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.
find all integers 
such that
, if the line connecting a point 
on a side of 
and its corresponding vertex bisects the perimeter, we write 
.
, let 
be points on sides 
, respectively, such that 
, 
, 
, 
, and 
, 
, 
.
.
be positive real numbers, and denote 
. Prove that
of positive integers, such that 
, and for 
,![$$ a_{n+1}=\sum_{i=2}^{n+1} \lfloor \sqrt[i]{a_n} \rfloor. $$](http://latex.artofproblemsolving.com/e/4/7/e47c377c88129c1dde6ed238ae59593f3026663b.png)
Show that for any prime 
, there are infinitely many terms in that are divisible by .
. Find the value of
where 
is a permutation of 
, and the summation runs over all such permutations.
be a set with (
) elements. Find all possible integers such that there exists a family 
of three-element subsets of , satisfying the following conditions:
, 
, exactly one element of contains 
.
, if 
, then 
.
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
be a scalene triangle with circumcircle 
and orthocenter 
, and let 
intersects at 
, respectively. The line passing through 
and parallel to 
intersects at 
. Define 
similarly. Let 
intersects 
at 
, 
intersects 
at 
, and 
intersects 
at 
. Prove that 
.
on vertices, where . Each edge is colored with one of three colors, and each color is used on at least one edge. Find the minimum positive integer 
such that for any such edge coloring and any color 
chosen from the three colors, it is possible to recolor at most edges to color so that the subgraph consisting of all edges of color is connected.
is a triangle, where the angle bisectors of 
and 
intersect at point . Let points 
and 
lie on sides 
and 
, respectively, such that 
passes through point and 
. If the areas of triangles 
, 
and 
are equal to 
, 
and 
, respectively, then what is the area, in 
, of ?
and using the integer 
to arrive at a contradiction.
for any real 
whose midpoint is 
and their connecting segment is represented by the line:
;
in the point ![$P\left(\frac{1/4-2ab-2m}{m-2(a+b)}, \frac{[-1 + 16b + 8a(2 + b)] m}{8 a + 8 b - 4 m}\right)$](http://latex.artofproblemsolving.com/a/7/d/a7d49a81d6add84b9aae1e2a212a6d398ae5f164.png)
.
are not symmetric with respect to , then we require 
or 
. We check these conditions via the abscissae of 
:
;
.
that satisfies:![$\textcolor{red}{m \in \left[\frac{1/4+2b^2}{2+b}, \frac{1/2+2(a^2+b^2)}{4+a+b}\right) \cup \left(\frac{1/2+2(a^2+b^2)}{4+a+b}, \frac{1/4+2a^2}{2+a}\right] }$](http://latex.artofproblemsolving.com/5/1/2/512a32311d00f7860829ddec9d6538bada922b23.png)
.
prove by Cauchy Inequality?
and 
.
.
so that any two different points in the graph of the quadratic function 
are not symmetrical to the straight line 