[/quote]
,
and 
are positive integers such that ![\[ 1 = \underbrace{\varphi( \varphi( \dots \varphi(}_{k\ \text{times}} n) \dots )). \]](http://latex.artofproblemsolving.com/b/6/4/b64fa5fbf6613768c78beb3ab2b712599bd7ea80.png)
Prove that 
.
denotes Euler's totient function, i.e. 
which are relatively prime to 
.
be the set of nonzero real numbers. Find all functions 
such that, for all 
,![\[(x-y)f(y^2)+f\left(xy\,f\left(\frac{x^2}{y}\right)\right)=f(y^2f(y)).\]](http://latex.artofproblemsolving.com/b/1/b/b1b1b156221baf8e9936027f352c50abcd2d05ff.png)
be a finite set of cells in a square grid. On each cell of 
, ![$\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{a} \geq \sqrt[n]{\frac{a^n + b^n}{2}} + \sqrt[n]{\frac{b^n + c^n}{2}} + \sqrt[n]{\frac{c^n + a^n}{2}}$](http://latex.artofproblemsolving.com/6/0/6/606cb6c4a384c641c90618f271d93b9a923ec8b8.png)
be an acute triangle with circumcenter 
and orthocenter 
.
to the circumcicle of 
.
be the midpoint of 
.
.
and 
Prove that
from the set of real numbers into the set of real numbers which satisfy for all 
,
the identity ![\[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\]](http://latex.artofproblemsolving.com/f/8/3/f83e465b212cbf72ab091fc49923b40938fde584.png)
and 
as ![\[\sqrt{2}=1.a_1a_2a_3\dots\quad\quad\sqrt{3}=1.b_1b_2b_3\dots\]](http://latex.artofproblemsolving.com/4/d/5/4d5049d5c3b26757a25c5dd34462c5462228b83d.png)
where each 
or 
is a digit between 0 and 9. Prove that there exist at least 1000 values of 
between 
and 
such that 
.
,
,
and 
,
.
intersects 
and 
.
is tangent to the circles with diameters 
and 
.![\[\left\lfloor\frac{2025}{1}\right\rfloor,\left\lfloor\frac{2025}{2}\right\rfloor,\left\lfloor\frac{2025}{3}\right\rfloor,\dots,\left\lfloor\frac{2025}{2024}\right\rfloor,\left\lfloor\frac{2025}{2025}\right\rfloor.\]](http://latex.artofproblemsolving.com/a/8/2/a8255ec218f896983d39044f8f777e2fd811479b.png)
Prove that
.
is tangent to the circumcircle of 
and 
,
.
through 
Y by Adventure10, Mango247, HWenslawski
In 
with 
,
is the midpoint of 
,
is the projection of onto 
and 
is arbitrary point on the side 
.Let 
be the intersection point of the parallel line through 
to 
with the parallel line through 
to .Prove that 
is the bisector of 
.
with 
,
is the midpoint of 
,
is the projection of onto 
and 
is arbitrary point on the side 
.Let 
be the intersection point of the parallel line through 
to 
with the parallel line through 
to .Prove that 
is the bisector of 
.
Thus 
is a rhombus with incircle 
and let the tangent from 
other than 
cut 
at 
Tangents 
of 
induce a proyectivity between 
thus if 
denote the points at infinity of 
where 
.
where 
is a fixed number.
where 
.
. We are talking of course about oriented angle.
axis lines 
are given by equation 
.
So 
and 
common denominator 
clearing
We are actually finished but it would be very nice to check that 
has no real roots (we don't want to divide by 
)
since 
be the reflection of 
is tangent to 
So, 
cyclic
be the contact points with the incircle. Let 
be the A-excenter. A-excircle touches the sides 
at 
respectively. Let the perpendicular to 
at 
at 
.
.
,
,
.
and 
be the inradius and A-exradius respectively. Heron's formula states that:
,
.
,
.
we will get that 
and we are done.
.
be the contact point of 
be the diametrically opposite point of 
be the another tangent line to 
,
.
.
.
.
,
,
.
where 
.
which is equivalent to 
,
is a rhombus then let 
such that 
which we will call 
at 
respectively then using pascal on 
and 
as desired, conclusion now follows from excenter propeties thus we are done 
.
.
,
.
(by easy angle chasing)
.
, 
so 
, 
so 
.
(by easy angle chasing)
