AoPS Academy
, let the incircle 
touch sides 
at 
, respectively. Let 
lie on the line through 
perpendicular to 
. Let 
be the intersections of 
with 
, respectively. Let 
be the projections of 
onto line . Let 
be the second intersections of 
with the incircle . Let 
be the intersection of 
and 
. Prove that the line 
bisects segment 
.
be a triangle with circumcircle 
, 
-angle bisector 
, and -median 
. Suppose that meets 
at and meets again at 
. A line 
parallel to meets , at 
, 
respectively, so that is between and . The circle with diameter 
meets again at 
. varies, show that 
is constant.
. 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.
grid consisting of 2025 unit squares. Within each unit square is a car, facing either up, down, left, or right. If the square in front of a car in the direction it is facing is empty, it can choose to move forward. Each car wishes to exit the grid.
cars from the grid in such a way that it becomes possible for all the remaining cars to eventually exit the grid. that guarantees that Kritesh's job is possible?
be defined by![\[
a_1 = 1, \quad a_{n+1} = a_n + \frac{1}{\sqrt[2024]{a_n}} \quad \text{for } n \geq 1, \, n \in \mathbb{N}
\]](http://latex.artofproblemsolving.com/8/8/f/88f423d87c08b20dc552703fcf30f2a8e9585902.png)
Prove that![\[
a_n^{2025} >n^{2024}
\]](http://latex.artofproblemsolving.com/a/6/8/a6806c817edb7ce29a399981137c1a3e93f63c63.png)
for all positive integers 
.
Find the range of values that can be taken by :
are positive reals.
, prove that![\[ a = b\cos C + c\cos B. \]](http://latex.artofproblemsolving.com/7/e/d/7ed699f18ede191e5b89bc79d7f0592294347818.png)
be an integer. In a configuration of an 
board, each of the 
cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate 
counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of 
, the maximum number of good cells over all possible starting configurations.
such that
for all reals 
satisfying 
at 
cut each others at 
. Then I found that 
and 
are concurrent.
-antipode of 
(below marked as ).
be a triangle with circumcentre . 
are the midpoints of 
. 
in 
such that 
. 
intersect 
at 
. Prove that 
.
intersect 
at 
is . is a point such that 
, 
intersects at 
.
. So 
be a triangle with circumcentre . are the midpoints of . in such that . intersect at . Prove that .
be a triangle with circumcentre . are the midpoints of . in such that . intersect at . Prove that . are the midpoint of 
intersects 
at we have:
are the reflection of about 
. so 
intersects 
at 
but 
is the radical axis of 2 circles. So we have the new problem
and 
. Prove that
Where 
and 
Prove that
Where 
be a triangle with circumcenter 
.
are midpoints of 
.
intersects 
at 
.
be the circumcenter of 
and 
.
and tangent line of 
at 
are concurrent
![[asy]
unitsize(80);
pair A,B,C,O,X,Y,K,E,F,N;
A=dir(110); B=dir(215); C=dir(-35); O = circumcenter(A,B,C); X = midpoint(A--B); Y = midpoint(A--C); K = extension(A,O,B,C); F = circumcenter(A,X,K); E = circumcenter(A,Y,K); N = extension(Y,F,X,E);
draw(A--B--C--cycle);
draw(X--K); draw(A--K);
draw(A--K); draw(K--Y);
draw(circumcircle(A,X,K), red);
draw(circumcircle(A,Y,K), red);
draw(circumcircle(A,B,C), red);
draw(circumcircle(A,N,K), red + linetype("4 4"));
draw(Y--N); draw(A--N);
draw(N--C); draw(E--N);
dot("$O$", O, NW * dir(10));
dot("$A$", A, N * dir(40));
dot("$B$", B, SW);
dot("$C$", C, SE);
dot("$X$", X, NW);
dot("$Y$", Y, E * dir(-40));
dot("$K$", K, SE);
dot("$E$", E, E);
dot("$F$", F, NW);
dot("$N$", N, NE);
[/asy]](http://latex.artofproblemsolving.com/c/5/f/c5fb0af000d91270f76df5ee9193c39a6ae2af4f.png)