AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
such that the sum of their cubes is equal to the square of their sum.
. Let the perpendicular line from 
to 
meets 
at points 
, respectively, and the foot from 
to 
is 
. 
intersects line 
at 
, 
intersects line at 
, and lines 
intersect at 
.
is tangent to 
.
, 
, 
is the point on such that 
. Let 
be the circle through 
tangent to 
at , and 
the circle through tangent to at . Let 
be the second intersection of and . Prove that 
lies on .
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., let 
be the excircle opposite to . Let 
and be the points where is tangent to 
, and , respectively. The circle 
intersects line at 
and 
. Let 
be the midpoint of . Prove that the circle 
is tangent to .
positive real numbers such that 
. Prove that :
such that for all 
,![\[
f(x + f(2y)) + f(x^2 - y) = f(f(x)) f(x + 1) + 2y - f(y).
\]](http://latex.artofproblemsolving.com/d/1/0/d10c01d7d416243614b24f04218d157e8b025ff2.png)
be an acute triangle. Points 
, and lie on a line in this order and satisfy 
. Let and be the midpoints of and 
, respectively. Suppose triangle 
is acute, and let 
be its orthocentre. Points and lie on lines 
and 
, respectively, such that 
and are concyclic and pairwise different, and 
and are concyclic and pairwise different. Prove that 
and are concyclic.
pairwise distinct complex numbers are written on a board. It turns out that there are exactly 760 ways to choose two numbers 
and 
from the board such that:![\[
a^2 + b^2 + 1 = 2ab
\]](http://latex.artofproblemsolving.com/8/f/c/8fc5df8dbc8e1647edfaa96dcfa8a0dfdac0c961.png)
Ways that differ by the order of selection are considered the same. Prove that there exist two numbers 
and 
from the board such that:![\[
c^2 + d^2 + 2025 = 2cd
\]](http://latex.artofproblemsolving.com/2/7/0/2706d72797c5e392b3f068a8174bd257e4d93fa1.png)
, 
are even?
be 
points on the plane, no three collinear and 
in which by the angle 
we mean the one which is less than 
(assume that 
and 
). Prove that some of these angles will add up to 
.
be points on the plane, no three collinear and in which by the angle we mean the one which is less than (assume that and ). Prove that some of these angles will add up to .
be points on the plane, no three collinear and in which by the angle we mean the one which is less than (assume that and ). Prove that some of these angles will add up to .
![$A_n=\left[n\right]^3{,}f\left(n\right)=\min \left|S\right|{,}S\ satisfy\ S\subseteq \ A_nand\ \forall \ x\in \ A_n{,}\exists \ y{,}z\in \ S\ {,}x{,}y{,}zcollinear$](http://latex.artofproblemsolving.com/a/a/6/aa671ae30c20a584500d6362790124ad6cbb67be.png)
and 
Let 
be the angle 
which is less than 
,
.
,
in clockwise or counterclockwise direction. After walking one round and returning back to the edge 
we have turned by a multiple of 
in total. Therefore, the signed sum of turning angles is a multiple of 
More formally, if we define 
and 
for the set of clockwise and counterclockwise angles, then for some integer
we have 
But the total number of angles we have is 
as much as possible from the above expression, we can conclude that 
On the other hand, by the assumption of the problem, the sum 
is equal to 
Therefore, the two sums should be 
