AoPS Academy
be an acute-angled triangle with 
, let 
be its circumcentre, and let 
be a point on the segment 
. The line through perpendicular to intersects the lines 
and 
at 
and 
respectively. The circumcircles of triangles 
and intersect again at 
.
and 
then 
is tangent to the circle 
, let 
and 
be a pair of isogonal conjugate points. The line 
intersects 
at 
, and the line 
intersects at 
. Let the circumcircle of 
and the circumcircle of 
intersect again at 
(other than ). Prove that the line 
bisects 
.
is right angled (for clarity, 
). Define 
as a point on line 
such that 
is parallel to . Additionally, since 
, 
meaning 
, 
, and 
. Also, since 
, by SSS, we have 
meaning 
. Since 
, we have 
or 
and we are done.
from the reals to the reals such that![\[f\left(f(x)+y\right)=2x+f\left(f(y)-x\right)\]](http://latex.artofproblemsolving.com/e/b/4/eb4e0d6580d9d3d6a627e39837edb3d1943d4596.png)
.
is called 
if
is a perfect square, and
, 
is the smallest positive integer such that 
is a perfect square., there exists a positive integer 
such that 
for all integers 
. be a scalene triangle with incircle 
. Denote by 
the midpoint of arc in the circumcircle of , and by the point where the 
-excircle touches . Suppose the circumcircle of 
meets again at 
and intersects at two points 
, 
.
or 
is tangent to .
, with 
, for which the edges 
are
with 
, for which the edges 
are all coloured blue.
be a positive integer. Alice and Bob play the following game. Alice considers a permutation 
of the set ![$[n]=\{1,2, \dots, n\}$](http://latex.artofproblemsolving.com/8/f/e/8fea25e7cf72aabc60f48c3b30f4d4818a084ae3.png)
and keeps it hidden from Bob. In a move, Bob tells Alice a permutation 
of ![$[n]$](http://latex.artofproblemsolving.com/0/d/e/0deff9086fd7840ef23860f5bc924f1d4d2d2310.png)
, and Alice tells Bob whether there exists an ![$i \in [n]$](http://latex.artofproblemsolving.com/e/6/7/e67206e47d5d5875105b339f177a3ab8f5151a5e.png)
such that 
. Bob wins if he ever tells Alice the permutation . Prove that Bob can win the game in at most 
moves.
denote the set of all real numbers. Find all functions 
such that
for all real numbers 
, 
.
for which there exist infinitely many positive integers 
such that ![\[ \frac{a^m+a-1}{a^n+a^2-1} \]](http://latex.artofproblemsolving.com/2/8/a/28a68e409451bbdcaee7c1286341f6aeea9ffdad.png)
is itself an integer.
is an acute angle triangle such that 
and 
. Let's denote by the center of the circumscribed circle of the triangle and 
the intersection of altitudes of this triangle. Line 
intersects in point 
and in point 
. Find the value of the ration 
.
is an acute angle triangle such that and . Let's denote by the center of the circumscribed circle of the triangle and the intersection of altitudes of this triangle. Line intersects in point and in point . Find the value of the ration .
is equlilateral. Now the nine point centre lies on the bisector of 
. Because if 
are the middle points of 
and 
the nine point centre, the quadrilateral 
is concyclic 
,So 
are symmetric points about the bisector of 
, So 
(since 
is an altitude)
(since 
is inscribed in the circle with center )
(since 
is isoceles since 
)
be the perpendicular from to . In any triangle, 
to meet the circumcircle at , then draw 
. 
since it's inscribed in a semicircle, so 
, and since 
. Now we have 
, and 
. Thus, 
, so 
is a parallelogram, so 
. be the perpendicular from to . Then 
so 
is a 
right triangle, so 
. Thus, since 
and 
, 
so 
.
of the line 
(
is incenter) with the circumcircle 
of 
. Since 
obtain that the points , , belong to the circle 
and the quadrilateral 
is a rhombus, 
a.s.o.
then prove that 
.
.
and 
are congruent, and so 
.
and so 
is cyclic and so by simple angle chasing we get that 
and 
claims we get that 
