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 
is called historic if 
. Show that the set of all nonnegative integers can be written as the union of pairwise disjoint historic sets.
+1 w
, the numbers 
are written on a blackboard. The players take turns making moves with Ingrid starting. A move consists of one of the players crossing out a number on the board that has not yet been crossed out. If the product of all currently crossed out numbers is 
after the move, the player whose move it was receives one point, otherwise, zero points are awarded. The game ends after all numbers have been crossed out., determine which player (if any) has a winning strategy
be a positive integer. Given is a subset 
of 
with 
elements. Prove that there exist three elements 
from such that 
.
be an acute triangle and 
its orthocenter. Let 
be a point on the parallel through to 
such that 
. Define 
and 
as points on the parallels through 
to 
and through 
to 
similarly. If 
are positioned around the sides of as in the given configuration, prove that are collinear.
be positive real numbers satisfying![\[ xy + yz + zx = 3xyz. \]](http://latex.artofproblemsolving.com/9/2/0/9200874fd6bf9a32ff0c4e82382a450be68cc444.png)
Prove that![\[
\sqrt{\frac{x}{3y^2z^2 + xyz}} + \sqrt{\frac{y}{3x^2z^2 + xyz}} + \sqrt{\frac{z}{3x^2y^2 + xyz}} \le \frac{3}{2}.
\]](http://latex.artofproblemsolving.com/f/5/f/f5fb85d81e93935601ceb5f30c25aea7aa3f23b2.png)
and 
be real numbers. Let 
be the matrix with entries ![\[a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;\\0,&\text{if }x_i + y_j < 0.\end{cases}\]](http://latex.artofproblemsolving.com/c/0/8/c0856fe7b2b113180659952c4d216a01d8849f64.png)
Suppose that is an 
matrix with entries 
, 
such that the sum of the elements in each row and each column of is equal to the corresponding sum for the matrix . Prove that 
.
. Find .
are collinear, and 
are arbitrary points
, let 
be the sum of the squares of the digits of . For example, 
. Determine all integers 
such that 
.
with centroid 
and circumcircle 
centred at 
, the extension of 
meets at 
; lines and 
intersect at ; and lines and 
intersect at . Suppose the circumcentre 
of the triangle 
lies on and 
are collinear. Prove that 
.
, let 
and 
be points on sides and , respectively, such that 
is cyclic. Let lines 
and 
intersect at point , and and 
be the midpoints of 
and 
, respectively. If 
is the foot of the perpendicular from to , and the circumcircles of triangles 
and 
intersect at second point 
different from , prove that 
and are collinear.
, such that 
for all integers 
.
and 
w.r.t center of the circle, be 
and 
respectively. then by simple angle chasing , we can see that 
forms a parallelogram . since diagonals of a parallelogram bisect each other 
midpoints of 
and 
concur at one point. (say at X)
thus the ratio turns out to be equal to
from the quadrilateral
and an inscribed circle, centre .
; then![\[r= 2 \cdot \sqrt{\frac{(M-abc)(M-bcd)(M-cda)(M-dab)}{abcd (ab+cd)(ac+bd)(ad+bc)}}\]](http://latex.artofproblemsolving.com/1/6/0/160b9c73f3e12ea7933112084d5bd75b95fa9868.png)
![\[AO = 5, BO =6, CO = 7, DO = 8.\]](http://latex.artofproblemsolving.com/6/b/5/6b5cece87816d6ca54c50ccd5a3128269a0bbbd8.png)
,
.
