Let $a,b,c,d,e,f$ be positive integers such that   $$\frac{a}{b} &lt; \frac{c}{d} &lt;\frac{e}{f}.$$      Suppose $af - be = -1$. Show that $d\ge b + f$.   Tried everything. I introduced constants $K_{1},K_{2}$ etc between $\frac{a}{b} , \frac{c}{d},\frac{e}{f}$ so that I have an equality, which can somehow be related to the equality given to us. I tried to use every special identity/inequality which I know, but to no avail. The problem always seems to be eliminating $c$ or $d$ from the inequality. I have been at it for 2 hours. This problem featured in a Regional Olympiad, using which after many stages the team for the IMO is selected.
Let $a,b,c,d,p$, and $q$ be natural numbers such that $ad-bc = 1$ and $\frac{a}{b} &gt; \frac{p}{q} &gt; \frac{c}{d}$.  How prove that  $q \geq b+d$?
short view distance in 3d view? single email limit exceeded salesforce very frequently..? my email limit is exceeded after sending 10-12 emails..what could be the problem..as i have seen the limit is 1000 emails/day.i am not able to find out what could be the problem..Can some give me some suggestions. As i also know we can send 10 emails in a single transaction.I am sending emails in a batch class.  I am calling this batch class on button click and filling emailList with emailsId's.                      String query='Select id from Lead';//No use of this query                     Batch_send_emails bse=new Batch_send_emails();                     bse.query=query;                                                         bse.resendEmails=emailList;                     database.executebatch(bse);           **Batch Class**           if(!resendEmails.isEmpty())           {               for(String s:resendEmails)                {                   emailsLeads.add(s);                }           }           Messaging.SingleEmailMessage mail = new Messaging.SingleEmailMessage();         mail.setToAddresses(emailsLeads);        // mail.setCcAddresses(ccAddresses);         String subject='GoToMeeting Invitation Details for '+subject;         String body='test';          mail.setHtmlBody(body);          mail.setSubject(subject);         Messaging.sendEmail(new Messaging.SingleEmailMessage[] { mail });          } Spurious whitespace with algpseudocode and noend I'm currently typesetting some pseudocode and was up to now pretty happy with the algorithmicx package and the algpseudocode command set. However, I would now like to use the noend option to omit end lines. That works, but unfortunately introduces additional whitespace (see lines 6 and 7 in the example below). I fiddled around with negative \vspaces but could not come up with a consistent value that works in all cases.    \documentclass{article} \usepackage[noend]{algpseudocode}  \begin{document}  \begin{algorithmic}[1]   \Function{lsolve}{$L$, $b$}     \State $x \gets b$     \For{$j \gets 0$ to $n - 1$}       \State $x_j \gets x_j / l_{jj}$       \ForAll{$i &gt; j$ where $l_{ij} \neq 0$}         \State $x_i \gets x_i - l_{ij} x_j$       \EndFor     \EndFor     \State \Return $x$   \EndFunction \end{algorithmic}  \end{document}   Is there any way to patch algorithmicx or do I have to finetune the pseudocode on a case by case basis? I would like someone to explain me how the sense of a sentence changes with using "for me" or "to me":  Some examples:     1a.  She is a good friend to me.   1b.  She is a good friend for me.        2a.  How special you are to me.   2b.  How special you are for me.        3a.  It's an interesting topic to me.   3b.  It's an interesting topic for me. Why do we age? or Do we have a theory of senescence? There seem to be a number of ideas about why we age. Hypotheses include the gradual accumulation of cell metabolic products affecting organism function and the reduction of  during cell division. My hand-wavey idea would be "wear and tear".  Are we anywhere near a consensus theory of senescence? Duo of alien xenologists study couple of Terran explorers, later save them Show that $\ker \hat{T} = \text{ann}(\text{range } T)$ There have been several discussions regarding code-only answers (answers that contain only code blocks with no textual explanation whatsoever):         Personally, I can't see why the quality filter should let such answers through at all. The main problem I have with code-only answers, especially once they're flagged as not an answer or very low quality, is that we can't determine at a glance whether these answers are:   Genuinely attempting to answer the question  answers Trying to correct an error in the question without actually answering it Random code dumps that may or may not relate to the question even by programming language or keyword(s) in the question   In particular, what exactly is an "informative code-only answer"? Does it mean the code is self-documenting? Does it mean the code substitutes code comments for a proper write-up? This is all very fuzzy. From  to the last question above:     I have never seen a code-only answer that couldn't be made better with the addition of appropriate text. They are of lower-quality than the same answer with some text.   This text makes all the difference, IMO, between an answer and a non-answer. Without it, it's often needlessly difficult to tell.  Existing code-only answers that are good-faith attempts to answer the question should be edited or downvoted, yes. But if the quality filter already tries to block code-only questions, why not block code-only answers too? These answers are often low quality, and (almost) always have ample room for improvement, just like questions. What are all possible ways to stop a zombie from despawning? Is there any way for which we are able to stop a zombie from de-spawning no matter how far we go? If so, what are all possible ways?  Bonus question: what are all the other techniques for which we can prevent a zombie from de-spawning within a specific block radius? Handling large numbers in C++? What is the best way to handle large numeric inputs in C++ (for example 10^100)?   For algorithms I usually switch over to ruby and I sometimes use strings.  Any other good methods? The use of “So X as to be Y” First of all, note that this is not a duplicate question of another one asking about the usage of so as to, since this one is asking about the structure of so . . . as to. . . .  I understand this might have been a better question at the , since I’m not a native English speaker. I am asking it here since I want to get some professional advice in regard to this question, since I don’t seem to have found anything similar to this on the Internet.  At the Oxford Dictionaries Online (OD) I have found the following definition of ludicrous:     So foolish, unreasonable, or out of place as to be amusing.   I understand the phrase so as to, simply meaning in order to and preceding the purpose for which an action is done.  So by itself is just therefore, hence, thus, or a way of emphasizing an adjective or adverb.  As to in itself means with regard to or according to, as pointed out in .  In the OD definition of ludicrous, the use of so looks fair to me, that is, it clearly emphasizes the high extent to which something is foolish or unreasonable.  The use of as to doesn’t work in any way I’ve mentioned in this case (neither as in with regard to nor according to). I think the dictionary I’ve referred to has missed one, if not more, meaning of the phrase as to.  In this case, it appears to have likely been used as an alternative to that when used in the form So X, that Y, meaning something is or happens X (a combination of adjectives or adverbs) enough to cause Y to happen.  Of course, I don’t mean that in the structure so X as to Y the phrase as to is absolutely equivalent to that, since substituting as to with that changes the structure of Y.  Does all this mean that the phrase as to is a great alternative to that (although changing the structure of Y) whenever the sentence is of the form so X, that Y (where X is a combination of one or more adjectives or adverbs)?  I am new here and so I’m not too familiar with the variety of existing tags here, so if anyone sees an additional one that would fit, please add it to this question. I would also be glad to see some other corrections of my grammar. What noun or adjective would one give to a person that is intolerant of stupidity? A person that becomes annoyed, angry, or of very short temper when conversing or interacting with people of a lower intelligence?  For example:       The man is very xyzxyz; his son can't do multiplication very well.     or     She is a(n) xyzxyz. She gets very frustrated when people forget things.   I'm looking almost for a scientific/medical/psychological term. One that would fall into the same kind of group as words like haemophiliac, masochist, nihilist, etc. Network reputation graph becomes nervous when clicking quickly, backs away slowly What is the justification for locking votes? Why am I not allowed to change my vote after some period of time? There are times when I go back to things I voted on, and decide they didn't deserve that specific vote. (I get wiser as I grow older) Recently, I accidentally deleted a partition off of my external hard disk. I had a copy of Minitool Power Data Recovery left over when the same drive had become RAW about a year ago. When I selected lost partition recovery, it immediately identified my deleted partition and allowed me to access and recover the files that were there.  My question is this. Is there any way to change the deleted partition back to NTFS without having to have a drive that you need to recover the files to and then move them back after formatting, as I don’t have any space left on either the computer or on my other external hard drive, so that I don’t need to buy a new one. The files on the disk were really important.  I am using Windows 7 Home Premium.  I appreciate any help in solving this crisis. Please be kind, I am an electronics nub.  This is in reference to getting an LED to emit photons.  From what I read (Getting Started in Electronics - Forrest Mims III and Make: Electronics) electrons flow from the more negative side to the more positive side.  In an example experiment (involving a primary dry cell, a SPDT switch, a resistor and an LED) it states that the resistor MUST be connected to the anode of the LED.  In my mind, if the electrons flow from negative to positive, wouldn't the electron flow run through the LED before the resistor; thereby making the resistor pointless? This thought occurred to me earlier and I'm surprised I hadn't considered it previously. I get the feeling that no meaningful generalization can occur in a non-separable Banach space but on the surface it seems like a meaningful generalization can be had if the Banach space is separable. Even if it's possible, convergence and other issues likely severely complicate the matter. Proving $\frac{1}{\sqrt{x}}\ge \frac{2}{x+1}$ for $x> 0$ Prove: $$\frac{1}{\sqrt{x}}\ge \frac{2}{x+1}, \quad\forall x&gt;0$$  Yeah, pretty much it. I've tried all manner of rearranging and just can't seem to get it. Thanks. Rolle's Theorem and the Mean Value Theorem Let $f\colon \mathbb{R} \to \mathbb{R}$ have derivatives of all orders. Suppose that $a &lt; b$ and that $f(a)=f(b)=f'(a)=f'(b) =0$. Prove that $f'''(c) = 0$ for some $c$ in $(a,b)$. I'm aware there are been several similar questions, and some great answers with hints on how to prove this, for example  and .  But I've never really seen a detailed proof of the absolute convergence of the Riemann zeta function in the half-plane $\Re(s)&gt; 1$.  I hope there's interest on a detailed proof of this for reference. Here is my attempt. Please, fill in any detail that might be missing, or point out any mistake!  On $\Re(s)=\sigma&gt; 1$, we have  $$\sum_{n=1}^{\infty}\bigg|\frac{1}{n^s}\bigg|=\sum_{n=1}^{\infty}\frac{1}{n^\sigma}$$  and  $$\frac{1}{n^\sigma}\leq\frac{1}{n^{1+\epsilon}}$$  for any $\epsilon &gt;0$, so by the direct comparison test, if   $$\sum_{n=1}^{\infty}\frac{1}{n^{1+\epsilon}}$$  converges absolutely, then so do $\sum_{n=1}^{\infty}|1/n^s|$ and $\zeta(s)$.  By the integral test, the convergence of $\sum_{n=1}^{\infty}1/n^{1+\epsilon}$ is equivalente to the finitude of the integral  $$\int_1^\infty \frac{1}{n^{1+\epsilon}}=\bigg[-\frac{1}{\epsilon x^\epsilon}\bigg]=\frac{1}{\epsilon}$$  which holds for any $\epsilon &lt; \infty$.  Any correction or improvement is welcomed! Thanks in advance.