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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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