[/quote]
,
such that 
.
be an acute triangle and let 
be an interior point for it such that 
.
When does equality hold?
be positive reals. Prove that there is a cyclic permutation 
of 
such that the inequality:![\[
\frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z}
\]](http://latex.artofproblemsolving.com/5/5/3/553527fc40a1b7d29dbdfb4b5e63635308f46c85.png)
and 
.
be real numbers such that 
.![\[
3 + (2 - \sqrt{3}) \cdot \frac{(b-c)^2}{b+(\sqrt{3}-1)c} \leq a+b+c
\]](http://latex.artofproblemsolving.com/5/a/8/5a83eaa7470bcca6546599ff4e296e4efec69b16.png)
and determine all the cases when the equality occurs.
have the same incenter 
.
is concyclic iff 
is concurrent
be the nonnegative integers in 
where 
.
.
.
such that 
where 
,
?
where 
,
be a scalene triangle with circumcircle 
and circumcircle 
.
and 
as the midpoints of 
and 
,
to 
.
meets 
,
intersects 
.
at 
and 
at 
.
intersect 
,
intersect 
.
is the reflection of 
and 
lies on the circumcircle of 
.
be real numbers such that 
and 
and determine the equality cases.
be positive real integer such that 
Prove that
such that 
and 
satisfying :![\[n^{2p}=m^2+n^2+p+1\]](http://latex.artofproblemsolving.com/7/4/e/74ee592cb8b3f3a3996ea34f40b78fde9d3672fc.png)
and 
are integers and 
is a prime number.
Approach: I'm trying to factor 
to get 
.
such that 
divides 
.
Y by Fouad-Almouine, Math-wiz, Atpar, megarnie
Let 
and 
be the sets of all lines and all points on the same fixed plane, respectively. Find all functions 
such that, for any two non-parallel lines 
, the points 
, 
and 
are collinear.
Proposed by Dorlir Ahmeti, Kosovo
and 
be the sets of all lines and all points on the same fixed plane, respectively. Find all functions 
such that, for any two non-parallel lines 
, the points 
, 
and 
are collinear.Proposed by Dorlir Ahmeti, Kosovo
This post has been edited 2 times. Last edited by dangerousliri, May 31, 2020, 10:55 AM
and a fixed point 
.
,
must collinear, which is true.
,
,
all lie on line 
.
.
.
we have 
.
(which are not concurrent, and no two of them are parallel) such that 
and 
.
collinear. This gives us 
.
.
.
.
are collinear, this is a contradiction as they are vertices of a non-degenerate triangle.
,
.
as the line which passes through 
and parallel to 
as the set of all lines that pass through 
.
collinear. However, both 
and 
are both on line 
.
such that 
.
are collinear.
.
must lie on the line passes through both 
and 
,
.
.
,
.
,
.
and 
in 
collinear gives us 
.
collinear gives us 
.
.
that doesn't pass through 
are collinear. This gives us 
.
.
will satisfy ![\[ f(\ell) = X, \forall \ell \ \text{which doesn't pass through } X \]](http://latex.artofproblemsolving.com/e/d/0/ed0f61895e20b8c0e82ae16d2097e5714b3fbdad.png)
Now, consider line 
that doesn't pass through 
,
,
are collinear. This implies 
collinear. This implies, 
.
.
such that 
is on 
and let 
are colinear so 
such that 
is between 
not in 
are collinear
are collinear
are collinear so 
contradiction
and 
are fixed
