be a triangle with circumcircle 
,
-
,
.
at 
and meets 
.
parallel to 
,
respectively, so that 
meets 
.
is constant.
.
be the feet of the altitudes from 
and from 
to 
respectively.
and 
as the reflections of 
and 
be the circumcircle of 
.
with 
the intersection of ray 
with 
is the circumcenter of 
can be inscribed within a circle.
and 
. Prove that
Where 
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 
cars from the 
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 
.
and 
Prove that
Where 
Find the range of values that can be taken by :
are positive reals.
![\[ a = b\cos C + c\cos B. \]](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 
,
such that
for all reals 
satisfying 
is a connected graph with 
edges. Prove that it is possible to label the edges 
in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1.
belongs to at most one edge. The graph 
is connected if for each pair of distinct vertices 
there is some sequence of vertices 
such that each pair 
is joined by an edge of 
.
Y by Adventure10
Let 
be a square with sides length 
. Let 
be a path within which does not meet itself and which is composed of line segments 
with 
. Suppose that for every point 
on the boundary of there is a point of at a distance from no greater than 
. Prove that there are two points 
and 
of such that the distance between and is not greater than 
and the length of the part of which lies between and is not smaller than 
.
be a square with sides length 
. Let 
be a path within which does not meet itself and which is composed of line segments 
with 
. Suppose that for every point 
on the boundary of there is a point of at a distance from no greater than 
. Prove that there are two points 
and 
of such that the distance between and is not greater than 
and the length of the part of which lies between and is not smaller than 
.
is quite trivial in one direction, but if they ean the shortest distance, we just take points very close to the boundary and with angles close to 
degrees except in the corners, what is the mistake in those ideas?
.
to 
along 
from the current point. Obviously at any point the red points are a finite union of red segments (with possibly length zero). Suppose such points 
are opposite sides of squares, then we must completely paint 
.
,
and 
painted before 
