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.
) for all their time and support. We received immense help from TestX01 for the proof of claim 2.19 and it's associated lemma.
by 
, 
for every positive integer 
. Prove that 
leave distinct remainders when divided by 
.
such that 
Prove that 
. Find all 
such that for all 
be real numbers such that 
find the largest value of 
be a triangle with 
, and let 
be the foot of the altitude from 
. A point 
is chosen inside the triangle 
so that 
bisects 
. Let 
be the intersection point of the lines 
and . Let 
be the semicircle with diameter that meets the segment 
at an interior point. A line through is tangent to at 
. Prove that the lines 
and meet on .
![\[ x^n + 1 = y^{n+1}, \]](http://latex.artofproblemsolving.com/4/a/8/4a80f616d2a091187cdd1c57a5b6e1f09cdc9396.png)
where is a positive integer not smaller then 2, has no positive integer solutions in 
and 
for which and 
are relatively prime.
, the sides 
and 
are parallel and the angles 
and 
are acute. Show that it is possible to divide the triangle into 4 disjoint triangle 
and the triangle 
into 4 disjoint triangles 
such that the triangles 
and 
are congruent for all 
.
is a parallelogram, its sides being parallel to the diagonals of 
and equal to half of each, hence its area is:
, but the area of is 
, consequently 
( 
is the angle between the diagonals 
and 
).
Prove that 
be real numbers.Prove that
and 
are 
and 
,
,
.