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 
be positive integers. If 
is divisible by 
for every positive integer 
, prove that 
.
with 
and 
. Lines 
and 
intersect at 
, and lines 
and 
intersect at 
. Let 
be the midpoints of sides 
, respectively. Let 
and 
be points on segment 
and 
, respectively, so that 
is the angle bisector of 
and 
is the angle bisector of 
. Prove that 
is parallel to 
if and only if 
divides into two triangles with equal area.
kawaii if it satisfies 
(
is the number of positive factors of , while 
is the number of integers not more than that are relatively prime with ). Find all that is kawaii.
+1 w
of positive integers with the following property: all of the terms of the sequence 
have a greatest common divisor 
, 
, 
be nonzero real numbers such that 
and 
. Find the value of
.
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 
be a function satisfying
for all integers and . Show that there exist positive integers 
and such that 
for all integers .
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 
![\[
f : \mathbb{R}^+ \to \mathbb{R}
\]](http://latex.artofproblemsolving.com/e/7/b/e7bfc5b236fb6f00ef328f850b1b14632cbf8416.png)
satisfying the condition that for all 
,![\[
f(x + y^2) \geq f(x) + y.
\]](http://latex.artofproblemsolving.com/3/a/a/3aa20083835682dbd81be692eba65cab19e923e5.png)
of positive integers with 
prime, 
, and 
, such that 
is a multiple of 
.
a point in the space and 
the centroid of a tetrahedron . Prove that:
be a logarithmic spiral centered at the origin (ie curve satisfying for any point on it, line 
makes a fixed angle with the tangent to at ). Let 
be a rectangular hyperbola centered at the origin, scaled such that it is tangent to the logarithmic spiral at some point.
on the spiral, the polar of wrt. is tangent to the spiral.
be a monic polynomial so that there exists another real coefficients 
that satisfy![\[P(x^2-2)=P(x)Q(x)\]](http://latex.artofproblemsolving.com/6/9/7/697faa929e4fda7a6e9b1cd97849bd42ffc14306.png)
Determine all complex roots that are possible from that satisfies (a), determine whether should have real coefficients or not.
are cyclic.
and 
are collinear.
and 
are collinear. and are collinear, we can say that 
are collinear. 
be orthocentre of triangle 
. Note that 
and 
are cyclic so by Simson's theorem on 
and 
: 
and 
are collinear.
with radius 
, and denote inverses with a 
.
and 
, so that 
. Moreover, from 
collinear, we get that 
, hence 
is a rectangle. From 
, we get the desired cyclicity, so inverting back yields collinear
and 
are tangent at point 
. Similarly 
and are tangent at point 
. Now using homothety wrt points and we find that 
It's obvious that and are tangent at and have common tangent line in . Now using this we have: 
. Hence 
are concyclic. Now again from the power of point 
and Thales' Theorem we have that 
. Now from 
, are collinear.
be collinear points, such that is between and . Let 
be circles with diametar 
respectively. Let 
be points on 
, both on the same side of . Let 
, 
, 
, 
. Then prove that are collinear 
are concyclic 
, where 
and are tangent at point ?
and are tangent at point ? with radius 
. Then and are each other's inverses, whiles also 
and are inverse pair. Now let 
be the inverse of . Since lies on the polar of , going through , must lines on the polar of , which is the perpendicular to at . So by inverting through the mentioned circle. goes to 
, while goes to . But they are parallel and hence they have only one intesecting point (at infinity), hence they are tangent at . Otherwise the lines would have to intersect at another point. by proving that they have a common tangent line at . Draw the tangent line to at C. Then the angle between the chord 
and the tangent line is 
. Hence by the Tangent and Chord Theorem they share a common tangent line, hence they are tangent to each other. There are lots of other way to prove this. It's easy to prove that is the center of homothety of the two circles, hence they have only one touching point.
and are tangent at point ?
and 
...
.So by the 
we have and are collinear.(My 100th post!!!!!!! 
)
be a triangle, and let 
and 
and 
and 
and 
and 
respectively. Prove that 
.