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.
, your right-neighbor’s is 
, and the next person over is 
, then for every trio in the circle they see
be a scalene triangle with incircle 
.
be the circle tangent to and passing through 
and 
, and denote by 
, 
the second intersection points of and 
, 
, resp. Define 
, 
, 
, 
cyclically.
be the circle internally tangent to 
, 
, 
. is tangent to 
.
non-negative integers whose sum is 
. Prove that there exists 
of them whose sum is at least .
, 
, 
, 
, 
satisfy
Find the minimum value of 
.
inscribed in 
. Two points 
lie on 
such that 
are isogonal in 
. 
is the midpoint of 
. 
lies on 
such that 
. 
intersects at 
. Prove that the line through parallel to 
passes through the Euler center of triangle .
be a nonempty subset of the points in the Cartesian plane such that for each 
exactly one of 
or 
also belongs to . Prove that for each positive integer 
there is a line in the plane (possibly different lines for different ) which contains at least points of .
be a function such that for all positive rational numbers 
, the following equation holds:![\[
f\left(\frac{f(x)}{y f(x) + 1}\right) = \frac{x}{x f(y) + 1}.
\]](http://latex.artofproblemsolving.com/e/e/a/eea34789c7b2fa698ddaff29d95f4af0a71850b5.png)
and 
, those two edges are in different colors. Note that it is permissible for and 
to be the same color. The directed-edge-chromatic-number of a tournament is defined to be the minimum total number of colors that can be used in order to create a proper directed-edge-coloring. For each , determine the minimum directed-edge-chromatic-number over all tournaments on vertices.
Prove that
Let 
Prove that![$$ (a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$$](http://latex.artofproblemsolving.com/f/5/e/f5e68204507aa9c3e71c28844bbee3a804546b94.png)
be an acute triangle. Let be the altitude on , and let 
be any interior point on . Lines 
, when extended, intersect 
at 
respectively. Prove that 
.
, the point is the midpoint of and 
is a point on the side such that 
. The cotangents of the angles 
, 
, and 
are positive integers 
.
is also an integer and equals 
.
, some of which consisting of Fibonacci numbers.
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.
and a positive integer such that for every integer 
, the number 
is a perfect square.
is inscribed in a circle with 
and 
. Let the point be the midpoint of the arc 
not containing point 
, and 
is the intersection point of the diagonals and 
. Prove that lines 
and are perpendicular.
is inscribed in a circle with and . Let the point be the midpoint of the arc not containing point , and is the intersection point of the diagonals and . Prove that lines and are perpendicular.
is the circle through , , and 
.
be centre of the circle . Let line 
and circle intersect at 
. Thus, 
is a diameter of the circle .
be intersection point of 
and 
.
. is defined similarly as , so 
. Furthermore, we get that 
is a rhombus and thus 
and thus 
and thus 
and thus 
is a parallelogram and hence, we have shown that .![[asy]size(14cm);
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such that 
coprime, 
and 
.
