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.
has been partitioned into disjoint pairs 
(
) so that for all 
, 
equals 
or 
. Prove that the sum ![\[ |a_1-b_1|+|a_2-b_2|+\cdots +|a_{999}-b_{999}| \]](http://latex.artofproblemsolving.com/0/4/4/04422a7d2266b92a6b3f215b698f078cda85091f.png)
ends in the digit 
.
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).
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 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 
.
has centroid 
. A line parallel to passing through intersects the circumcircle of at 
. Let lines and intersect at . Suppose a point is chosen on such that the tangent of the circumcircle of 
at , the tangent of the circumcircle of at 
and concur. Prove that 
.
be the number of positive integers that divide the integer . For all positive integral divisors 
of 
, what is the sum of numbers 
?
.
.