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.
such that the equation
be a positive integer and 
be a polynomial with integer coefficients such that 
.
is divisible by 
.
such that for every positive real 
, there exists an integer 
satisfying![\[
\frac{\varphi(n!)}{n^\alpha\,(n-1)!} \;>\; c.
\]](http://latex.artofproblemsolving.com/4/4/4/4443cf7166cc65e67e2e504a826c4ac5f91aab2b.png)
be a positive integer and let 
be a strictly increasing sequence of positive real numbers with sum equal to 2. Let 
be a subset of 
such that the value of![\[
\left|1-\sum_{i \in X} a_{i}\right|
\]](http://latex.artofproblemsolving.com/2/5/1/2515e074de1782f38453e1fce8ab7c2ec7cfe4c0.png)
is minimised. Prove that there exists a strictly increasing sequence of positive real numbers 
with sum equal to 2 such that![\[
\sum_{i \in X} b_{i}=1.
\]](http://latex.artofproblemsolving.com/3/7/4/37410820fc4c546f11d57212cfce53a7ec4941e3.png)
be a scalene triangle with circumcircle 
, and suppose the incircle of touches 
at 
. The angle bisector of 
meets and at 
and 
. The circumcircle of 
intersects the 
-excircle at 
, 
, and at 
. Prove that line 
passes through either or .
be positive real numbers such that : 
. Prove that : 
![$P, Q \in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/c/3/0/c301578bade5b93056d77454921cba0a9812212c.png)
be relatively prime nonconstant polynomials. Show that there can be at most three real numbers 
such that 
is the square of a polynomial.
be real numbers less than 2.Then prove that 
and 
intersect each other at 
and 
. The common tangent to two circles nearer to touch and at 
and 
respectively. Let 
and 
be the reflection of and respectively with respect to . The circumcircle of the triangle 
intersect circles and respectively at points 
and 
(both distinct from ). Show that the line 
is the second tangent to and .
has circumcircle and circumcenter 
. A circle 
with center intersects the segment at points and , such that 
, , , and 
are all different and lie on line in this order. Let and 
be the points of intersection of and , such that , , , , and lie on in this order. Let 
be the second point of intersection of the circumcircle of triangle 
and the segment 
. Let 
be the second point of intersection of the circumcircle of triangle 
and the segment 
.
and 
are different and intersect at the point . Prove that lies on the line 
.
be the diameter of semicircle 
. Circle is internally tangent to and tangent to at . Let be a point on and a point on such that 
and is tangent to . Prove that 
bisects 
is a point on the plane of 
Lines 
cut 
at 
Define 
and let 
be the orthogonal projection of 
onto 
Then lines 
bisect 
and 
intersect the parallel through to at and 
respectively and, as 
, we need to prove 
.
and 
, or, dividing side by side the two equalities: 
, but, as from Ceva for 
with , the right side of the last equality is 
, hence our goal has been achieved.
, 
, 
, 
and 
. lies on and 
, so for some 
,
Comparing real and imaginary coefficients, we have 
and 
, so
Similarly,
, that is (with directed angles)
so it suffices to show that 
is real, and expanding,
which is real, so this holds.
is equivalent to saying 
is real as we use directed angles and won't have configuration issues because of the acute ? If not, you'd probably have to prove that it's negative too)
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
,
is the foot of the altitude from 
,
a point on segment 
.
and 
meet 
and 
at 
and 
respectively. Prove that 
.
.
.
.
,
,
is the internal bisector of 
