IMO $2011$ problem $3$, how can I prove that $f(x) = 0, \; \forall \; x \leq 0 $?
If $f(x + y) \leq yf(x) + f(f(x))$ for all real numbers $x$ and $y$, then prove that $f(x) = 0$ for all $x ≤ 0$.
FICA was not withheld from my paycheck Hello personal finance,  I was filing my taxes for 2016 and I just found out that I transitioned from Non-Resident-Alien to Resident Alien around September last year by passing substantial presence test. After doing some research, I have found out that I qualify as a resident alien for the whole tax year (from January 2016).   Unfortunately, my employer did not know about this change in status, so for all of 2016 no Social Security or Medicare taxes were withheld.   I have found out that I need to correct this, and I need help figuring out a few things.    Should I have a W-2 re-issued? I spoke to two accountants and one said I should, the other said I shouldn't. I don't know which one to believe.  Who pays for the FICA I should have paid last year? Is it my employer's problem to fix or mine?  Should I file the taxes with the old or the new W-2?  As best as I understand it, the new W-2 will be written as if I made this contributions. So there will be a discrepancy between what I actually earned and what the W-2 says. How should I handle this discrepancy? Declare it as extra income? Or claim it back in case I earned less than I should have? centering column headers in siunitx table column Sum of reciprocals of numbers with certain terms omitted I know that the harmonic series $1 + \frac12 + \frac13 + \frac14 + \cdots$ diverges. I also know that the sum of the inverse of prime numbers $\frac12 + \frac13 + \frac15 + \frac17 + \frac1{11} + \cdots$ diverges too, even if really slowly since it's $O(\log \log n)$.  But I think I read that if we consider the numbers whose decimal representation does not have a certain digit (say, 7) and sum the inverse of these numbers, the sum is finite (usually between 19 and 20, it depends from the missing digit). Does anybody know the result, and some way to prove that the sum is finite? Particularly, does proc always list the VmRSS value in kB? I can't find a solid answer on the , although it appears that it sticks to one unit. How do I make Endermen spawn more frequently? I'm playing vanilla Minecraft, and am getting ready to head to the end.  I need Eyes of Ender (and hence Ender Pearls), and in my main hunting grounds (a large desert area) as little as one Enderman may spawn per night.  Is there any way to "promote" Enderman spawning so I can get pearls more quickly? How can I test whether my camera is working after it was dropped? Issue : How can I evaluate my cameras performance synthetically?  Scenario : Bought a Nikon D3100 an year back. Photos looked good enough. The camera fell, zoom lens effected.  But works. Cleaned lens using soap, water etc.  Carefully. Now photos look much worse.  How do I know if Photos are really bad?  Is there a tool?    Sample Photo  Now (Bad)   Original (Good)   Please help me realize whether my camera has gone bad.  Can you tell your personal opinion also? For $D$ a GCD domain , let $a,b,x \in D \setminus \{0\}$ , then is it true that $\gcd (ax,bx)=x \cdot \gcd (a,b)$? $g\in G$ maximal order in $G$ abelian then $G=\left\oplus H$ If $g\in G$ is an element of maximal order in a finite abelian group $G$ then exists $H\leq G$ such that $G=\left&lt;g\right&gt;\oplus H$  Attempt: Using fundamental theorem I know that $G=C_{n_1}\times\cdots\times C_{n_k}$ where $n_i|n_{i+1}$. With some work I proved $|\left&lt;g\right&gt;|=n_k$, so $C_{n_k}\cong \left&lt;g\right&gt;$. Then $G\cong H\times \left&lt;g\right&gt;$, but this is not an equality. Is there a way to stream Netflix content to my Ubuntu machine? Removing Ubuntu from a dual-boot setup with Windows How can I scale all images in a folder to the same width? I want to scale all images in a given folder to the same width (but different appropriately scaled heights). How can I do this using  a GUI-based or command-line tool?  For bonus points, is it possible to restrict which images are scaled based on their initial width (that is, only scale images that have a width > x and/or a width &lt; y)? Storing output of command in shell variable Is it possible to use a hyphen in a listing (in a sentence) for abbreviation, even if the compound word consists of two separate words Can you set custom ringtones for contact groups? In iPhone, you can easily change the default ringtone (Settings → Sounds → Ringtone) and set a custom ringtone for individual people (Contacts → [some person] → Edit → Ringtone).   But can you set a custom tone for a group of contacts? (E.g. "family" or "workmates".)  I think that would be handy—and quite basic functionality, so to speak—but I couldn't find a way to do this through the iPhone UI. Is it possible at all (without jailbreaking)? Infinite Geometric Series Formula Derivation Why can I only change my Stack Overflow name once every 30 days? What is the difference between building from source and using an install package? How well-known is the theology of Middle-earth, in Middle-earth? One thing that is notably absent in Middle-earth is any mention of temples, churches, priests or religious rites.  There are passing mentions of Men who would "worship" Sauron, but generally the world of Middle-earth seems much less religious than medieval Europe, or even classical antiquity.  One never hears about "the cult of Aule" or "the cult of Yavanna," although historically there were many "cults of Athena" or "cults of Poseidon."  Aragorn doesn't (to my memory) make an offering to Tulkas before battle, nor does he pray to Eru Ilúvatar for victory, and all the holidays and solemnities seem to commemorate historical figures, not religious ones.  This is especially strange in a world where there are still living witnesses, like Galadriel, to the actions of the Valar and other "divine" beings, and where there are still Maiar wandering around talking to people and putting on flashy firework shows.  Now, a lot of people talk about how Tolkien didn't want the Valar to be considered "gods," but at the same time he did use phrases like "the gods of old" numerous times.  Clearly, these beings would have qualified for worship in most cultures.  And even without the Valar being considered a true pantheon, there is still Eru, a closer analogue to the Christian God of Tolkien's beliefs, and we know the denizens of Middle-earth knew of him (Him?) because of the line "Eru, who in Arda is called Ilúvatar."  And yet, Ilúvatar seems not to be terribly important to anyone on a daily basis.  So my question is this: how familiar with the "theology" of Middle-earth were the citizens of Middle-earth?  Would the average merchant in Rohan know about the Valar and the Maiar and Iluvatar and the Music of the Ainur?  The average noble in Minas Tirith?  The younger Elves, like Legolas?  The Dwarf Lords?  Were the theological truths about the formation of the world, the Valar and the Maiar just so well known that they weren't worth commenting on, or had they been forgotten by most?  And if they had been forgotten, why wasn't the knowledge (whether through deliberate teachings or casual conversation) re-invigorated by witnesses like Gandalf or Galadriel? What is the difference between Θ(n) and O(n)? Why can't I run two mysqli queries? The second one fails