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 
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
such that 
and exist 
odd numbers 
satisfied all conditions below:![$\sqrt[k]{a_1}+\sqrt[k]{a_2} + ... + \sqrt[k]{a_n} = \sqrt[k]{b_1}+\sqrt[k]{b_2} + ... + \sqrt[k]{b_m} $](http://latex.artofproblemsolving.com/c/9/e/c9e051a562376b2b9e02e82d46a41126874cdf60.png)
for all positive number 
with 
.