AoPS Academy
be a positive integer. For a set of points in the plane 
, we call 
distinct points 
connected if the line 
contains exactly 
points from .
such that there exists a set of points in the plane with the property that any point 
is connected with exactly 
other points from .
be a point in the plane, distinct from the vertices of 
. Consider 
the reflections of with respect to lines 
and 
, in this order.
are collinear if and only if lies on the circumcircle of . does not lie on the circumcircle of and the centroids of triangles and 
coincide, prove that is equilateral.
then prove \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{2}\cdot\frac{abc}{ab+bc+ca}\ge \frac{5}{3}.$$
and 
Prove that
Let and 
Prove that
Let and 
Prove that
Let and 
Prove that
is the incentre and point 
is defined such that 
and 
is the perpendicular bisector of 
let 
and 
be the circles passing through 
and 
respectively such that 
is tangent to and define 
to be a point such that it lies on both the circles and prove that 
and 
are perpendicular.
weights, each weighing 
and another 
weights with 
each. Write a algorithm that determines the ways in which a scale can be balanced with a weight 
on the left pan, and display the number of possible solutions. (The weights can be placed on both pans and the program starts with the numbers 
. What will be displayed after three successive runs: 5,2,5,1,4 | 5,2,5,1,11 | 5,2,5,1,20?
with orthocenter 
. Extend 
to meet at 
. The angle bisector of 
meets the midpoint of 
. If 
, compute the area of 
.
with side length of 
. The point 
is selected on the diagonal 
such that 
. Denote the circumcenters of 
as 
. Find the length of 
be a triangle with 
and let 
be the midpoint of . 
( incenter) intersects at and 
intersects the circumcircle of at 
. Prove that 
be a triangle with 
and let be the midpoint of . ( incenter) intersects at and intersects the circumcircle of at . Prove that 
be a triangle with and let be the midpoint of . ( incenter) intersects at and intersects the circumcircle of at . Prove that be a triangle with and let be the midpoint of . ( incenter) intersects at and intersects the circumcircle of at . Prove that 
be a triangle with and let be the midpoint of . ( incenter) intersects at and intersects the circumcircle of at . Prove that be a triangle with and let be the midpoint of . ( incenter) intersects at and intersects the circumcircle of at . Prove that on the segment 
we need to take 
otherwise will be on the half-line 
but beyond 
but that doesn't change the result
be a triangle with and let be the midpoint of . ( incenter) intersects at and intersects the circumcircle of at . Prove that be a triangle with and let be the midpoint of . ( incenter) intersects at and intersects the circumcircle of at . Prove that
be a triangle with and let be the midpoint of . ( incenter) intersects at and intersects the circumcircle of at . Prove that be a triangle with and let be the midpoint of . ( incenter) intersects at and intersects the circumcircle of at . Prove that
:
we have 
but from Menelaus we get 
so 
but 
so 
be the 
-excenter of , 
be the reflection of over , and be the midpoint of 
. Note that 
is a parallelogram, so it suffices to show that 
, or 
.![[asy]
size(9cm);
defaultpen(fontsize(10pt));
pair A=(-43.2,51.3), B=(-60,-30), C=(60,-30), M=(0,-30), D=(-48,27.9), E=(-65.7,13.6), F=(-52,8.8), I=(-24.2,-0.8), O=(0,0), X=(24.2,-59.2), Y=(-107.2,27.9), P=(4,-10.6);
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
dot(F);
dot(I);
dot(O);
dot(P);
dot(M);
dot(X);
dot(Y);
draw(circumcircle(A,B,C), linewidth(0.5));
draw(A--B--C--cycle, linewidth(0.5));
draw(M--D, linewidth(0.5));
label("$A$", A, (-0.5,1));
label("$B$", B, SW);
label("$C$", C, SE);
label("$M$", M, (-0.5,-1));
label("$E$", E, (-1,-0.5));
draw(C--E, linewidth(0.5));
draw(E--Y, linewidth(0.4)+dashed);
draw(B--I, linewidth(0.5));
draw(D--E, linewidth(0.5));
draw(I--X--C, linewidth(0.4)+dashed);
label("$D$", D, (-0.6,0.6));
label("$I$", I, (0,-1.5));
label("$I'$", X, (-1,-0.5));
draw(circumcircle(A,I,B), linewidth(0.7)+dashed);
draw(Y--D, linewidth(0.4)+dashed);
label("$I_C$", Y, (-1,0));
label("$F$", F, SW);
label("$P$", P, (0.5,-1.5));
draw(M--P, linewidth(0.4)+dashed);
[/asy]](http://latex.artofproblemsolving.com/c/5/a/c5a08fcaa75a85e525e1f24ee22d18db797a7d29.png)
; it follows by a well-known property of harmonic bundles that
Using this in conjunction with the fact that 
and 
yields that 
, which immediately implies the desired conclusion.
and 
Prove that
such that ![\[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]](http://latex.artofproblemsolving.com/0/6/2/0623b7c328af6d088cfc5dd288afe3e17a94ff29.png)
for all reals 
and 
.
and the triangles 
are similar.
come from?
is the main struggle, though it's easy to prove that fact using barycentric coordinates:
,
.
.
,