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.
+2 w
we are given :
and one side length of the triangle equals 
, then under these conditions find the maximum area of the triangle 
.
be a trapezoid with parallel sides 
and 
, where 
. A circle passing through 
and 
intersects 
again at 
respectively. Prove that 
are concurrent.
in it. Prove that there are an infinite number of perfect cubes in this progression. (A perfect cube is an integer of the form 
, where 
is an integer. For example, 
, 
and are perfect cubes.)
+1 w
there are elements 
in an arithmetic progression in the order 
such that 
.
days later (but perhaps earlier, as well). Hedvig received the trophy today. Determine all integers 
for which we can be absolutely certain that she cannot receive the trophy again in 
days, given the above information.
such that 
. Prove that 
are positive integers pairwise distinct and is also a positive integer 
.![$\sqrt[n]{p},\sqrt[n]{q},\sqrt[n]{r}$](http://latex.artofproblemsolving.com/3/1/3/3133b60c921168020d149a9d28ef69b756a704d9.png)
form terms of a same arithmetic progression.
and 
. Prove that
Let 
be reals such that 
. Prove that
real numbers 
, we can select a real number 
and transform the sequence into 
. This transformation can be performed multiple times, with each chosen real number potentially being different
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 four positive integers in arithmetic progression (that is, 
) and with 
. Can the product 
ever be a number of the form 
for some 
and some 
, with 
?
the sides and are parallel. Let 
be the midpoint of diagonal 
. Suppose that triangles 
and 
have equal area. Prove that 
.
Prove that
?
to 
.
be the midpoint of 
is a parallelogram so 
passes through 
is a parallelogram as well so 
.