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.
+1 w
with![\[2x^3zf(z)+yf(y) \ge 3yz^2f(x)\]](http://latex.artofproblemsolving.com/a/b/f/abf10d7e679f6f04db556424a0a0eda26f371fd4.png)
for all 
.
, 
, 
on 
, 
, 
be the touch points of the incircle of 
. Line 
intersects 
at 
, 
. The incircle of and 
intersect again at 
. If 
is the tangent point of the 
−mixtilinear incircle and , prove that , , are collinear.
be a function taking in positive integers and outputting nonnegative integers, defined as follows:
is the number of positive integers 
with 
such that the equation 
has an integer solution 
.
such that
and 
Adit Aggarwal, India.
such that the following inequality holds for all 
:![\[
ab + bc + ca \leq \frac{(a + b + c)^2}{3} + k \cdot \max \{ (a - b)^2, (b - c)^2, (c - a)^2 \} \leq a^2 + b^2 + c^2
\]](http://latex.artofproblemsolving.com/1/c/e/1ce66163068b499383adb4a4ecec86c5f4edaec3.png)
are smaller than the square root of , i.e., 
for all proper positive divisors 
of different from .
and 
for 
. Let 
. Prove that there exist infinitely many natural numbers 
for which 
is divisible by 
. Does there exist for which 
is divisible by ?
satisfy,
Where {
} is a permutation of . Prove that 
be a fixed square and 
a variable point on segment 
. The square 
is constructed such that 
is on segment 
and 
is on segment 
. Let be the intersection point of lines 
and 
. Find the locus of as varies along segment .
holds 
. does equality occur?
and be chords , with 
. Find the length of the circle chord that divides angle 
in half.
-bisector intersect 
at and Let bisector 
intersect the circle at 
, we have our cosine rule, 
, 
such that:
.
