AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
![$P(x) \in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/4/5/3/453a624c3b002c1b0e78e0023b24dd22ddd03557.png)
such that:
with 
is an 
th prime and 
is an arbitrary positive integer.
of 
touch 
at 
, 
, let 
denote the line tangent to through 
. Define 
such that 
. Prove that the circumcenter 
of lies on 
.
be a cubic polynomial given by 
, where 
are integers and 
. Suppose that 
for infinitely many pairs 
of integers with 
. Prove that the equation 
has an integer root.
be real numbers such that 
. Determine the minimum value of 
and determine all values of 
such that the minimum value is achived.
cities. The airline Starways wants to establish some one-way flights between pairs of cities in such a way that each city has exactly one flight out of it. Find the smallest positive integer 
such that no matter how Starways establishes its flights, the cities can always be partitioned into groups so that from any city it is not possible to reach another city in the same group by using at most 
flights.
distinct circles in a plane, with radius 
.
circles, which form a set 
, which satisfy the following., they intersect with each other or, they don't intersect with each other.
satisfy:
.
with incentre 
, let 
and 
by the midpoints of 
and 
respectively, and 
and 
be the feet of the altitudes from 
and to the respective sides. Denote by 
the line being tangent tot he circumcircle of triangle and passing through 
, and denote by 
the reflection of in 
. Let 
by the intersection of 
and , and let 
be the intersection of 
and . Defined 
analogously. If 
is the intersection of 
and 
, prove that 
.
with two altitudes 
and 
.Let be the point on the line such that 
.The lines 
and 
intersect at point 
, and 
is the point on segment 
such that 
.Suppose that bisects .Prove that triangle 
is isosceles.
s such that:
such that 
be the set of positive real numbers.
such that there exists a function 
with the property:![\[
f(y^2f(x) + y + c) = xf(x+y^2)
\]](http://latex.artofproblemsolving.com/e/3/0/e30ee6d9f69cc43422a5c2b9813192a632d359cb.png)
for all 
.
satisfying
Prove
is non-increasing.
.
and 
gives![\[f((y-a)f(a)+\sqrt{y-a}+c)=af(y)<bf(y)=f((y-b)f(b)+\sqrt{y-b}+c)\]](http://latex.artofproblemsolving.com/f/e/6/fe6b840479f1eb218e9418aa9e165ba8aa137f3d.png)
for all 
.
.
,
gives LHS<RHS. Hence by IVT, such 
exists, contradiction. 
,
,
and 
,
.
,
.
and 
gives![\[af(a+y)=bf(b+y)\]](http://latex.artofproblemsolving.com/f/e/e/fee275f8872b721d726b266b56ba1b170c6ae7ce.png)
So for sufficiently large 
,![\[f(k(a-b)+r)=\left(\frac{b}{a}\right)^kf(r)\]](http://latex.artofproblemsolving.com/f/0/f/f0f76f5796fa93149b629665f52b79087296d182.png)
![\[\log f(k(a-b)+r)=\log f(r) - k(\log a - \log b)\]](http://latex.artofproblemsolving.com/8/2/0/8204717fddbbdaa57cb1d3bd442ec4be837b1cb5.png)
And similarly, we have![\[\log f(k(c-d)+r)=\log f(r) - k(\log c - \log d)\]](http://latex.artofproblemsolving.com/f/0/a/f0a66b4e6fc459bb3de67c09924a2c7e79f9977e.png)
But note that for all reals 
,
such that 
,![\[\log f(r) - p(\log a - \log b)=\log f(p(a-b)+r)\le \log f(q(c-d)+r)= \log f(r) - q(\log c - \log d)\]](http://latex.artofproblemsolving.com/3/b/d/3bda4f73665cbb920b8ca552b524f9dabcee8a74.png)
![\[ \frac{p}{q} \ge\frac{\log c - \log d}{\log a - \log b}\]](http://latex.artofproblemsolving.com/b/6/7/b671d04a5b26903be42cc8b41e46bbc52abfdf69.png)
![\[\frac{c-d}{a-b}+\epsilon \ge\frac{\log c - \log d}{\log a - \log b}\]](http://latex.artofproblemsolving.com/2/f/b/2fbba7b83531358f94b729fe60d461cb8d902429.png)
and taking 
gives 
.
.
,
.![\[\frac{y^2f(1)+y+c-(1+y^2)}{\log\left(\frac{y^2f(1)+y+c}{1+y^2}\right)}=k\]](http://latex.artofproblemsolving.com/5/7/3/573bd5365a8c5a0605d8ac8a2828a9a3dc3ba9e1.png)
for all 
except for finitely many values, as 
only holds for finitely many values. Then,![\[\lim_{y\rightarrow\infty}\frac{y^2f(1)+y+c-(1+y^2)}{\log\left(\frac{y^2f(1)+y+c}{1+y^2}\right)}=k\]](http://latex.artofproblemsolving.com/a/a/b/aab3866fbbc6e4f3872a157ce8b392f3b77ecc91.png)
If 
,
whereas the numerator tends to 
,![\[\lim_{y\rightarrow\infty}\frac{y+c-1}{\log\left(1+\frac{y+c-1}{y^{2}+1}\right)}=k\]](http://latex.artofproblemsolving.com/c/f/8/cf8ce8970303560d514ecebf499c47f78b1d3362.png)
But by L'Hôpital's rule,![\[\lim_{y\rightarrow\infty}\frac{(y^2+1)(y^2+1+y+c-1)}{y^2+1-2y(y+c-1)}=k\]](http://latex.artofproblemsolving.com/e/7/a/e7a5a5ebfba833736b77a628e535d3c7cf517825.png)
and the numerator has degree 
,
,