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.
be an acute triangle. Points 
, and 
lie on a line in this order and satisfy 
. Let 
and 
be the midpoints of 
and 
, respectively. Suppose triangle 
is acute, and let 
be its orthocentre. Points 
and 
lie on lines 
and 
, respectively, such that 
and are concyclic and pairwise different, and 
and are concyclic and pairwise different. Prove that 
and are concyclic.
+2 w
be reals such that 
and 
. Prove that
, let 
be all the positive integers smaller than that are coprime to . Find all 
such that
for all 
.
+1 w
be a quadrilateral inscribed in a circle 
Let the tangent to 
at 
meet rays 
and 
at 
and 
respectively. A point 
is chosen inside 
so that 
and 
Let 
be a point on segment 
satisfying 
Prove that lines 
and 
are concurrent.
has 2 roots 
and equation 
has 2 roots called 
. Prove that 
be an acute triangle with incentre 
and 
. Let lines 
and 
intersect the circumcircle of at 
and 
, respectively. Consider points 
and 
such that 
and 
are parallelograms (with 
, and 
). Let be the point of intersection of lines 
and 
. Prove that points 
, and are concyclic.
+1 w
, find all possible positive integer values of![\[
\frac{a}{b} + \frac{b}{c} + \frac{c}{a}.
\]](http://latex.artofproblemsolving.com/3/2/b/32b41635c159921e11245369e66185ec69b0d785.png)
be reals such that 
and . Prove that
Let be reals such that 
and . Prove that
be the side lengths of a triangle. Prove that![\[\frac{ab+bc+ca}{(a+b)^2} + \frac{ab+bc+ca}{(b+c)^2} + \frac{ab+bc+ca}{(c+a)^2} \leqslant \frac{85}{36}.\]](http://latex.artofproblemsolving.com/f/8/0/f80aed170f4f4bd78a2e881ce93b6b7ddfa03c22.png)
be a triangle with its orthocenter. The circle with diameter ![$[AC]$](http://latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
cuts the circumcircle of triangle 
at 
. Prove that the point of intersection of the lines 
and 
is the midpoint of the segment ![$[BH]$](http://latex.artofproblemsolving.com/a/9/f/a9fd89d48264ff1c3f9b77f62ef66f916978d99d.png)
of positive integers is called central if for every positive integer 
, the arithmetic mean of the first 
terms of the sequence is equal to .
, 
, 
, 
of positive integers such that for every central sequence 
, 
, 
, , there are infinitely many positive integers with 
.
Let 
and 
implying 
or 
implying 
Thus 
and 
.
be a function satisfying![\[
f(f(x)) = x^2 - 6x + 6
\quad \text{for all} \quad x \in \mathbb{Z}.
\]](http://latex.artofproblemsolving.com/8/a/e/8aebed1e260316d675ca7d104d83fabcd8b5717e.png)
Given that![\[
f(i) < f(i+1) \quad \text{for} \quad i = 0, 1, 2, 3, 4, 5,
\]](http://latex.artofproblemsolving.com/9/f/9/9f92574f9fe14e2e30c3e51e194c661fbd4ece07.png)
find the value of![\[
f(0) + f(1) + f(2) + \cdots + f(6).
\]](http://latex.artofproblemsolving.com/4/0/7/4075d7efa78dda4c46a5cf764fb8852e8c0fb09b.png)
and 
Prove that
with 
lie in this order on a line. Let 
be a square. The circle with diameter 
intersects the line 
at points 
and 
intersect at 
.
clearly 
is a tangent to the circle with diameter 
.
and 
congruent since they're already both right-angled and 
.
,
,
,
,
and 
are congruent,so 
for some 
.
and finally
QED