AoPS Academy
![\[\sum_{cyc}(x+y)\sqrt{(z+x)(z+y)} \geq 4(xy+yz+zx),\]](http://latex.artofproblemsolving.com/0/4/7/047adb616a6be9f9ed52f2385fa592fb5464e420.png)
and 
.
and 
be elements of the set 
such that 
, 
is a divisor of 
, and 
is a divisor of 
. Determine the largest value that 
can take.
such that the following equation holds:![\[
7^a + 1 = 2b^2
\]](http://latex.artofproblemsolving.com/e/0/8/e08d3fc7feefd7b4acc4a0f9c7e8bcfa328b5791.png)
such that![\[\lim_{n \to \infty} \left( \frac{1 + 2^{1/3} + 3^{1/3} + \dots + n^{1/3}}{n^t} \right ) = c\]](http://latex.artofproblemsolving.com/e/1/a/e1a87bd7bee41a7d656e2ff067a87dff0c9929ec.png)
for some real 
. Also find the corresponding values of 
.
![$f\in C^1[a,b]$](http://latex.artofproblemsolving.com/8/4/8/848da12534d564560fe1cb016142159234be98b4.png)
, 
and suppose that 
, 
, is such that
for all ![$x\in [a,b]$](http://latex.artofproblemsolving.com/7/8/d/78d9dcd969a2c45f7cc9234aa1377b6ec1c2d5d3.png)
. Is it true that 
for all ?
is compact?
be a family of functions from 
. It is known that for all 
, there exists 
such that for all 
, the following equation holds:![\[
y^2 \cdot f\left(\frac{g(x)}{y}\right) = h(xy)
\]](http://latex.artofproblemsolving.com/f/7/1/f7196164e0639695529b06788ea7cb1e50eddbe7.png)
and all 
, the following identity is satisfied:![\[
f\left(\frac{x}{f(x)}\right) = 1.
\]](http://latex.artofproblemsolving.com/1/5/b/15b363bf1754a4edd72423c3d0ea631173897beb.png)
. By hypothesis, there exists a function 
such that![\[
y^2\cdot f\left(\frac{f(x)}{y}\right) = g(xy), \quad \forall x,y\in\mathbb{R}^+
\]](http://latex.artofproblemsolving.com/4/d/e/4de11b6a7f4cdaae20e844da9ed8dd948db338bd.png)
Substituting 
gives![\[
f^2(x) f(1) = g(x f(x)), \quad \forall x \in \mathbb{R}^+
\]](http://latex.artofproblemsolving.com/c/a/3/ca3e23d972be3e9aacb9e6e4e757ee23529306c2.png)
Similarly, another function 
satisfies![\[
y^2 g\left(\frac{f(x)}{y}\right) = h(xy), \quad \forall x,y\in\mathbb{R}^+
\]](http://latex.artofproblemsolving.com/6/3/2/6329b9ac5863c54ef921dc081c23d38585bfe534.png)
Setting 
, we obtain![\[
g(x f(x)) = h(1) x^2, \quad \forall x \in \mathbb{R}^+
\]](http://latex.artofproblemsolving.com/8/b/0/8b015c73d1dd88542a49c1cf7a46a29f5156b7e1.png)
From the previous equation,![\[
f^2(x) f(1) = h(1) x^2
\]](http://latex.artofproblemsolving.com/6/e/a/6eae3c8779d1e0b12e1b4d99a91c925fd4454a88.png)
Rearranging,![\[
f^2(x) = \frac{h(1)}{f(1)} x^2
\]](http://latex.artofproblemsolving.com/f/4/a/f4af16541a2c512c7a9d83f077afac0142939394.png)
which implies![\[
f(x) = c x, \quad \forall x\in\mathbb{R}^+
\]](http://latex.artofproblemsolving.com/8/1/1/8119b15f1885facd586b4bbf47a3b0aea2d51588.png)
for some constant 
. Finally,![\[
f\left(\frac{x}{f(x)}\right) = f\left(\frac{1}{c}\right) = 1
\]](http://latex.artofproblemsolving.com/f/1/a/f1ab747b7e7d797d498ed99f4babf933b23dc6ed.png)
as required.
we get an 
and a equation, in which we set 
to get that 
, now by creating an 
for the queried 
we get by letting 
that 
which now means that 
and thus 
for all positive reals 
, and this is the shape of any random function in 
, so the condition now holds trivially as 
, and thus we are done 
:D :play_ball:
