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.
and 
be two circles with respective centres 
and 
intersecting in two distinct points 
and 
such that 
is an obtuse angle. Let the circumcircle of 
intersect and respectively in points 
and 
. Let the line 
intersect in 
; let the line 
intersect in 
. Prove that, the points 
are concyclic.
from the set of positive integers to the set of positive integers such that, for all positive integers 
and 
, there exists a non-degenerate triangle with sides of lengths![\[ a, f(b) \text{ and } f(b + f(a) - 1).\]](http://latex.artofproblemsolving.com/4/a/b/4ab97da27e05d0f9e7762f74d0d0bd2dc8b9842f.png)
and 
.Find the minimum of ![$S=\sqrt[3]{\frac{a^3+b^3+c^3}{3}}+\frac{a+b+c}{9}$](http://latex.artofproblemsolving.com/9/d/9/9d93ff9d9f7355e452f1007aeb87423e3491353a.png)
and 
be the incircle and circumcircle of the acute triangle 
, respectively. Draw a square 
so that all of its sides are tangent to , and 
, 
are both on 
. Extend 
and 
, intersecting at 
and 
, respectively. Prove that 
and 
intersects on .
converge?
be a cyclic quadrilateral with circumcentre . The diagonals 
and 
meet at point . Suppose the four incentres of triangles 
lie on a circle with centre . Prove that 
are collinear.
such that, for all positive integers 
and 
,
be a positive integer. We say that is bao-good if there exist integers 
such that:
for which![\[
a \mid s^b - l.
\]](http://latex.artofproblemsolving.com/5/d/9/5d9db7678c54d1c6a909b4aca39a329cadd71e2c.png)
2. For every proper divisor 
of (that is, 
and 
), there exists a positive integer such that![\[
a' \mid s^b - l.
\]](http://latex.artofproblemsolving.com/4/b/2/4b26b9bd3da8958287703c3f575bf5d5da7fd477.png)
Determine all bao-good positive integers .
such that![\[(a^2 + b^2 + c^2)^2 = 3(a^3b + b^3c + c^3a) + 1. \]](http://latex.artofproblemsolving.com/9/e/4/9e400fd807a9535ede983b75dcc307d8b75a35b6.png)
![$f \colon [0,1] \to \mathbb{R}$](http://latex.artofproblemsolving.com/a/1/6/a16e0dd7aead8a308ccb37da016854116ec5be61.png)
be a continuous function which is differentiable on 
. Prove that either 
for all ![$x \in [0,1]$](http://latex.artofproblemsolving.com/d/3/4/d3423a3200d0ad7660fc4265d59d5598a53d551d.png)
for some constants 
or there exists 
such that 
.
be an endomorphism of 
It follows from the condition given in your problem that is a field and so 
i.e. is injective. Choosing 
in the condition, shows that the characteristic of is 
Let 
So 
for some positive integers 
Consider the polynomial ![$g(x):=(x^m+1)^n-x \in \mathbb{F}_p[x]$](http://latex.artofproblemsolving.com/0/f/4/0f4cb6baabdae533391243bc41ad00961093a763.png)
and let 
be the set of roots of 
in Then is finite, 
and 
for all 
Thus, since is injective, 
and so 
for some 
proving that is surjective hence an automorphism.
.
such that for any 
there exist positive integers 
such that 
Prove that any endomorphism of 