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.
of a prime number 
and a positive integer 
for which 
is an integer.
such that the equation in the unknown 
:
admits 
roots: 
s.t. 
.
does there exist an 
matrix 
whose entries are all in 
, such that 
is the matrix of all ones?
the point 
is the centre of the excircle opposite the vertex 
This excircle is tangent to the side 
at 
, and to the lines 
and 
at 
and 
, respectively. The lines 
and 
meet at 
, and the lines 
and 
meet at 
Let 
be the point of intersection of the lines 
and , and let 
be the point of intersection of the lines 
and 
Prove that is the midpoint of 
opposite the vertex is the circle that is tangent to the line segment , to the ray beyond 
, and to the ray beyond 
.)
such that if 
and 
are distinct rational numbers satisfying 
or 
, then 
? Justify your answer.
be a function, two times derivable on 
for which there exist 
such that![\[ \frac { f(b)-f(a) }{b-a} \neq f'(c) ,\]](http://latex.artofproblemsolving.com/a/3/1/a31861ef909a6e17ed8b495b9b9b16d887ba573f.png)
for all 
.
.
be a positive integer. Consider the following equation: ![\[ \{x\}+\{2x\}+ \dots + \{nx\} = \lfloor x \rfloor + \lfloor 2x \rfloor + \dots + \lfloor 2nx \rfloor\]](http://latex.artofproblemsolving.com/8/2/9/8299b3ceb1650418c0a373536cb1d2931900efd9.png)
a) For 
, solve the given equation in 
., the equation has at most 
real solutions.
be a set with a binary operation 
such that
for all 
.
for all . is commutative and give an example where is not associative.
and 
intersect at two points 
and 
. Let 
be the line tangent to these circles at 
and 
, respectively, so that lies closer to than . Let 
be the line parallel to and passing through the point , with 
on and 
on . Lines 
and 
meet at 
; lines 
and meet at 
; lines 
and meet at 
. Show that 
.
is such that 
for any positive integers 
. Assume there exists a positive integer such that 
for all positive integers 
. Determine all possible values of 
.
satisfy that 
. Prove that: 
be a positive root of the equation 
. Prove that the following sequence converges: 
, just don’t know how to prove it.
and 
.
.
hence ![${{n}^{2}}(\sqrt[n]{a}-\sqrt[n+1]{a})={{n}^{2}}\sqrt[n+1]{a}\left( \frac{\sqrt[n]{a}}{\sqrt[n+1]{a}}-1 \right)={{n}^{2}}\sqrt[n+1]{a}\left( {{a}^{\frac{1}{n(n+1)}}}-1 \right)=$](http://latex.artofproblemsolving.com/d/f/e/dfe9afef2115ded44a29597baa9c9884a9b8f3e6.png)
![$=\frac{n}{n+1}\sqrt[n+1]{a}\frac{{{a}^{\frac{1}{n(n+1)}}}-1}{\frac{1}{n(n+1)}}\to \ln a=\ln ({{x}^{2}}+x+1)$](http://latex.artofproblemsolving.com/e/f/6/ef670bda07317faa7fd305ee1a089ef8d71ab387.png)
