[/quote]
,
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 
.
.
kawaii if it satisfies 
is the number of integers not more than 
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 
.
and 
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, 
,
,
is a multiple of 
.
a point in the space and 
the centroid of a tetrahedron 
.
be a 
on it, line 
makes a fixed angle with the tangent to 
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 
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 
with exactly 9 elements that is subset of 
.
and 
that satisfy the following:
is an empty subset.
is 
and 
.
to 
and 
.
and 
is 
is 
is tangent to the 
Y by Mango247
Let 
be a triangle with 
and orthocenter 
. If 
are the 
altitudes, consider the circle through , tangent to 
at 
, the circle through tangent to 
at 
and the circle through tangent to 
at 
. Prove that is the center of the circle circumscribed by the triangle formed by the intersections, different from , of two of these circles.
be a triangle with 
and orthocenter 
. If 
are the 
altitudes, consider the circle through , tangent to 
at 
, the circle through tangent to 
at 
and the circle through tangent to 
at 
. Prove that is the center of the circle circumscribed by the triangle formed by the intersections, different from , of two of these circles.This post has been edited 3 times. Last edited by parmenides51, Aug 17, 2020, 7:22 PM
