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.
be a triangle with centroid 
. Points 
and 
are chosen on rays 
and 
, respectively, such that![\[ \angle ABS=\angle ACR=180^\circ-\angle BGC.\]](http://latex.artofproblemsolving.com/0/a/d/0ad01e29194358ea0d3e10cb74b0769d1bd3a656.png)
Prove that 
.
and 
for which![\[ \left \lfloor \frac{a^2}{b} \right \rfloor + \left \lfloor \frac{b^2}{a} \right \rfloor = \left \lfloor \frac{a^2 + b^2}{ab} \right \rfloor + ab.\]](http://latex.artofproblemsolving.com/c/3/6/c3682397d21443cb720e56251c260b984392a84a.png)
be a triangle with 
, incenter 
, circumcenter 
, and circumcircle 
. The line 
meets 
at 
, and meets again at 
. Let 
be the reflection of over 
. Let the line through perpendicular to meet at 
. 
meets 
and 
at 
and 
, respectively. Suppose that the circumcircle of 
meets again at 
. Prove that 
, , 
, are concurrent if and only if the sides of 
form an arithmetic progression.
= 
for integers,
= 
= = 
, as 
the 
which gets us the 
form therefore L'hopital came to my mind , which might be a completely wrong intuition, anyway what should i do to find that limit
such that 
for all 
.
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
, where![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](http://latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
with 
right and 
, 
. On the ray 
we take two points and 
so that 
and 
. Consider the squares 
and 
containing inside 
and 
, respectively. If 
and 
, 
and 
, prove that 
and that 
, and are collinear.
inside a square 
we raise a segment 
perpendicular to the plane of the square. Show that the projections of on the planes 
, 
, 
and 
are coplanar if and only if ![$O\in [AC]\cup [BD]$](http://latex.artofproblemsolving.com/6/6/6/666df8f2afa87f15c79f9c96d9d1cea17a2a7af0.png)
.
. Proof that![\[
(2^x - 1)(5^x - 1) = y^n
\]](http://latex.artofproblemsolving.com/f/8/f/f8f013a96863bc5f39aa23edb98eed9aec54a17f.png)
have no positive integer solution 
.
be defined by 
(a) Show that 
has exactly one point of discontinuity. at its point of discontinuity.
given by 
Show that 
exists everywhere and find the direction along which the absolute value of the directional derivative is maximized at the point 
given by Show that exists everywhere and find the direction along which the absolute value of the directional derivative is maximized at the point 
. Since 
are each defined for all 
exists everywhere. The direction derivative 
at the point 
computes to: 
. Thus, 
, which is maximized for 
(i.e. along the forward and the reverse directions of the gradient).
in coordinates, then along any direction given by the unit vector 
the directional derivative of 
is 