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 positive real numbers such that![\[
\log_x 64 + \log_{y^2} 16 = \frac{5}{3} \quad \text{and} \quad \log_y 64 + \log_{x^2} 16 = 1
\]](http://latex.artofproblemsolving.com/d/7/9/d79facd90d9c5b53cdc4df9db18923ad34f472df.png)
What is the value of 
? and be positive real numbers such that![\[\log_x64+\log_{y^2}16=\dfrac{5}{3}\text{ and }\log_y64+\log_{x^2}16=1 \]](http://latex.artofproblemsolving.com/6/6/6/666e220af0dd4e1f77283dc5109a59d796c81bda.png)
Find 
be positive integers. How many pairs of 
are there such that 
is rational?
satisfy the equality, 
prove that the triangle must have a right angle.
such that for any real numbers 
and 
, the following inequality holds:![\[
f\left(x^2+2y f(x)\right) + (f(y))^2 \leq (f(x+y))^2
\]](http://latex.artofproblemsolving.com/8/4/3/8437449cd3f8f0da24d17a60b55dcb563e6b5655.png)
defined by 
and![\[
u_{n+1} = \frac{1}{2}u_n^2 - 4 \quad \text{for all } n \in \mathbb{N}.
\]](http://latex.artofproblemsolving.com/9/9/4/994aa754cc1288ce4f28a95a0276e64282fb5f66.png)
a) Prove that there exist infinitely many positive integers 
such that 
.![\[
\lim_{n \to \infty} \frac{2u_{n+1}}{u_0u_1\cdots u_n}.
\]](http://latex.artofproblemsolving.com/f/f/1/ff174e7431cbfbc17c650d109651241286756a1a.png)
+1 w
is a cyclic quadrilateral. Let 
and 
. Denote by 
and 
the orthocenters of triangles 
and 
, respectively. Prove that the points 
, , are collinear.
be a triangle. A circle passing through 
and 
intersects the sides 
and 
again at 
and 
respectively. Prove that 
, 
and 
are concurrent, where 
and 
are the orthocentres of triangles and 
respectively.
.The Incircle touches sides 
and 
in 
and 
,respectively. Lines CI and EF intersect at 
. The point 
≠ is on the line AI so that 
=
.If 
is the foot of the altitude from 
in triangle ,prove that points , and are colinear.
, , 
and 
positive integers such that:
+ 
+ 
= 
,
,
be real numbers different from 0 for which 
+ 
+ 
= 0 holds+)(+)(+)≠ 0
= + + and 
= 
+ 
+ 
. Prove that numbers and are both positive or both negative.
, or 
.
such that 
and 
that are greater than or even equal to 
.