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.
Prove that
Where 
Prove that
Let 
Prove that
is tangent to sides 
and 
at 
and 
respectively, and 
is the circumcenter of 
. Prove that 
.
denote the maximum number of 
-tetrominoes that can be placed on an 
board such that each -tetromino covers at least one cell that is not covered by any other -tetromino.
such that![\[
f(N) \leq cN^2
\]](http://latex.artofproblemsolving.com/7/4/8/748109c3d63203f9a67a167434e10a909f38e83f.png)
for all positive integers 
.
into 
or 
can sometimes be quite complex. That's why I’ve written a program to assist with this process.
such that for all positive real numbers 
and 
, the following equation holds:![\[
\frac{x + f(y)}{x f(y)} = f\left( \frac{1}{y} + f\left( \frac{1}{x} \right) \right)
\]](http://latex.artofproblemsolving.com/7/d/f/7df25755bc33bef970fffb392fb5b6b789390881.png)
such that for all 
and 
,
such that 
or 
, there exist integers![\[
1 \leq a_1 < a_2 < \cdots < a_{(p-1)/2} < p
\]](http://latex.artofproblemsolving.com/6/8/e/68e3d73edec43b2cc480832adf23bbbb5f4c7bee.png)
such that![\[
\prod_{\substack{1 \leq i < j \leq (p-1)/2}} (a_i + a_j)^2 \equiv 1 \pmod{p}.
\]](http://latex.artofproblemsolving.com/2/9/e/29edf6ab8cee2a526e7753a140c5a291e987856a.png)
be distinct real numbers such that![\[
a + b + c + \frac{1}{abc} = \frac{19}{2}
\]](http://latex.artofproblemsolving.com/5/1/e/51eef77358db2b6c98f0fb0fb4e30db837a03642.png)
Find the maximum possible value of 
.
.The solution to 
is only 
for some 
, consider the two circles on 
plane: 
. Find the range of 
for which these circles have two intersection points, then find the equation of the line passing through these points.
on the perimeter of a circle with radius 1. Let the length of the cord 
. When two points move on the circle, find the maximum area of 
.
satisfies 
. Prove that 
.
without using the following method. I know that you can solve it using 
but I'm searching for an algebraic proof.
without using the following method. I know that you can solve it using 
but I'm searching for an algebraic proof.
,![\[
\sum_{k=1}^{\infty}\frac{1}{(1+k)^2} = \frac{\pi^2}{6} - 1 \neq \frac{1}{2}.
\]](http://latex.artofproblemsolving.com/7/b/b/7bbaae881538cbaceb4327c60dcd0c78916c1053.png)
Do you intend to compute![\[
\lim_{n\to\infty}\sum_{k=1}^{\infty}\frac{n}{(n+k)^2}\,?
\]](http://latex.artofproblemsolving.com/4/1/1/41150cdf57e2f5e7aa10892a726a5f79ebc987b5.png)
![\[
\frac{n}{n+k} - \frac{n}{n+k+1} < \frac{n}{(n+k)^2} < \frac{n}{n+k-1} - \frac{n}{n+k},
\]](http://latex.artofproblemsolving.com/6/f/f/6ff255d1fd6fd29bc20bc47fea55bc5c0fa60e24.png)
and summing obtains the desired result.