If one is given eleven integers, and told that for any ten of them it is possible to split them into two sets of five with the same total value, show that the eleven integers must be equal.  My thoughts so far:   -if you had 10 equal numbers and one different one, you could swap the different one in so it wouldn't work  -if you had nine equal numbers, then clearly those nine plus a different number couldn't be divided up into two equal sets  -if there were eight equal numbers, then calling the equal numbers N and the other numbers $n_1$, $n_2$, $n_3$. If we exclude $n_3$, then either    $N&lt;n_1,n_2$ or $N&gt;n_1, n_2$, in which case we would require $4N + n_1 = 4 N + n_2$ which implies that $n_1=n_2$. By swapping $n_3$ for $n_1$, we can see that this also implies $n_1=n_2=n_3$. However, then the case of 7$N$s and 3$n$s wouldn't be able to be split into two equal sets.  $n_1&lt;N&lt;n_2$, in which case $5N = 3N + n_1 + n_2$ so $2N=n_1 +n_2$. Similarly $N=n_2+n_3=n_1+n+n_3$. However, this isn't possible because either $n_3&gt;N$ in which case $n_2 + n_3 &gt; N$ or $n_3&lt;N$ in which case $n_1+n_3&lt;N$.
Let $S$ be a finite collection of real numbers (not necessarily distinct). If any element of $S$ is removed then the remaining real numbers can be divided into two collections with same size and same sum ; then is it true that all elements of $S$ are equal ?  (I know the result is true if the &quot;reals&quot; are replaced by &quot;integers&quot;.)
How does the lens diameter influence photo quality? I have tested two different 50mm lenses in my camera. One was a Nikkor 50mm ∅52mm. The other one was a Sigma 50mm ∅72mm. I took some pictures with both lenses using the same setup for aperture and shutter speed, but couldn't notice significant differences in the quality of the pictures.   So, how does the diameter affects the photo quality, if it does? What advantages would the ∅72mm lens have over the ∅52mm one? Are the following sentences grammatical or not? If not, does substituting a to-infinite make them grammatical? (Optional: designate if the sentence expresses purpose, reason, aim, or function)  1 I have decided to purchase a jet for getting to NYC and back more quickly. 2 I have decided to buy a jet for having a quicker way to get to NYC and back.  This is my only reason and aim; this will be the jet's only purpose and function.  3 I hate hairdryers, so I am buying a space heater for drying my hair more naturally, and that is my only aim &amp; purpose &amp; reason and that will be its only function 4 For having a lasting shine in your hair, buy ShineOn Shampoo. 5 My neighbor and I have decided to cooperate more for having a better life together.   Research done: ELL  (Note the  here is a restatement of Cambridge Dictionary's ) ELL  ELL  BBC English  (Currently the best site I've found.) ELU  All of which have answers that I find  (i) incomplete (ii) contradictory (iii) terribly confusing, if not (iv) wrong. To sort through the haze, I am focusing on one issue. Background Some sources say that you need to use a to-infinitive for expressing purpose, aim, function, reason, etc. I find these to be rather hazy distinctions, given that the ODO defines  as The reason for which something is done or created or for which something exists and the same work lists as : aim, reason, function, etc. It is my assertion that at least some of the following sentences demonstrate that one can use the -ing form (be it a gerund, verbal noun, participle) to express individual purpose (aim, reason, function, intention, whatever you want to call it). Do you agree, or not? Why (not)?  In shorter terms, are the sentences grammatical? or do we have to substitute a to-infinitive (including in order to + infinitive) to make them grammatical? (Optional: designate if the sentence expresses purpose, reason, aim, function, etc.) Idiom: People caring about minor stuff while something terrible is happening Imagine a situation in which the whole place is on fire, a bomb is about to explode, everyone is running for their lives and someone is checking his looks on the mirror... pretty inappropriate for the situation, don't you think? I'm looking for an idiom or colloquial/slang/informal expression to describe such an indifferent stance. How to copy some, but not all files? So, you can use the * as a wild card for all files when using cp within context of a directory.  Is there a way to copy all files except x file? In the movie Interstellar,  Cooper (played by ) enters a black hole and sends a message to Murph (his daughter). Later, he is found by rangers, but I don't understand how he gets out of the black hole?  As far as I know, nothing can get out of a black hole, as it has extremely high gravity. Short story about aliens that are parts of a spaceship Scan vs. Map vs. Apply fetching list of friends using facebook 2.0 Need verification - Prove a Hermitian matrix $(\textbf{A}^\ast = \textbf{A})$ has only real eigenvalues Proof: Let eigenvalue $\lambda \neq 0$ such as  $$\textbf{A}\vec{v}  = \lambda\vec{v}$$ $$\Rightarrow (\textbf{A}\vec{v})^\ast = (\lambda\vec{v})^\ast$$ $$\Rightarrow (\vec{v}^\ast\textbf{A}^\ast)=(\lambda^\ast\vec{v}^\ast)$$ Right-multiply both sides by $\color{orangered}{\vec{v}}$$$\Rightarrow (\vec{v}^\ast\textbf{A}^\ast \color{orangered}{\vec{v}} )=(\lambda^\ast\vec{v}^\ast \color{orangered}{\vec{v}} )$$ $$\textbf{A}^\ast=\textbf{A}$$ $$\Rightarrow(\vec{v}^\ast\textbf{A}\vec{v})=(\lambda^\ast\vec{v}^\ast\vec{v})$$ $$\Rightarrow(\vec{v}^\ast\lambda\vec{v}) = (\lambda^\ast\vec{v}^\ast\vec{v})$$ $$\Rightarrow(\lambda\vec{v}^\ast\vec{v}) =  (\lambda^\ast\vec{v}^\ast\vec{v})$$ $$\Rightarrow \lambda = \lambda^\ast$$ $$\Rightarrow \lambda\in\mathbb{R}$$ Global and local variables in R linux find command to move files Windows 7 low and critical battery levels cannot be set lower than 10% Is the following correct, or is there more to it?  "I forgot his name" — I knew his name, but I forgot it.  "I forget his name" — I keep forgetting his name. Where using "forget" basically means that you tried to remember this information before as well, but you couldn't then either. I am using 301 redirects for all URLs.  Should I change sitemap URLs now?  The sitemap currently contains the following URLs:   example.com/cat/1.html example.com/cat/2.html example.com/cat/3.html   Should I change the sitemap's URLs to a new destination?   example.com/cat/year/1.html example.com/cat/year/2.html example.com/cat/year/3.html Take an FF body and a crop body, with similar MP rating - and the FF body will give a sharper image of the same object. At least, that seems to be the case - as here:    I assume a low ISO was used in both images.  Why does the FF body, with the same lens and almost the same MP, give an image that is so much sharper?   My best guess is that the diffraction / airy still play a role, and the larger pixels reduce effective overlap - thus increasing contrast/sharpness by displaying less of the neighboring diffraction, as below? Why couldn't they omit the Batgirl narratives and scene in The Killing Joke movie? I loved the comic book. I've read it multiple times. And given that the purpose of the book and consequently, the movie, was       for the Joker to prove that anyone can be driven mad simply because of "one bad day", all he needed was the Commissioner right?    Why would they need to have us, as viewers, bond with Batgirl emotionally for around 25-30 minutes, have her frolic around with Batman and try to prove herself to him, etc.?     Sure, you feel deep sorrow for Barbara when she gets shot and molested by Joker. And this obviously is what affects Commissioner Gordon (in part),   But it isn't essential to the crux and the essence of the movie in my opinion (and going by the book, so to speak). Did Sam Liu or Brian Bolland offer any explanation to this end? Citing @inbook with biblatex: how to put authors/editors before book title? I wanted to discuss my intuition of why we formulate the concept of continuity in terms of pre-image of open set is open instead of images for example if we consider $f(x) = c$ where $c$ is some constant, then that should be continuous, but if we formulated continuity in terms of images it won't be continuous. Is the reason we do it that way, because pre-images will always guarantee that function in the domain can only have $1$ value? Take a look on a website with specified viewport size, that is bigger than physical size? My display driver's maximum screen width is 1280 pixels.   I need to test a website with resolution 1920&times;1080 pixels. Or 3000&times;5000.   Different browsers is optional. How does vendor restocking work in Fallout 4?