[/quote]
,
and 
.
be a function such that
for all 
.
.
lies inside of scalene triangle 
with incenter 
such that 
Lines 
and 
intersect line 
respectively at 
and 
.
intersects 
at 
and line through 
at 
.
and 
be non-negative real numbers such that ![\[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]](http://latex.artofproblemsolving.com/b/9/e/b9e86e898c0536b7323a03611d5bdbf679caa710.png)
Prove that 
such that for any positive integer 
there exists a positive integer 
such that the last 
are equal to 
of the triangle 
which doesn't coincide with 
or 
intersectes the altitudes from 
and 
at 
and 
respectively. Let 
be the reflection of 
be the reflection of 
Prove that the circles 
and 
are tangent.
of the vertices of the regular polygon 
we write the number 
and in the remaining ones we write the number 
Let 
be the number written on the vertex 
A vertex is good if ![\[x_i+x_{i+1}+\cdots+x_j>0\quad\text{and}\quad x_i+x_{i-1}+\cdots+x_k>0,\]](http://latex.artofproblemsolving.com/b/f/4/bf4cf984af7f310c1eb90b6b37126bfca9181305.png)
for any integers 
and 
Note that the indices are taken modulo 
Determine the greatest possible value of 
such that:
is the 
-
is the circumcircle of this triangle. 
is the middle of arc 
.
meets 
.
,
is marked. Point 
is on segment 
,
is on segment 
such that the circumcircles of triangles 
and 
are tangent to line 
.
and 
passing through the center of the circumcircle of triangle 
,
.
such that ![\[f(xy-1)+f(x)f(y)=2xy-1,\]](http://latex.artofproblemsolving.com/8/8/8/888ca39f2b7f8cec6d6426bee28d40eade40a66e.png)
for any real numbers 
and 
of positive integers with the following property: all of the terms of the sequence 
have a greatest common divisor 
Y by HWenslawski, Mango247, Mango247, Mango247
An acute-angled triangle 
is given, and 
are the midpoints of sides 
respectively. The internal angle bisector of 
meets 
at 
, and the internal angle bisector of 
meets 
at 
. The line 
intersects 
at 
, and 
at 
. Prove that the lines 
are concurrent.
is given, and 
are the midpoints of sides 
respectively. The internal angle bisector of 
meets 
at 
, and the internal angle bisector of 
meets 
at 
. The line 
intersects 
at 
, and 
at 
. Prove that the lines 
are concurrent.
,
.
be the lengths of the three sides and 
be the length of 
.
.
,
.
.
.
,
,
,
at 
at 
.
.
then angle-chase further in 
to find 
.
which equals 
?
so 
