AoPS Academy
lie the following points: 
and 
on 
, 
on 
, and 
on 
. Let![\[
P = AM\cap BN,\quad
R = KM\cap LN,\quad
S = KN\cap LM,
\]](http://latex.artofproblemsolving.com/7/e/a/7ea90a0f1261ed7445bbac58fb0cdd76810259f3.png)
and let the line 
meet at 
. Prove that the points 
, , and 
are collinear.
be inscribed in the circle 
. A line through point 
intersects 
and 
at points 
and 
, respectively. Let 
be the reflection of across the midpoint of , and 
be the reflection of across the midpoint of . Prove that:
of triangle across line 
lies on the circle .
and 
coincide.
be a triangle and 
and 
be two points on 
, 
such that 
and let 
. Let the parallel through 
to the interior angle bisector of 
intersect 
in 
. Prove that 
.
be an isosceles triangle with 
, and let 
be its incenter. Incircle touches sides 
at 
, respectively. Foot of altitudes from 
to 
are 
, respectively. Rays 
intersect 
at 
, respectively. Prove that 
touches incircle at 
.
be an acute traingle with 
. Point 
is a foot of altitude through vertex 
. Point is a circumcenter of . Prove that areas of quadrilaterals 
and 
are equal.
. 
amount of people stand on coordinates 
where 
. Every person got a water cup and two people are considered to be neighbour if the distance between them is 
. At the first minute, the person standing on coordinates 
got litres of water, and the other 
people's water cup is empty. Every minute, two neighbouring people are chosen that does not have the same amount of water in their water cups, and they equalize the amount of water in their water cups. will not have more than 
litres of water.
such that ![$1+[\sqrt{2n}]~$](http://latex.artofproblemsolving.com/e/6/a/e6aa1bf5d761758196027add39f2e3cd77db1c1c.png)
divides 
.
, ![$[x]$](http://latex.artofproblemsolving.com/b/c/e/bceb7b14e55d33a8bca29b7863ad3cdae95afce4.png)
denotes the largest integer not exceeding . )
let denote the intersection of diagonals 
and 
, and let denote the intersection of diagonals 
and 
. Prove that if 
and 
, then bisects angle 
.x table such that the unit squares are colored arbitrary in black and white, such that exactly three of the squares placed in the corners of the table are white, and the other one is black. Prove that there exists a 
x square which contains an odd number of unit squares white colored.
be a scalene triangle with circumcenter , incenter , and incircle 
. Let touch the sides 
, 
, and 
at points , , and respectively. Let 
be the projection of to 
. The line 
intersects the circumcircle of again at point 
. The circumcircles of 
and 
intersect again at points 
and 
respectively. Prove that the lines 
, 
, and 
are concurrent.
for which the following statement holds: there exists at least one triple 
of integers such that
and all triples of real numbers, satisfying the equations, are such that 
are integers.
be a triangle with 
and 
. Let be the point on the line with 
. Let 
and 
be the centers of the circumcircles of the triangles 
and 
respectively. Show that the area of the square 
is equal to the area of the triangle .
be a triangle with and . Let be the point on the line with . Let and be the centers of the circumcircles of the triangles and respectively. Show that the area of the square is equal to the area of the triangle .
be a triangle with and . Let be the point on the line with . Let and be the centers of the circumcircles of the triangles and respectively. Show that the area of the square is equal to the area of the triangle .
be a triangle with and . Let be the point on the line with . Let and be the centers of the circumcircles of the triangles and respectively. Show that the area of the square is equal to the area of the triangle . be a triangle and 
be the Kiepert triangle corresponding to 
. The circle with center at 
and passing through 
and again cuts at 
, and at 
.
and 
be the centers of the circles (
) and (
), respectively. The points 
and 
are considered. Points 
and 
are defined cyclically. The six points 
lie on a conic, whose center is X(20303).![$$
\Sigma_{abc,xyz}(a^8-2 a^4 (3 b^4+2 b^2 c^2+3 c^4)+8 a^2 (b^6+c^6)-(b^2-c^2)^2 (3 b^4+2 b^2 c^2+3 c^4)) x^2+2 (a^8-(b^2-c^2)^4-2 a^6 (b^2+c^2)+2 a^2 (b^2-c^2)^2 (b^2+c^2)) y z] = 0.
$$](http://latex.artofproblemsolving.com/a/0/0/a005d35ae2abc0ed8d67b6d9c1b50f73faa299ac.png)
= 
.
satisfying
Prove
and 
,
.
,
.![$\therefore [ABC]=2[O_1O_2C]=[CO_1PO_2]$](http://latex.artofproblemsolving.com/8/9/8/8982b8638ecddbc2017ea4e72284ca1964b630b0.png)
.![$[X]$](http://latex.artofproblemsolving.com/f/6/3/f631f7a53d834de52b718ed6b496689e9658eadf.png)
means the area of 
.