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 an isosceles trapezoid with 
. Let 
be the midpoint of 
. Denote by 
and 
the circumcircles of the triangles 
and 
, respectively. Let 
be the crossing point of the tangent to at 
with the tangent to at 
. Prove that 
is tangent to .
, let 
denote the feet of the altitudes from 
, respectively, and let be the circumcircle of 
. Let 
and 
be the circles through tangent to at and 
, respectively. Show that and meet at a point on 
other than .
such that![\[ f(x+f(y))=y+f(x+1),\]](http://latex.artofproblemsolving.com/b/a/3/ba35828951778c0d35f50b5b557796bd2cf9e338.png)
for all 
.
is defined by 
, and 
for 
. Consider the set 
of ordered pairs 
for which there is a finite set 
of positive integers such that 
, 
. Prove that there exist real numbers 
, 
, and 
with the following property: An ordered pair of nonnegative integers satisfies the inequality ![\[m < \alpha x+\beta y < M\]](http://latex.artofproblemsolving.com/a/9/b/a9bb35658173fdab821cc8180420e47d5437e0a2.png)
if and only if 
.
.
, satisfying ![\[
f(xy)(f(x) - f(y)) = (x-y)f(x)f(y)
\]](http://latex.artofproblemsolving.com/0/9/b/09b8ed4e133ccf33b3f902fd9a9aa8e23c264341.png)
for all 
.
be 
positive integers such that the 
products![\[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\]](http://latex.artofproblemsolving.com/0/2/c/02cbff15c4d0118156a9d24be09360facb264d5a.png)
form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.
+1 w
be a positive integer. Define a sequence by setting 
and, for each 
, letting 
be the unique integer in the range 
for which 
is divisible by 
. For instance, when 
the obtained sequence is 
. Prove that for any the sequence 
eventually becomes constant.
be an acute triangle with 
. The bisector of intersects the lines 
and at points and , respectively. The circumcircle of triangle 
has center and intersects the circumcircle of triangle at point with different to .
is tangent to the circumcircle of triangle .
satisfying
and 
such that 
is divisible by 
? 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.
such that![\[
2^a \cdot 3^b + 1 = 5^c.
\]](http://latex.artofproblemsolving.com/0/a/c/0ac1f22162eacf698847bceb33f9a337bd53c44b.png)
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)
