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.
. Let 
be its incenter, 
be its 
-excenter, 
be midpoint of arc 
, 
is the midpoint of 
.
be the intersection of the line 
with 
, 
be the intersection of the angle bisector of 
with the line .
,the intersection of line 
with 
and 
,the intersection of 
and 
such that 
. Ana wins if the points 
are collinear. Who has a winning strategy?
+1 w
be an acute angled triangle with orthocenter 
and 
. Let 
be any other point on the arc 
of the circumcircle of 
and let line 
intersect line 
at 
and let line 
intersect line 
at 
. Let the circumcircles of 
and intersect again at 
(
). Let the lines 
intersect the circumcircle of 
again at 
(
). Prove that the lines 
concur.
be real numbers satisfying 
Prove that 
and determine all cases of equality.
with no obtuse angles.![$$\frac1{\sqrt3}\cdot\left(\cot A+\cot B+\cot C\right)+\left(2-\sqrt 3\right)\sqrt[3]{\cot A\cot B\cot C}\geq\frac2{\sqrt3}.$$](http://latex.artofproblemsolving.com/d/1/4/d14154281d7ccf60a2dcfe1d937b974ae1d74f97.png)
such that:
for any positive integer 
, the numbers 
are all distinct, and for any positive integers and 
, 
?
and let 
be positive integers such that 
. Prove that 
and determine when equality holds.
different ropes. She then performs a experiment that causes 
children, each connected to each other with the same type of rope, to be cured. Two experiments are said to be of the same type, if each of the ropes connecting the children has the same medicine imbued in it. She then unties them and lets them go back home.
be the minimum m such that Myla can perform at least 
experiments of the same type. Prove that:
For every 
there exists a 
and 
such that for all 
, ![\[f(n, k) = a_kn + b_k.\]](http://latex.artofproblemsolving.com/6/6/0/660699c6cec811eee25f727076665a6df3257f57.png)
Find the value of 
for every .
inscribed in a circle with center 
. 
is any point not on (O). 
intersect 
at 
. Let 
be the Lemoine points of triangle 
respectively. Prove that 
are collinear.
be a sequence of positive real numbers such that for every 
, we have:![\[
a_n = \max_{1 \leq i \leq 2025} a_{n-i} - \min_{1 \leq i \leq 2025} a_{n-i}
\]](http://latex.artofproblemsolving.com/e/3/0/e30e9e533eb9a41cf847484e676ac4522edda665.png)
Prove that there exists a natural number 
such that for all 
, the following holds:![\[
a_n < \frac{1}{1404}
\]](http://latex.artofproblemsolving.com/0/1/f/01f1a81eed2230332d51a15501ef2a6ad3a7ec82.png)
, points 
and lie on and respectively such that 
. Points 
and 
are the reflections of and with respect to the midpoints of sides and , respectively. Point lies on segment 
such that the circumcenter of triangle 
coincides with the circumcenter of triangle . passes through the midpoint of segment 
.
-th move, Jumpy swaps the positions of the two walnuts adjacent to walnut . such that, on the -th move, Jumpy swaps some walnuts 
and 
such that 
.
written on the first page and empty pages. Suppose in each of the next seconds, Alice can flip to the next page, and write down the sum or product of two numbers (possibly the same) which are already written in her grimoire.
be the largest possible number such that for any 
, Alice can write down the number on the last page of her grimoire. Prove that there exists a positive integer 
such that for all 
, we have that ![\[n^{0.99n}\leqslant F(n)\leqslant n^{1.01n}.\]](http://latex.artofproblemsolving.com/5/7/e/57e2b7e3d97564665d8de28a3afd74bdcf74ba5b.png)
, and its area is thus 
. The volume we are looking for must then be 
.
such that : 
such that : 
by the trihedron with vertex in the center of that ball, it its dihedral angles are 