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.
and 
such that 
is divisible by 
?
+1 w
be a triangle with incenter 
, and suppose the incircle is tangent to 
and 
at 
and 
. Denote by 
and 
the reflections of and over . Let 
be the intersection of 
with 
, and let 
be the midpoint of . Prove that 
and 
are perpendicular.
be a convex quadrilateral with perpendicular diagonals intersecting at 
. Prove that the reflections of across , BC, CD, DA$ are concyclic.
be three diameters of the circumcircle of an acute triangle . Let 
be an arbitrary point in the interior of 
, and let 
be the orthogonal projection of on 
, respectively. Let 
be the point such that 
is the midpoint of 
, let 
be the point such that is the midpoint of 
, and similarly let 
be the point such that is the midpoint of 
. Prove that triangle 
is similar to triangle .
such that for any 
, and for any 
real numbers 
there exists real numbers 
such that 
for at least 
indices 
. Here, 
and 
denotes the fractional part of 
.
be an acute-angled scalene triangle. Let 
and 
be points on the rays 
and 
, respectively, such that 
. Similarly, let 
and 
be points on the rays 
and 
, respectively, such that 
. Prove that if 
and 
intersect at 
, then 
is parallel to .
be the circumcircle of a triangle . Denote by and 
the midpoints of the sides and 
, respectively, and denote by 
the midpoint of the arc of not containing 
. The circumcircles of the triangles 
and 
intersect the perpendicular bisectors of and at points and , respectively; assume that and lie inside the triangle . The lines 
and 
intersect at 
. Prove that 
.
such that the largest prime divisor of 
is equal to the largest prime divisor of 
.
be the set of positive real numbers. Find all functions 
such that, for all positive real numbers , 
, 
and 
, the following three conditions are satisfied:
,
,
.
of positive integers that satisfy the equation: 
.
be the set of real numbers. Determine all functions 
such that ![\[ f(x^2 - y^2) = x f(x) - y f(y) \]](http://latex.artofproblemsolving.com/6/8/f/68f78e0df627f6ac04052433c01627bf16f0af8e.png)
for all pairs of real numbers and .
and 
, define 
to be the total number of times the digit appears when all the multiples of 
between 
and inclusive are written out in base 
. such that there are precisely two distinct values among 
, 
, 
, 
.
be distinct real numbers such that 
Find the value of 
Find the value of 
Find the value of 
Prove that 
,
,
be defined as![\[
x=\frac{a}{b-c},\quad y=\frac{b}{c-a},\quad z=\frac{c}{a-b}.
\]](http://latex.artofproblemsolving.com/c/6/b/c6b2629dfafd73409a6d911d1dd984ff4c70a94c.png)
Then, the system of equations![\[
a - xb + xc = 0,
\]](http://latex.artofproblemsolving.com/d/9/b/d9b088945fa707c2bfd4cad278308b35642dbc49.png)
![\[
ya + b - yc = 0,
\]](http://latex.artofproblemsolving.com/1/6/e/16eace37b2c4c09aa59968e0d299e1f982e0213d.png)
![\[
-za + zb + c = 0,
\]](http://latex.artofproblemsolving.com/8/6/f/86f4f7da9c109d2621815fef961b7f16c5d164fc.png)
admits a nonzero solution 
.![\[
\begin{vmatrix}
1 & -x & x \\
y & 1 & -y \\
-z & z & 1
\end{vmatrix}
= xy + yz + zx + 1 = 0.
\]](http://latex.artofproblemsolving.com/3/a/1/3a10ec43739f905365053fb18558037bda2cce8e.png)
It follows that![\[
(x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx)=2-2=0.
\]](http://latex.artofproblemsolving.com/4/6/0/4605588d4b8c63e3de8fdbeec43959607c06c5c3.png)
