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.
such that 
for all positive integers 
.
be a triangle with 
Line bisector of 
intersects 
at point 
. Prove that 
.
, and for all 
, define the sequence 
by the recurrence![\[
a_{n+1} = a_n^2 + 1
\]](http://latex.artofproblemsolving.com/3/1/1/3114595ded8c8961d611afaab9ad19e254973001.png)
Prove that there is no natural number 
such that![\[
\prod_{k=1}^{n} \left( a_k^2 + a_k + 1 \right)
\]](http://latex.artofproblemsolving.com/a/4/f/a4f88ee7e3d4e53e369319f74cd9b12be26c9681.png)
is a perfect square.
be a triangle. 
is the midpoint of segment 
, and points 
, 
are selected on sides 
, 
respectively such that , , are collinear. The circumcircles 
and 
intersect at a point 
. The circumcircle 
intersects line again at a point 
. Show that the lines 
, 
and the tangent to at point 
concur.
such that 
for all 
people playing a game at Cheesy's math casino, where is an odd prime number. Let 
be a positive integer. A subset of length 
from the set of integers from 
to inclusive is randomly chosen, with an equal probability (
and is fixed). The winner of Cheesy's game is person 
, if the sum of the chosen numbers are congruent to 
for 
., find all values of such that no one will sue Cheesy for creating unfair games (i.e. all the winning outcomes are equally likely).
where 
is a parallelogram.
.
is connected to the midpoint of ![$[CT]$](http://latex.artofproblemsolving.com/f/3/f/f3f8e9913dc696e543e6a34d169ec3deb276c84d.png)
, point 
to the midpoint of ![$[DT]$](http://latex.artofproblemsolving.com/6/d/d/6dd49550ab42ec2f047eaf4b357ac4e7d1387b1e.png)
,
to the midpoint of ![$[AT]$](http://latex.artofproblemsolving.com/6/e/3/6e397da3c2a38b457f7b3504c5cb2a488a64f3a6.png)
and point 
to the midpoint of ![$[BT]$](http://latex.artofproblemsolving.com/6/a/4/6a4b1bcea7921e762a359321ab861b6937f04782.png)
..
.
be a positive integer and let 
, 
, ..., 
be a sequence of non-negative real numbers. Find the maximum value of 
in terms of .
are randomly arranged in the cells of a 
square (
). For any pair of numbers situated on the same row or on the same column the ratio of the greater number to the smaller number is calculated. Let us call the characteristic of the arrangement the smallest of these 
fractions. What is the highest possible value of the characteristic ?
be a triangle of orthocenter 
. The circle of diameter 
and the circumcircle of triangle 
intersect a second time in 
.
passes through the midpoint of segment 
.
be a trunk of a triangular pyramid. Consider points 
and 
so that the planes 
and 
are parallel. Prove that the planes 
and 
are parallel.
.
