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.
that satisfies![\[
a^3+b^3=1957
\]](http://latex.artofproblemsolving.com/5/d/8/5d8d2a40734ec6b0a9149f9a4ae3a5828d5055f3.png)
![\[
(a+b)(a+1)(b+1)=2014
\]](http://latex.artofproblemsolving.com/3/0/0/300f2d16e669a1d8d9c54b357a6eaf7a8eb5d826.png)
Hence, find the value of 
and n a positive integer 
, Show that : 
be a primes ,
be positive integers such that 
and 
.
be an acute triangle with incentre 
and 
. Let lines 
and 
intersect the circumcircle of at 
and 
, respectively. Consider points 
and 
such that 
and 
are parallelograms (with 
, and 
). Let 
be the point of intersection of lines 
and 
. Prove that points 
, and are concyclic.
, let 
be all positive integers smaller than that are coprime to . Find all 
such that 
for all 
is the largest positive integer that divides both 
and 
. Integers and are coprime if 
., 
, 
, and 
is the point such that 
and 
, where and 
lie on the same side of 
. Let 
be the point on such that 
, and let 
be the midpoint of 
. Prove that the line through parallel to 
passes through the midpoint of .
.
, there exist indices 
such that 
is divisible by .
, find the minimum possible value of 
.
be an acute triangle with the 
mixtilinear incircles are 
respectively. 
is tangent to the circumcircle of at 
. 
are the centers of circles 
respectively. Suppose the reflection of line 
over 
intersects the reflection of line 
over 
at point 
Prove that: 
, no two of which are 
. Prove that:
and 
such that
be a sequence of positive real numbers such that for every 
, we have:![\[
a_n = \max_{1 \leq i \leq 2025} a_{n-i} - \min_{1 \leq i \leq 2025} a_{n-i}
\]](http://latex.artofproblemsolving.com/e/3/0/e30e9e533eb9a41cf847484e676ac4522edda665.png)
Prove that there exists a natural number 
such that for all 
, the following holds:![\[
a_n < \frac{1}{1404}
\]](http://latex.artofproblemsolving.com/0/1/f/01f1a81eed2230332d51a15501ef2a6ad3a7ec82.png)
such that 
and 
, where and are constant. How can we construct a circle perfectly tangent to the two lines and the 
-axis in Desmos?
where 
and 
Prove that
.
