The above applies $\forall x,y \in \mathbb{R}$  I've tried: $x + y \ge 0$  $$x + y \ge x$$  $$ (x + y)^2 \ge 2xy$$  $$\frac{(x + y)^2}{2} \ge xy$$  But the closest I get is $\dfrac{x+y}{\sqrt{2}} \ge \sqrt{xy}$  Any ideas?
4198
The arithmetic - geometric mean inequality states that $$\frac{x_1+ \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n}$$ I'm looking for some original proofs of this inequality. I can find the usual proofs on the internet but I was wondering if someone knew a proof that is unexpected in some way. e.g. can you link the theorem to some famous theorem, can you find a non-trivial geometric proof (I can find some of those), proofs that use theory that doesn't link to this inequality at first sight (e.g. differential equations …)?  Induction, backward induction, use of Jensen inequality, swapping terms, use of Lagrange multiplier, proof using thermodynamics (yeah, I know, it's rather some physical argument that this theorem might be true, not really a proof), convexity, … are some of the proofs I know.
103450
Prove that $r \in R$ such that $0 \lt r \lt 1$ then $\frac{1}{r (1 - r)} \ge 4$  My method:  Assume that $r \gt 0$, and $1 - r \gt 0$. Hence $r(1 - r) \gt 0$. So, $1 \le 4r (1 - r)$. Hence, $1 \le 4r - 4r^2$. Hence, $4r^2 - 4r + 1 \ge 0$. Thus, $(2r - 1)^2 \ge 0$.  I think I might be missing something but I am not sure.
92174
If $x,y\in\mathbb R$, I have problems to show that  $$\lfloor x\rfloor+\lfloor y\rfloor\le \lfloor x+y\rfloor\le \lfloor x\rfloor+\lfloor y\rfloor + 1 $$  Can someone help me?
81693
Let $x,y$ be positive numbers. Prove that $x^x+y^y \ge x^y +y^x$.   This question appeared in Summer 1991 Russian Olympiad team test. Apparently, I tried to come up with different approach such as Jensen, Karamata's inequality and nothing works so far. I just need a discussion here. Hints are not necessary.
53169
It is given that $X,Y \overset{\text{i.i.d.}}{\sim} N(0,1)$  Show that $\frac{XY}{\sqrt{X^2+Y^2}},\frac{X^2-Y^2}{2\sqrt{X^2+Y^2}} \overset{\text{i.i.d.}}{\sim} N(0,\frac{1}{4})$  I was thinking of making polar transformations $X=r \cos \theta, Y=r \sin \theta$  Then I am getting stuck at ranges of $\theta$
43823
In general, $\frac{d}{dx}(f(x) \cdot g(x)) \neq  \frac{d}{dx}f(x) \cdot \frac{d}{dx}g(x)$   When does this result hold true? My first try is to use product rule on left side and compare the two sides, but this hasn't helped at all. Any suggestions?
50944
Prove that $(\frac{bc+ac+ab}{a+b+c})^{a+b+c} \ge \sqrt{(bc)^a(ac)^b(ab)^c}$  I tried it to do using $AM \ge GM$ but don't know how to proceed.  Please help.
106975
Show that $$\lim_{(x,y) \to (0, 0)}\frac{xy^2}{x^2+y^2} = 0$$  I have tried switching to polar coordinates but I'm not getting a single term.  This is what I did.   Putting $$x=r\sin θ,\quad  y=r\cos θ$$ we obtain   $$\left|\frac{xy^2}{x^2+y^2}\right|=|r\cos^2 θ \sin θ|                        =|r\sin θ(1-\sin^2 θ)|                        = |r\sin θ-r\sin^3 θ|$$
88475
let $ y=\displaystyle \sqrt{x+\sqrt{{x}+\sqrt{x+\cdots}}}$, i'm really interesting to know how do I find :  $\displaystyle \frac{dy}{dx}$ ?.  Note: I have used the definition of derivative of the square root function but i don't succed .  Thank you for any help
114686
How to deal with $m+\frac{4}{m^2}\geq3$ for every $m &gt; 0$ ? I multiplied both sides by $m^2$ and got $m^3+4-3m^2\geq0$ and have no idea how to continue with this
91198
Let $x&gt;0$, $y&gt;0$ and $z&gt;0$. Show that   $$x^3y+y^3z+z^3x\ge xyz(x+y+z).$$   I known we can't WLOG: $x\ge y\ge z$, if this, I can use rearrangement inequality, But other I can't it. Thanks?
67934
$$\frac{\sum_{i=0}^nx_iy_i-\bar y\sum_{i=0}^nx_i}{\sum_{i=0}^nx_i^2-\bar x\sum_{i=0}^nx_i}= \frac{\sum_{i=0}^n(x_i-\bar x)(y_i-\bar y)}{\sum_{i=0}^n(x_i-\bar x)^2}$$  Maybe it refers some calculation rules about summation. I cannot solve that. Thanks!
43470
I have the folowing exercise (which I've been thinking quite a while and couldn't figure out):  Show that $\forall (x,y)$ in the first quadrant: $$\frac {x^2+y^2}{4}\leq e^{x+y-2}$$  My idea was to work with maxima and minima, but I'm stuck... Any help will be much appreciated!
17114
As in the title. If $x,y&gt;0$ and $x^2+y^3\ge x^3+y^4$, prove that $$x^3+y^3 \le 2.$$  This seems to be a very tricky one. I tried applying various inequalities like AM-GM, unfortunately, none of techniques I'm familiar with seem to work here.  I'd greatly appreciate any hints.
122977
I have to prove that the result of $x * y \in G$ so $\frac{x+y}{1+xy} \in (-1, 1)$. So $x &gt; -1$ and $y &gt; -1$ at the same time $x &lt; 1$ and $y &lt; 1$.  If I multiply the first 2 expressions I obtain $xy &lt; 1$ which is true, and if I sum up the last 2 I get $x + y &lt; 2$. At this point I am not sure if I am doing well. How should I continue this problem?
52355
For $a,b,c,d \in \mathbb{R}$, is $\sqrt{a^2+b^2} + \sqrt{c^2+d^2} \geq \sqrt{(a+c)^2 + (b+d)^2}$? I have tried to show this using the fact that $\sqrt{x+y}\leq\sqrt{x}+\sqrt{y}\; $  for non-negative $x,y$ but haven't been able to arrive at a proof.
82829
I know that for any nonzero $x,y\in\mathbf{R}$,  $$ax^2+by^2+cxy &gt; 0,$$  where $a,b,c\in\mathbf{R}$. What can I deduce about $a$, $b$, and $c$?    For example, letting $x=1$ and $y=0$, I know that $\boxed{a&gt;0}$. Letting $x=0$ and $y=1$, I know that $\boxed{b&gt;0}$. Letting $x=y=1$, I know that $\boxed{c&gt;-(a+b)}$.  What else can I deduce about $a$, $b$, and $c$? How will I know when it's time to stop looking? (Is that last question even answerable?)