsymbol{{v}}\:\mathrm{and}\:\boldsymbol{{k}}\:\mathrm{are}\:\mathrm{orthoganal}. \\ $$$$\mathrm{Do}\:\mathrm{so}\:\mathrm{in}\:\mathrm{the}\:\mathrm{most}\:\mathrm{creative}\:\mathrm{way}\left(\mathrm{s}\right)\:\mathrm{or}\:\mathrm{most} \\ $$$$\mathrm{complex}. \\ $$$$ \\ $$$$\mathrm{Let}\:\mathrm{this}\:\mathrm{post}\:\mathrm{be}\:\mathrm{a}\:{trial}\:\mathrm{competition}. \\ $$$$\mathrm{If}\:\mathrm{this}\:\mathrm{goes}\:\mathrm{well},\:\mathrm{we}\:\mathrm{should}\:\mathrm{continue} \\ $$$$\mathrm{doing}\:\mathrm{more}\:\mathrm{in}\:\mathrm{the}\:\mathrm{future}! \\ $$$$ \\ $$$${Good}\:{Luck} \\ $$$$ \\ $$$${Edit}: \\ $$$$\mathrm{Let}\:\mathrm{me}\:\mathrm{know}\:\mathrm{what}\:\mathrm{you}\:\mathrm{all}\:\mathrm{think}! \\ $$
Commented by prakash jain last updated on 24/Nov/15
How about we set up a topic at the beginning of the month and challenge on that topic at the end of the month. This way we cover one new topic every month. Topic can be for example series convergence, etc. Everybody can always put question on other topic anytime.
$$\mathrm{How}\:\mathrm{about}\:\mathrm{we}\:\mathrm{set}\:\mathrm{up}\:\mathrm{a}\:\mathrm{topic}\:\mathrm{at}\:\mathrm{the}\:\mathrm{beginning} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{month}\:\mathrm{and}\:\mathrm{challenge}\:\mathrm{on}\:\mathrm{that}\:\mathrm{topic} \\ $$$$\mathrm{at}\:\mathrm{the}\:\mathrm{end}\:\mathrm{of}\:\mathrm{the}\:\mathrm{month}. \\ $$$$\mathrm{This}\:\mathrm{way}\:\mathrm{we}\:\mathrm{cover}\:\mathrm{one}\:\mathrm{new}\:\mathrm{topic}\:\mathrm{every}\:\mathrm{month}. \\ $$$$\mathrm{Topic}\:\mathrm{can}\:\mathrm{be}\:\mathrm{for}\:\mathrm{example}\:\mathrm{series}\:\mathrm{convergence}, \\ $$$$\mathrm{etc}. \\ $$$$\mathrm{Everybody}\:\mathrm{can}\:\mathrm{always}\:\mathrm{put}\:\mathrm{question}\:\mathrm{on} \\ $$$$\mathrm{other}\:\mathrm{topic}\:\mathrm{anytime}. \\ $$
Commented by RasheedAhmad last updated on 24/Nov/15
Participants here are of different levels and also they belong to different areas of maths. They can′t be forced to answer suggested question/s.For example I am not in position now to answer your Q because at the instant I have forgotten Vectors. Anyway I appreciate your passions.
$${Participants}\:{here}\:{are}\:{of}\:{different} \\ $$$${levels}\:{and}\:{also}\:{they}\:{belong}\:{to} \\ $$$${different}\:{areas}\:{of}\:{maths}. \\ $$$${They}\:{can}'{t}\:{be}\:{forced}\:{to}\:{answer} \\ $$$${suggested}\:{question}/{s}.{For}\:{example} \\ $$$${I}\:{am}\:{not}\:{in}\:{position}\:{now}\:{to}\: \\ $$$${answer}\:{your}\:{Q}\:{because}\:{at}\:{the} \\ $$$${instant}\:{I}\:{have}\:{forgotten}\:{Vectors}. \\ $$$${Anyway}\:{I}\:{appreciate}\:{your}\:{passions}. \\ $$
Commented by Filup last updated on 24/Nov/15
Thats a good point! Maybe we can offer different levels of difficulty or something?
$$\mathrm{Thats}\:\mathrm{a}\:\mathrm{good}\:\mathrm{point}!\:{M}\mathrm{aybe}\:\mathrm{we}\:\mathrm{can}\:\mathrm{offer} \\ $$$$\mathrm{different}\:\mathrm{levels}\:\mathrm{of}\:\mathrm{difficulty}\:\mathrm{or}\:\mathrm{something}? \\ $$
Commented by Yozzi last updated on 24/Nov/15
I′m no fan of mathematics competitions (probably because I′m terribly slow at problem solving in unknown areas) but I′ll try the questions for the benefit of improving my abilities.
$${I}'{m}\:{no}\:{fan}\:{of}\:{mathematics}\:{competitions} \\ $$$$\left({probably}\:{because}\:{I}'{m}\:{terribly}\:{slow}\:{at}\:{problem}\right. \\ $$$$\left.{solving}\:{in}\:{unknown}\:{areas}\right)\:{but}\:{I}'{ll}\:{try}\:{the}\:{questions}\:{for} \\ $$$${the}\:{benefit}\:{of}\:{improving}\:{my}\:{abilities}. \\ $$$$ \\ $$
Commented by Filup last updated on 24/Nov/15
It wouldnt be a race... more of a way of finding the most unique answers?
$$\mathrm{I}{t}\:\mathrm{wouldnt}\:\mathrm{be}\:\mathrm{a}\:\mathrm{race}…\:\mathrm{more}\:\mathrm{of}\:\mathrm{a}\:\mathrm{way}\:\mathrm{of} \\ $$$$\mathrm{finding}\:\mathrm{the}\:\mathrm{most}\:\mathrm{unique}\:\mathrm{answers}? \\ $$
Commented by Yozzi last updated on 24/Nov/15
That seems more sensible: a search for the different solutions for a given problem (if there are a finite number of distinct methods).
$${That}\:{seems}\:{more}\:{sensible}: \\ $$$${a}\:{search}\:{for}\:{the}\:{different}\:{solutions} \\ $$$${for}\:{a}\:{given}\:{problem}\:\left({if}\:{there}\:{are}\:{a}\right. \\ $$$$\left.{finite}\:{number}\:{of}\:{distinct}\:{methods}\right). \\ $$
Commented by 123456 last updated on 24/Nov/15
nice.
$$\mathrm{nice}. \\ $$
Commented by Yozzi last updated on 24/Nov/15
I like that idea.
$${I}\:{like}\:{that}\:{idea}. \\ $$
Commented by 123456 last updated on 01/Dec/15
the new topic will be opened on the first day of mouth or 1 mouth after the other?
$$\mathrm{the}\:\mathrm{new}\:\mathrm{topic}\:\mathrm{will}\:\mathrm{be}\:\mathrm{opened}\:\mathrm{on}\:\mathrm{the} \\ $$$$\mathrm{first}\:\mathrm{day}\:\mathrm{of}\:\mathrm{mouth}\:\mathrm{or}\:\mathrm{1}\:\mathrm{mouth} \\ $$$$\mathrm{after}\:\mathrm{the}\:\mathrm{other}? \\ $$
Commented by prakash jain last updated on 09/Dec/15
Let us try at the beginning of the month.
$$\mathrm{Let}\:\mathrm{us}\:\mathrm{try}\:\mathrm{at}\:\mathrm{the}\:\mathrm{beginning}\:\mathrm{of}\:\mathrm{the}\:\mathrm{month}. \\ $$

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