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 positive integer. Dominoes are placed on a 
board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each , determine the largest number of dominoes that can be placed in this way.
or 
. Dominoes are placed on the board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. Two cells are said to be adjacent if they are different and share a common side.)
of positive real numbers that satisfy the system:![\[
\begin{aligned}
11bc - 36b - 15c &= abc \\
12ca - 10c - 28a &= abc \\
13ab - 21a - 6b &= abc.
\end{aligned}
\]](http://latex.artofproblemsolving.com/1/7/0/170e1cfaee90f5c37428fa75192ac889261a8f83.png)
be positive reals. Prove that there is a cyclic permutation 
of 
such that the inequality:![\[
\frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z}
\]](http://latex.artofproblemsolving.com/5/5/3/553527fc40a1b7d29dbdfb4b5e63635308f46c85.png)
and 
.
be a scalene triangle with orthocenter 
and circumcenter 
. Denote by 
, 
the midpoints of 
, 
. Suppose the circle 
with diameter meets the circumcircle of at 
, and meets line 
at a point 
. The tangent to at 
meets line 
at 
. Show that the circumcircles of 
and 
intersect at a point 
on 
.
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of square numbers. Prove that: 
is the sum of square numbers
be a triangle with circumcircle 
and incenter 
and let be the midpoint of . The points 
, 
, 
are selected on sides , 
, 
such that 
, 
, and 
. Suppose that the circumcircle of 
intersects at a point 
other than 
. Prove that lines 
and 
meet on .
, 
, 
, 
, 
, 
initially contains one coin. The following operations are allowed
, 
, remove one coin from and add two coins to 
;
, 
, remove one coin from and swap the contents (maybe empty) of the boxes 
and 
., , , , become empty, while box contains exactly 
coins. be a triangle, its orthocenter, its circumcenter, and 
its circumradius. Let be the reflection of the point across the line 
, let be the reflection of the point 
across the line 
, and let be the reflection of the point 
across the line 
. Prove that the points , and are collinear if and only if 
.
be a real number and let 
be arbitrary real numbers in 
. The sequences 
and 
are then defined recursively by
. Prove that 
for all sufficiently large .
is a prime number
is inscribed in circle 
. Let denote its incenter and 
its -excenter. Let denote the midpoint of arc 
. Line 
meets a second time at . The perpendicular to 
at meets sides 
and at and respectively. The circumcircle of 
meets 
a second time at , and the circumcircle of 
meets 
a second time at 
. Prove that 
passes through the antipodal to on .
be different in pairs, 
, 
, 
, 
. If 
then 
.
be different in pairs, 
. If 
then 
is an equilateral triangle.
be different in pairs, , , , . If then . be different in pairs, . If then is an equilateral triangle.
Prove that
