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 
odd, which satisfy the equation:
is the foot of the altitude from 
of triangle 
. On the lines 
and 
points 
and 
are marked such that the circumcircles of triangles 
and 
are tangent, call this circles 
and 
respectively. Tangent lines to circles and at and intersect at 
.
.
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
and 
be positive real numbers such that 
. Prove that
When equality holds?
satisfy the equality, 
prove that the triangle must have a right angle.
is a quadratic trinomial such that there exists 
real numbers that satisfies 
and 
. Prove that 
are the only numbers satisfying 
and 
with centroid 
, such that 
. Let 
and be the second points of intersection of lines 
and 
with the circumcircle of triangle , respectively. Prove that the line 
is tangent to the Euler circle of triangle .
questions, each with 
answer choices. A participant who answers all questions correctly will receive a gift. To ensure that at least one member of my family answers all questions correctly, how many family members need to take the quiz?
, each of our twin children also begins to have twins every year. If this pattern continues, how many years will it take for my family to grow large enough to have the required number of members to guarantee winning the quiz gift?
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.
are written on a board. and plays the following game: They take turns choosing a number from the board and deleting them. starts first. They sum all the deleted numbers. If after a player's turn (after he deletes a number on the board) the sum of the deleted numbers can't be expressed as difference of two perfect squares,then he loses, if not, then the game continues as usual. Which player got a winning strategy?
be an acute triangle with 
. be 
the circumcircle circumscribed to the triangle and the midpoint of the smallest arc of this circle. Let and points of the segments and respectively such that 
. Let 
be the second intersection point of the circumcircle circumscribed to 
with . Let and be the intersections of lines 
and 
with other than 
, respectively. Let 
and 
be the intersection points of lines 
and 
with lines and respectively. Show that the 
line passes through the midpoint of
in the 
-
plane, for the constants 
, draw the domain of the point 
such that the minimum value of 
is positive.
for real ![$x,y \in [-1,1]$](http://latex.artofproblemsolving.com/3/3/4/334fe866f1ee0278e7085e880adc7bc125024665.png)
such that 
yields the critical point:
(i);
(ii) (for 
);
(a maximum for all real ![$x,y \in [-1,1], a \neq 0$](http://latex.artofproblemsolving.com/7/2/f/72f5d56e3f3b8a798af49dc74fcd10b825a2e96f.png)
).
. If 
, then 
. Recall that we also require 
by (ii), which gives the required set:
for 
.
be six positive real numbers. Prove that
