In Re: C: Fill & Download for Free

Well, it’s kinda cheating, but let [math]x=a^2+\frac1{a^2}[/math] ([math]a[/math] is a complex number, so I won’t care about sign). Then[math]\sqrt{2+x}=\sqrt{2+a^2+\frac1{a^2}}=a+\frac1a[/math]provided [math]Re(a)>0[/math].Now let [math]a=-b^2[/math]. Then[math]\sqrt{2-\sqrt{2+x}}=\sqrt{2+b^2+\frac1{b^2}}=b+\frac1b[/math]And again, we need [math]Re(b)>0[/math]. You know where we’re going, right? Now let [math]b=c^2[/math]:[math]\sqrt{2+\sqrt{2-\sqrt{2+x}}}=\sqrt{2+c^2+\frac1{c^2}}=c+\frac1c[/math]with [math]Re(c)>0[/math].Thus[math]a^2+\frac1{a^2}=c+\frac1c\iff c=a^2[/math] or [math]c=\frac1{a^2}[/math],and [math]Re(a^2)>0.[/math] Since [math]a=-b^2=-c^4[/math],[math]c=c^8[/math] or [math]c=\frac1{c^8}\iff c^7=1[/math] or [math]c^9=1[/math].So [math]x=c^8+\frac1{c^8} = c+\frac1c=2Re(c)[/math], in which [math]c[/math][math][/math] is any seventh or ninth root of unity with [math]Re(c)>0[/math], [math]Re(c^2)>0[/math], [math]Re(c^4)<0[/math] and [math]Re(c^8)>0[/math]. We don’t need to check the last inequality since [math]Re(c)=Re(c^8)[/math]. In other words, using arguments between [math]-\pi[/math] and [math]pi[/math], we progressively obtain [math]-\frac\pi2 < arg(c) < \frac\pi2[/math], [math]-\frac\pi4 < arg(c) < \frac\pi4[/math], and [math]-\frac\pi4< arg(c)<-\frac\pi8[/math] or [math]\frac\pi8<arg(c)<\frac\pi4[/math]. The arguments are [math]\frac{2k\pi}9[/math] or [math]\frac{2k\pi}7[/math]. Only [math]\pm\frac{2\pi}9[/math] qualifies, and they give the same answer.So [math]x=2Re(c)[/math], in which [math]c[/math][math][/math] is the primitive ninth root of the unity. Incidentally, this means that [math]x=2\cos\frac{2\pi}9[/math].I stated that this is kinda cheating because the trigonometric substitution uses essentially the identity [math]\cos^2\alpha=\frac{1+\cos(2\alpha)}2[/math], which reduces to [math]\left(w+\frac1w\right)^2=w^2+\frac1{w^2}+2[/math] if we let [math]w=\cos\alpha+i\sin\alpha[/math]. So the substitutions we made are actually trigonometric substitutions in disguise.

Absolutely! What’s life without catfishes?As to how I respond, it depends on my mood. But my usual responses are something similar too:Nigerian prince/Bank Manager scam (I have loads of $$$and want to share it with you):I thank them for their kind words and the opportunity and then let them know that I’m so incredibly wealthy that I just couldn’t be the kind of person who would be so ungrateful that I’d take more not meant for me. They usually mention something about God and being Christian, so start mentioning giving to charities and spam them with charitable links.Their usual response is to say something like “you could do a lot of good charitable work with this money if you send me $1400 USD (or whatever) at which point, you let them know that you’re incredibly wealthy because you charge for any and all work and their consulting bill for the information regarding charities already totals $1755 USD and you couldn’t possibly continue the conversation until they settle the bill… upon which they usually write something similar to F*CK YOU YOU MOTHER F@$#!ing WH*RE C*NT which isn’t very charitable or Christian at all but *is* amusing.Sometimes the Nigerian prince isn’t that Nigerian… Image source: linkA US soldier trying to get leave or a solo-father soldier with a daughter looking for a mother for his child (sigh).I act interested. I try to guess which way the scam is going to go and then I avoid all mentions. If they raise a daughter, I never mention the daughter - I ignore all references and change the subject while acting vaguely so they still think they have a mark.When they’re invested in the exchange (they start getting shrill and raising the daughter repeatedly) I let them know that I have a phobia of children and forbid them from discussing the make-believe daughter which spokes their wheel and ruins the scam.If they’re trying to get money for leave so that they can come and visit me (oh please! after 4 damn messages??) I let them know that the dates just aren’t convenient. I won’t agree to send money unless they send me a detailed itinerary of what we’ll be doing when he comes to visit me.HOT TIP: If they baulk at continuing, just accuse them of not loving you enough and fake sulk. You can string it out for days… until they eventually figure it out and write something like F*CK YOU YOU MOTHER F@$#!ing WH*RE C*NT which isn’t very parental or ‘professional soldier like’ but *is* amusing.That’s it - those a really the only catfish scams I get.I don't even get that many of these anymore since I shut down my Quora inbox. I have to rely on facebook to get my scam fix. If it’s an obvious scam profile, I nearly always accept the invite.It’s worthwhile keeping in mind that every minute you waste of these peoples time is a minute that they can't use to actually scam someone who’s less gullible making it a win/win/win/win.You waste their time, you gain amusement, you learn new ways to string together insults and you potentially save someone else for getting financially assaulted. #notallheroeswearcapes

(A). we have to choose the way. the way of “vision” whether its aggressiveness or its calm, restful or relaxing nature. so, he is a monk and he belongs to the second category. he is more into civilisational values.(B). neither Modii nor Adityanath are aggressive both are great leaders of today's work and leaders of new India. yes, I won't deny that yogi is stricter than modi to some extent.(C). see if we call a PM, CM aggressive for the following then I think we haven't seen the real aggressiveness by radical Islamists.both are working to get back civilisation framework. see how yogi asked the State to release its official documents in Sanskrit also.on the other side, both had hands in re-modelling the ram temple after years of debate and fight.giving back the real soil to the sons in Kashmir (to pandits) is much more than anything else. I really felt bad when I read about their cleaning in Kashmir by Nehru plannings.yogi is much more direct in doing his things. he speaks very bluntly and does things directly. no feel no shame and don't care about appeasement.UP has been seeing developments and lowest riots ever under yogi and the same on a national level under modi.MY OPINIONboth are good leaders but we have to choose them again and again as we are witnessing our disrespects and against our culture.