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.
be a circle with centre 
, and 
a convex quadrilateral such that each of the segments 
and 
is tangent to . Let 
be the circumcircle of the triangle 
. The extension of 
beyond 
meets at 
, and the extension of 
beyond 
meets at 
. The extensions of 
and 
beyond 
meet at 
and 
, respectively. Prove that ![\[A D+D T+T X+X A=C D+D Y+Y Z+Z C.\]](http://latex.artofproblemsolving.com/8/2/6/8261276053071cc6cc06dda824a883426bf1d4ac.png)
and 
, a parameter 
, and a real log-exponent 
. For every real 
define
(independent of 
and of 
) such that![$$
\lim_{x\to 1^{-}}
F_{a,\beta}^{(k,r)}(x)\,
e^{-\Lambda_{k,r,\lambda}\,(1-x)^{-\gamma}}
\;=\;
\begin{cases}
0, & a<c,\\[6pt]
\infty, & a>c.
\end{cases}
$$](http://latex.artofproblemsolving.com/e/3/8/e38c6b43d513bcc697587fff610a846b3a1b0e0a.png)
explicitly and verify that it coincides with the classical case 
, namely 
.
(your answer may depend on 
but not on ).
such that for every 
is satisfied? Explain the answer.
be a random point on the smaller arc 
of the circumcircle of square 
, and let 
be the intersection point of segments 
and 
. The feet of the tangents from point to the circumcircle of the triangle 
are 
and 
, where 
is the center of the square. Prove that points , , and lie on a single circle.
and 
such that
be an integer, and let 
be positive real numbers such that 
. Prove that![\[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\]](http://latex.artofproblemsolving.com/7/7/f/77fb719e975225b59c64f1f3aa102c98a6c25f50.png)
there exist nonnegative integers 
and integers 
such that 
colors, there exists a monochromatic regular hexagon.
is the smallest number such that if 
, for each coloring of the set 
with two colors there exists a monochromatic arithmetic progression of length 
. Prove that
.
points in plane. Prove that the number of isosceles triangles having their vertices among these points is 
. Find a configuration of points in plane such that the number of equilateral triangles with vertices among these points is 
.
be a field of characteristic 
, 
.
is the square of an element from 
from can be written as the sum of three squares, each , of elements from .
be written in the same way?![$\{x_n\} \subseteq [0,1]$](http://latex.artofproblemsolving.com/8/e/4/8e4206325387485d73765c9d7070ecb1ce18ff60.png)
, there exists some constant , for any 
, among the sequence 
and 
could be chosen to satisfy 
and 
.
such that: 
and 
.
