AoPS Academy
, where 
is a prime number. Show that if 
is an even number, then the polynomial ![\[-1+\prod_{k=0}^{n-1} P\left(X^{p^k}\right)\]](http://latex.artofproblemsolving.com/5/d/f/5df80f885e63852e4f53a6145dafd18d63dc3526.png)
is divisible by 
.
are positive integers such that for every natural number the set of prime factors of 
are same. prove that 
.
and let 
be reals such that 
.
be a continuous function defined on ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
with 
and 
. We define the point 
to be the projection of point 
on the x-axis. Prove that there exist points 
such that 
is a square.
![$f : [0,1] \longrightarrow \mathbb{R}$](http://latex.artofproblemsolving.com/9/a/4/9a4b7997c35990f839742d29866122791ade8b7c.png)
is differentiable with 
. If 
for all ![$x \in [0,1]$](http://latex.artofproblemsolving.com/d/3/4/d3423a3200d0ad7660fc4265d59d5598a53d551d.png)
, then show that 
for all 
.
increasing with
? increasing with
?
and the distribution function is 
then
and find the expression of 
over all triples 
of distinct real numbers such that
are on the side 
such that 
and let 
be the projections of 
onto 
respectively. The line 
intersects 
at 
(different from point 
).
.
.
, it's easy to know that 
, from which we can get 
.
, which is the area of 
, as desired.
. We have![\begin{align*}
2[AMDN] &= 2([AMD]+[AND]) \\
&= AD(AM \sin \theta + AN \sin(A-\theta)) \\
&= AD(AF \sin \theta \cos(A-\theta) + \cos \theta \sin(A-\theta)) \\
&= AD \cdot AF \cdot \sin \theta.
\end{align*}](http://latex.artofproblemsolving.com/9/a/f/9afd7bdeab076d2d8de1ddae51b2a174a41c222e.png)
So it suffices to show that 
, but this is simply true because .
Now dividing both sides by 
gives the result.
Divide by 
and use 
.
Using Lemma 1: 
This is equivalent to 
Which is just Ptolemy theorem for 
.![[asy]
import graph; size(10cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -14.241848214741799, xmax = 11.317889835069966, ymin = -5.96603640835851, ymax = 9.940734075836872; /* image dimensions */
pen rvwvcq = rgb(0.08235294117647059,0.396078431372549,0.7529411764705882); pen dtsfsf = rgb(0.8274509803921568,0.1843137254901961,0.1843137254901961);
/* draw figures */
draw((1.7619209254634167,8.7237821962927)--(-7.69687926965853,-0.48362131590244306), linewidth(1));
draw((-7.69687926965853,-0.48362131590244306)--(4.904379462048781,-0.5778950720000042), linewidth(1));
draw((4.904379462048781,-0.5778950720000042)--(1.7619209254634167,8.7237821962927), linewidth(1));
draw(circle((-1.3737046562445991,2.482789896605576), 6.984420711370107), linewidth(1) + rvwvcq);
draw((1.7619209254634167,8.7237821962927)--(-3.6152839988300443,-0.5141569413699879), linewidth(1) + dtsfsf);
draw((1.7619209254634167,8.7237821962927)--(2.4172067581662664,-3.3833096014855637), linewidth(1) + dtsfsf);
draw((-5.616382898360874,1.5415794707760124)--(4.01227895316257,2.0627224343031854), linewidth(1) + rvwvcq);
draw((-5.616382898360874,1.5415794707760124)--(-3.6152839988300443,-0.5141569413699879), linewidth(1) + rvwvcq);
draw((-3.6152839988300443,-0.5141569413699879)--(4.01227895316257,2.0627224343031854), linewidth(1) + rvwvcq);
/* dots and labels */
dot((-7.69687926965853,-0.48362131590244306),dotstyle);
label("$B$", (-7.625027712498727,-0.3227630623057912), NE * labelscalefactor);
dot((4.904379462048781,-0.5778950720000042),dotstyle);
label("$C$", (4.965082120946319,-0.40526705859311163), NE * labelscalefactor);
dot((-3.6152839988300443,-0.5141569413699879),dotstyle);
label("$F$", (-3.549330295905114,-0.3557646608207194), NE * labelscalefactor);
dot((2.4172067581662664,-3.3833096014855637),linewidth(4pt) + dotstyle);
label("$D$", (2.4899622323267168,-3.243404530876935), NE * labelscalefactor);
dot((-5.616382898360874,1.5415794707760124),linewidth(4pt) + dotstyle);
label("$M$", (-5.54592700605826,1.6738336478473634), NE * labelscalefactor);
dot((4.01227895316257,2.0627224343031854),linewidth(4pt) + dotstyle);
label("$N$", (4.074038961043263,2.2018592240862143), NE * labelscalefactor);
dot((2.2642971136055023,-0.5581438324854661),linewidth(4pt) + dotstyle);
label("$E$", (2.324954239752077,-0.42176785785057574), NE * labelscalefactor);
dot((1.761920925463417,8.7237821962927),linewidth(4pt) + dotstyle);
label("$A$", (1.8299302620281561,8.851681324844241), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
[/asy]](http://latex.artofproblemsolving.com/a/0/b/a0b5111ee09630046735ad62d794f07764d2fd5a.png)
what is it?
.
is perpendicular to 
，
,
,
,
