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 circumcircle. Let 
, 
are lie on the sides 
, 
respectively, such that 
. 
and 
is incenter of the triangle 
, 
is K-excenter of the triangle 
(opposite to 
and tangents to 
). If 
is midpoint of the arc of then prove that 
.
be a sequence of positive real numbers, and 
be a positive integer, such that![\[a_n = \max \{ a_k + a_{n-k} \mid 1 \leq k \leq n-1 \} \ \textrm{ for all } \ n > s.\]](http://latex.artofproblemsolving.com/f/1/2/f12c9c5e0f154549118a33a2f359329333f432a7.png)
and , such that![\[a_n = a_{\ell} + a_{n - \ell} \ \textrm{ for all } \ n \geq N.\]](http://latex.artofproblemsolving.com/b/0/b/b0b2bd4507621302eeee27ae50ed9c0afd72c6ef.png)
and 
be two triangles inscribed in the same circle, centered at 
, and sharing the same orthocenter 
. The Simson lines of the points 
with respect to triangle form a non-degenerate triangle 
. lies on the circle with diameter 
.
lie in this order on the circle and form a convex, non-degenerate hexagon.
+2 w
, the player whose turn is chooses a prime divisor 
of and writes the numbers 
. In the board, is written at the start number 
and Anna plays first. The game is won by whom who shall be first able to write a number bigger or equal to 
.
be the circumcircle of 
and let 
be A-excircle .Let the two common tangents of 
cut in 
.Prove that 
.
satisfying the following condition: 
, centered at 
, is internally tangent to circle 
, centered at 
, at 
. Let and 
be variable points on and , respectively, such that line 
is tangent to (at ). Determine the locus of 
-- the circumcenter of triangle 
.
such that has a multiple which is alternating.
and be the incircle and circumcircle of the acute triangle , respectively. Draw a square 
so that all of its sides are tangent to , and , are both on . Extend 
and 
, intersecting at and , respectively. Prove that 
and 
intersects on .
and 
be touch points of the incenter of at 
and , respectively. Let and be the circumcenter of triangles 
and 
, respectively. Show that 
and 
concurrent.
are such that 
, find the minimum value of expression 
be a sequence of strictly positive real numbers. For each nonzero positive integer 
, define![\[
s_n = a_1 + a_2 + \cdots + a_n \quad \text{and} \quad
\sigma_n = \frac{a_1}{1 + a_1} + \frac{a_2}{1 + a_2} + \cdots + \frac{a_n}{1 + a_n}.
\]](http://latex.artofproblemsolving.com/1/9/1/191a2ebd41e3d236c90e9022ed65323461083815.png)
Show that if the sequence 
is unbounded, then the sequence 
is also unbounded.
be positive real numbers such that 
. Find the minimum value of the following sum :
knowing that the denominators are positive real numbers.
![\[
f : \mathbb{R}^+ \to \mathbb{R}
\]](http://latex.artofproblemsolving.com/e/7/b/e7bfc5b236fb6f00ef328f850b1b14632cbf8416.png)
satisfying the condition that for all 
,![\[
f(x + y^2) \geq f(x) + y.
\]](http://latex.artofproblemsolving.com/3/a/a/3aa20083835682dbd81be692eba65cab19e923e5.png)
is the set of nonnegative integers
is not always true. Check 
.
and 
?
and 
.
for all positive integers 
,![\[f(x+1)\ge f(x)+N\]](http://latex.artofproblemsolving.com/e/6/d/e6d7a6bcac7cb561489adbf005fd4c784afac66d.png)
so 
is infinitely large, contradiction.