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.
is call 
if it satisfies the following properties:
and 
for all indices 
.
.
for which: Every sequence, there always exist two terms whose diffence is not less than 
. (where is given positive integer)
be a triangle with incentre 
and circumcentre 
. Let 
be the touchpoints of the incircle with 
, 
, 
respectively. Prove that 
is the Euler line of 
.
of integers that has the following property:
(possibly 
) then 
for every integer .
of nonzero integers such that the only admissible set containing both 
and is the set of all integers.
satisfies 
, and its incenter is . 
is the foot from to 
，and 
is a point on segment such that 
. Line 
meets 
at point 
, and the orthocenter of 
is 
.
is tangent to 
.
be a non-cyclic convex quadrilateral. The feet of perpendiculars from to 
are 
respectively, where 
lie on segments 
and 
lies on 
extended. The feet of perpendiculars from to 
are 
respectively, where 
lie on segments 
and 
lies on 
extended. Let the orthocenter of 
be . Prove that the common chord of circumcircles of 
and 
bisects 
.
![\[
f : \mathbb{R}^+ \to \mathbb{R}
\]](http://latex.artofproblemsolving.com/e/7/b/e7bfc5b236fb6f00ef328f850b1b14632cbf8416.png)
satisfying the condition that for all 
,![\[
f(x^2 + y) \geq f(x) + y.
\]](http://latex.artofproblemsolving.com/c/7/c/c7cd7c9a4d4372afc23e4503bb1c05fc7d07e788.png)
be an acute-angled triangle with 
and let be the foot of the-angle bisector on . The reflections of lines and in line meet and at points and 
respectively. A line through meets and at 
and respectively such that and 
while lies strictly between 
and . Prove that the circumcircles of
and 
are tangent to each other.
of natural numbers such that
be reals such that 
. Prove that
Let be reals such that 
. Prove that
and 
intersect at exactly one point. Prove that for all real numbers 
, the inequality 
is always valid.
and intersect at exactly one point. Prove that for all real numbers , the inequality is always valid.
and 
.
such that: 
.
and 
.
