AoPS Academy
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.
is called 
if there exists a permutation 
of the numbers 
, such that:
and 
have different parities for every 
;
the sum 
is a quadratic residue modulo for every 
.
th power.
are and 
, respectively. The excenter and exradius relative to vertex 
is 
and 
, respectively. Let 
denote the circumradius. Prove that if 
, then 
.
-gon, and at each vertex there is a black or a white cap. (Caps of the same colour are indistinguishable from each other.) Under one of the caps 
dollars are hidden, and there is nothing under the other caps. The host rotates the table, and then Ádám chooses a cap, and take what is underneath. Ádám's accomplice, Béla is working at the company behind this game. Béla is responsible for the placement of the dollars under the caps, however, the colors of the caps are chosen by a different collegaue. After placing the money under a cap, Béla
such that for all positive integers 
,![\[
f(mn) = \mathrm{lcm}(m, n) \cdot \gcd(f(m), f(n)),
\]](http://latex.artofproblemsolving.com/6/0/7/6071692e2716901e88cc1cfc19ed60388c10701d.png)
where 
and 
denote the least common multiple and the greatest common divisor of 
and 
, respectively.
be the midpoint of . It is well known that 
are cyclic, and call this circle 
. Note that is the center of . Let 
be the Miquel point of quadrilateral 
. (
) And rename 
as 
. Note that 
by cyclic, 
lies on . And by Miquel’s theorem, 
lies on 
, and we are done.
is cyclic, let its center be 
be the antipode of 
, and be the antipode of 
gives that 
lies on 
, so problem solved
is cyclic, let its center be be the antipode of , and be the antipode of gives that lies on , so problem solved
and 
be their respective pedal triangles. Let![\[ X_1=P_2Q_3\cap P_3Q_2,\quad X_2=P_1Q_3\cap P_3Q_1,\quad X_3=P_1Q_2\cap P_2Q_1 \]](http://latex.artofproblemsolving.com/7/f/e/7fe3ec4f69ec074635b2202daad9371e3bb8c72d.png)
Prove that the points 
,
and 
lie on the line 
are collinear is trivial by Pascal on 
.
by cyclic, 
be cyclic quadrilateral inscribed on circle 
be the Miquel point of quadrilateral 
,
,
are concurrent
,
,