AoPS Academy
be an acute-angled scalene triangle with circumcircle 
and circumcenter 
. Suppose 
. Let 
be the orthocenter and 
be the incenter of triangle . Let 
be the midpoint of the arc 
of the circumcircle of triangle 
, containing .
be a point on the arc 
of not containing 
, such that 
. Let 
be the circumcenter of triangle 
. Prove that the lines 
and 
meet on .
be a positive integer. In an 
table, an upright path is a sequence of adjacent cells starting from the southwest corner to the northeast corner such that the next cell is either on the top or on the right of the previous cell. Find the smallest number of grids one needs to color in an table such that there exists only one possible upright path not containing any colored cells.
. Let 
and 
be the feet of the altitudes from 
to and from 
to 
respectively.
and 
as the reflections of across lines and , respectively. Let be the circumcircle of 
. Denote by 
the second intersection of line 
with , and by 
the intersection of ray 
with . is the circumcenter of , prove that , , and are collinear if and only if quadrilateral 
can be inscribed within a circle.
is given. Find all the pairs of points 
so that is on the sides of the triangle, 
is inside the triangle, and four non-intersecting segments from the set 
divide the triangle into four triangles with equal areas.
be on . There are two ways 
can be divided into four triangles.
![[asy]
draw((0,0)--(4,0)--(3,3)--(0,0));
draw((0,0)--(2.5,1.5)--(4,0));
draw((2,0)--(2.5,1.5)--(3,3));
label("A",(0.1,0.1),SW);
label("B",(3.97,0.1),SE);
label("C",(3,2.98),N);
label("X",(2,0.02),S);
label("Y",(2.6,1.5),NW);
[/asy]](http://latex.artofproblemsolving.com/e/e/2/ee29e0346632c61c8fb43e74babf99eb6b22e3e0.png)
![$[\bigtriangleup AXY]=[\bigtriangleup BXY]$](http://latex.artofproblemsolving.com/2/d/0/2d0c5a870a2b1beec80707fa41b6c5dd880d3c9a.png)
, 
and 
. Next, ![$[\bigtriangleup ACY]=[\bigtriangleup BCY]=\frac{1}{4}[\bigtriangleup ABC]$](http://latex.artofproblemsolving.com/b/e/8/be8c73dca226bf07c4e1100845ac02cd70cd1b33.png)
and ![$[\bigtriangleup ABY]=\frac{1}{2}[\bigtriangleup ABC]$](http://latex.artofproblemsolving.com/3/8/6/38632cba9ef0593a85c517d4468fc139509d0cf9.png)
, so 
. Considering the permutations, there are 3 total pairs of points in this case.
(WLOG 
)![[asy]
draw((0,0)--(4,0)--(3,3)--(0,0));
draw((0,0)--(2,1));
draw((2.7,0)--(2,1)--(3,3)--(2.7,0));
label("A",(0.1,0.1),SW);
label("B",(3.97,0.1),SE);
label("C",(3,2.98),N);
label("X",(2.7,0.02),S);
label("Y",(2.1,1),NW);
[/asy]](http://latex.artofproblemsolving.com/b/f/c/bfc34370dea0858e50e7c5c5b8ffb14d8bce2306.png)
![$[\bigtriangleup BCX]=\frac{1}{4}[\bigtriangleup ABC]$](http://latex.artofproblemsolving.com/c/1/4/c141eed2b015931625841d8d2dc78aa2d6c6456d.png)
, so 
and 
. Clearly must be the centroid of 
. Let the intersection of 
and meet at . Then 
and 
, so 
. There are 6 pairs of points in this case.
![\[
\text{Given that } 0 < \theta < 90^\circ,\ \text{solve the equation: } \sin(\theta - 60^\circ)\sin\theta + \sin(54^\circ - \theta)\sin 54^\circ = 0
\]](http://latex.artofproblemsolving.com/b/2/4/b24276c1fbddbcc97e50cef52db03ea9b264d7ee.png)
![\[
\text{What is the value of } \theta\ (\text{in degrees})\ \text{that satisfies the equation?}
\]](http://latex.artofproblemsolving.com/0/0/f/00fbcd5839fedced0e5636c1def5360d65fc0444.png)
mean? Is it the coordinates of the point Y in the space?!