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.
defined by:![\[
T_0 = 0,\quad T_1 = T_2 = 1,
\]](http://latex.artofproblemsolving.com/a/8/a/a8a2935b678607acac4180530e4b485673c3ced6.png)
and![\[
T_{n+3} = T_{n+2} + T_{n+1} + T_n \quad \text{for all } n \geq 0.
\]](http://latex.artofproblemsolving.com/8/b/0/8b044e593142ee6b2b75f28575c9b9eaa1ec7b4c.png)
Prove that, for every positive integer 
, we have:![\[
T_{3n} = a_0 T_0 + a_1 T_1 + a_2 T_2 + \cdots + a_{2n} T_{2n},
\]](http://latex.artofproblemsolving.com/f/b/c/fbcbfd6edb17f4c7d7e30d2f6abc75718183acd6.png)
where, for each 
, 
is the coefficient of 
in the expansion of the polynomial 
.
be a triangle such that 
. Let 
be the point on the side 
such that 
. The segment 
is extended to 
so that 
. Prove that ![\[ \angle ECB+180^{\circ }=2\angle EBC. \]](http://latex.artofproblemsolving.com/4/9/1/4919c773385bed5335f1cb0779e2cac0c782889d.png)
be triangle with point 
and 
on 
and 
are on the same side of 
be midpoint of segment and 
be midpoint of segment 
be intersection of 
with 
and 
such that, for all positive integers 
and 
, if divides 
, then divides 
.
be a triangle with incentre 
. The circle through 
tangent to 
at meets side 
again at 
. The circle through 
tangent to at meets side 
again at 
. Prove that 
is tangent to the incircle of 
is a prime number
, 
be the orthocenter of it and 
be any point on the side 
. The points 
are on the segments 
, respectively, such that the points 
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
. Prove that the points 
and lie on the same line. be a positive integer, and let be the set 
.
and 
be difined as:
Which is larger, 
or 
?
be a square with diagonals and 
, and a point in one of the sides of the square. Show that the sum of the distances from P to the diagonals is constant.
touch sides 
and 
at 
and 
respectively. Let 
and 
denote the circumcircles of triangle 
and 
respectively.
and 
intersect at 
and 
and 
intersect at 
and 
and intersect at 
and 
Prove that 
and intersect in a common point.
Show that 
and 
are concurrent.
be a trapezoid inscribed in circle 
, 
. Let is the symmetric point of across , 
intersects at 
. Let is midpoint of and 
is orthocenter of triangle . On take a point so that 
. Prove that: 
coincide
pairwise distinct complex numbers are written on a board. It turns out that there are exactly 760 ways to choose two numbers 
and 
from the board such that:![\[
a^2 + b^2 + 1 = 2ab
\]](http://latex.artofproblemsolving.com/8/f/c/8fc5df8dbc8e1647edfaa96dcfa8a0dfdac0c961.png)
Ways that differ by the order of selection are considered the same. Prove that there exist two numbers 
and 
from the board such that:![\[
c^2 + d^2 + 2025 = 2cd
\]](http://latex.artofproblemsolving.com/2/7/0/2706d72797c5e392b3f068a8174bd257e4d93fa1.png)
pairwise distinct complex numbers are written on a board. It turns out that there are exactly 760 ways to choose two numbers and from the board such that:Ways that differ by the order of selection are considered the same. Prove that there exist two numbers and from the board such that:
of nodes, one for each complex number and draw edges if the condition is held.
is form by exactly 
disconnected and simple path.
Condition is equivalent to 
meaning that can only be connected to 
and 
, this ensure that is a union of disconnected and simple path.
of these path, each of length 
. The we have that 
and 
, then 
.
nodes. So we have a complex number 
and 
and its easy to check that![\[
c^2+(c+45i)^2+2025 = 2c(c+45i)
\]](http://latex.artofproblemsolving.com/a/e/b/aeb34e72265ec8195327125c6953716ab047ab4b.png)
.
, such that for all 
:![\[ f(x+f(y))=\dfrac{f(x)}{1+f(xy)}\]](http://latex.artofproblemsolving.com/1/4/1/1418fad9fd00b31c38bfbc5657e2a322d66ef450.png)
