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.
with an area of 
, with 
. If 
, Hence find the value of 
.
be real numbers such that ![\[ \frac{1}{x^2-1} + \frac{1}{y^2-1} + \frac{1}{z^2-1} = 1. \]](http://latex.artofproblemsolving.com/8/f/5/8f5d49c9b44e3db3cd85fea7b6521e1dc02d4269.png)
Prove that ![\[ \frac{1}{x+1} + \frac{1}{y+1} + \frac{1}{z+1} \le 1. \]](http://latex.artofproblemsolving.com/c/0/4/c0438d6eba686d0a11fd4decb4afa9a14aa50bac.png)
such that, for every positive integer 
the integer
is a multiple of 
(Note that 
denotes the greatest integer less than or equal to 
For example, 
and 
)
.
, there exist indices 
such that 
is divisible by .
, find the minimum possible value of 
.
. The point 
is in the interior of . The following ratio equalities hold:![\[\angle PAD:\angle PBA:\angle DPA=1:2:3=\angle CBP:\angle BAP:\angle BPC\]](http://latex.artofproblemsolving.com/9/3/8/9388a91325842f1ec67a4693f0f10af46ed66c6d.png)
Prove that the following three lines meet in a point: the internal bisectors of angles 
and 
and the perpendicular bisector of segment 
.
for which there exists an integer 
such that 
is a divisor of 
.
and 
, there exists a positive perfect square with sum of digits equal to 
.
, where the number of 
s is 
(one can also just cite 2003 n4 for 
) 
satisfy 
is the smallest positive perfect square with sum of digits equal to for all positive integers 
. It's clear that doesn't divide , that 
for all , and since is a perfect square, 
.
for some positive integer 
with 
. The values 
are exactly 
, which are clearly all distinct perfect squares.
.
.
.
such that 
.![$$2 \sqrt[4]{\frac{a}{b+c-a}}\ge 2 +\frac{2a^2-b^2-c^2}{(a+b)(a+c)}.$$](http://latex.artofproblemsolving.com/f/0/7/f0731fbe5c83aedc8474975f930f7194c08cc070.png)
and 
Prove that![$$3\left(\sqrt[3] 2+\frac{1}{\sqrt[3] 2} -1\right) \geq a+b+c\geq \frac{3+\sqrt{11}}{2}$$](http://latex.artofproblemsolving.com/f/6/6/f66042b4d07212741245953a8940280c94a52988.png)
![$$\frac{3}{2}\left(4+\sqrt[3] 4-\sqrt[3] 2\right) \geq a+b+c+ab+bc+ca\geq \frac{3(1+\sqrt{11})}{2}$$](http://latex.artofproblemsolving.com/0/5/6/056db8bf4d80242d705c265b6a64e063c8fca15a.png)
Prove that
denote the sum of digits function. Does there exist ![\[
n, s(n), \dots, s^{2024}(n)
\]](http://latex.artofproblemsolving.com/c/3/3/c33803a77912ed4dde5a5c0ae05934496f04707f.png)
are all distinct perfect squares?
,
such that 
.
times.![\[y=\sum_{i=0}^{x-1} 10^{2^i} > x.\]](http://latex.artofproblemsolving.com/c/6/0/c60c3b85bac8e4a3ccaaa66830f9b0b496895f38.png)
Observe that
Here, any term of the form 
with 
uniquely determines 
and cannot be a power of 
.
,
:![\[81, 9801, 998001, 99980001, 9999800001 \dots\]](http://latex.artofproblemsolving.com/9/8/2/9829753bc3cf7feda72593b745c8cf1e3316cfe2.png)
This finishes.