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 the property that ![\[f(x-f(y))=f(f(x))-f(y)-1\]](http://latex.artofproblemsolving.com/f/2/5/f25cc1e8ae0be1fd02b347fd94be4fab88af1d46.png)
holds for all 
.
be positive integers. Prove that it is impossible to have all of the three numbers 
to be perfect squares.
be a fixed triangle, and let 
, 
, 
be the midpoints of sides 
, 
, 
, respectively. Let 
be a variable point on the circumcircle. Let lines 
, 
, 
meet the circumcircle again at 
, 
, 
, respectively. Assume that the points 
, 
, 
, , , are distinct, and lines 
, 
, 
form a triangle. Prove that the area of this triangle does not depend on .
, and 
be positive real numbers, such that 
. Prove the inequality![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](http://latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
be an acute-angled triangle with circumcircle 
. A circle 
is internally tangent to at 
and also tangent to 
at 
. Let 
and 
intersect at 
and 
respectively. Let 
and 
be points on line such that 
is the midpoint of 
and 
is the midpoint of 
. Lines 
and 
meet at 
and intersect again at 
and 
respectively. The ray 
meets the circumcircle of triangle 
again at 
.
.
,
,. . . ,
is 
if for every index 
that holds 
, there exist at least 
terms of the serie such that they are all less or equal to .
, where ,,. . . , is an serie.
be a triangle with 
. The angle bisectors of 
and 
meet the sides 
and 
at 
and 
, respectively. Let 
be the incentre of triangle 
. Suppose that 
. Find all possible values of .
be a fixed positive integer. There are 
people, numbered 
, participating in a tennis tournament. For any two positive integers 
with 
, player has a higher skill level than player 
. Prior to the first round, the players are paired arbitrarily and each pair is assigned a unique court among courts, numbered 
.
, the winner of court moves to court 
and the loser of court stays on court ; however, the winner of court 1 stays on court 1 and the loser of court 1 moves to court . such that, regardless of the initial pairing, the players 
all change courts immediately after the th round. be positive reals. Prove that![\[ \frac{(a-b)(a-c)}{2a^2 + (b+c)^2} + \frac{(b-c)(b-a)}{2b^2 + (c+a)^2} + \frac{(c-a)(c-b)}{2c^2 + (a+b)^2} \geq 0. \]](http://latex.artofproblemsolving.com/6/9/3/6937791380a8642edca3e773dc72dbf3f0e0f206.png)
be points on the sides 
respectively of a triangle . Suppose that 
and 
meet at 
and meet at 
, and 
and meet at 
, such that 
, and 
. Compute the ratio of the area of 
to the area of 
.
and 
intersect at different points and . The straight lines tangents to that pass through and intersect at 
. Let be a point on that is out of . The 
line intersects at again, the 
line intersects again to in and the line intersects again to the circumference in 
. Prove that the 
line passes through the midpoint of the 
segment.
are positive real numbers such that 
, prove that ![\[ (1+a)(1+b)(1+c) \geq 8 (1-a)(1-b)(1-c) . \]](http://latex.artofproblemsolving.com/0/b/e/0bedd9ed8362ac5202845f8a4d0d24519423e9b4.png)
,such that 
P
.
intersects three of the lines 
,
,
,
,
