[/quote]
,
committees such that each senator is on exactly one committee. Each senator hates exactly three other senators. (If A hates B, then B does not necessarily hate A.) Find the smallest 
.
and 
,
is not less than the area of 
with circumcenter 
.
be a point on 
such that 
.
the point on 
.
is the projection of 
,
passes through the orthocenter of 
.
,
be the feet of altitudes from 
,
to sides 
,
,
and 
the incircles of triangles 
and 
,
and 
at 
and 
meet circles 
and 
,
.
intersects sides 
of cyclic quadrilateral 
in its interior points 
and 
,
beyond point 
,
beyond point 
at 
.
and 
intersect at 
,
and 
intersect at 
.
and 
meet at point 
,
meet at point 
and lines 
.
and 
be 
and 
respectively. Let the intersection point of the line that passes through 
and the line that passes through 
be 
,
,
,![\[ \frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab} \geq a+b+c \]](http://latex.artofproblemsolving.com/9/f/a/9fa6cca20b3620fbb32ee782196e96f6fffa0063.png)
satisfies the following condition.![$\sum_{k=1}^{n}\frac{1}{2}\left(1 - (-1)^{\left[\frac{n}{k}\right]}\right)a_k=1$](http://latex.artofproblemsolving.com/3/1/0/3109e3f024dc59c2cdad60efa270494d6095c330.png)
holds.
,
.
satisfy that 
.
![\[\left\lfloor\frac{2025}{1}\right\rfloor,\left\lfloor\frac{2025}{2}\right\rfloor,\left\lfloor\frac{2025}{3}\right\rfloor,\dots,\left\lfloor\frac{2025}{2024}\right\rfloor,\left\lfloor\frac{2025}{2025}\right\rfloor.\]](http://latex.artofproblemsolving.com/a/8/2/a8255ec218f896983d39044f8f777e2fd811479b.png)
,
,
.
and 
are located on 
and 
respectively, such that 
and 
.
to 
.
be the circumcircle of a triangle 
the midpoint of the arc 
and 
intersect the perpendicular bisectors of 
,
intersect at 
.
which satisfy the following relationship for all real numbers 
and 
![\[f(xf(y)-f(x))=2f(x)+xy.\]](http://latex.artofproblemsolving.com/d/0/e/d0e9e56267aa6241a0323c7b2245e4752c338785.png)
Y by Adventure10, Mango247
A set of points of the plane is called obtuse-angled if every three of it's points are not collinear and every triangle with vertices inside the set has one angle 
. Is it correct that every finite obtuse-angled set can be extended to an infinite obtuse-angled set?
(UK)
. Is it correct that every finite obtuse-angled set can be extended to an infinite obtuse-angled set?(UK)
This post has been edited 2 times. Last edited by parmenides51, Apr 11, 2022, 7:28 PM
.
and 
for small enough 
.
has an angle larger than 
for 
because 
does and 
is small enough. For the triangles 
,
and for 
we have that 
and ![$\angle XA_{1}A_{2} \not\in [89^{\circ}, 91^{\circ}]$](http://latex.artofproblemsolving.com/d/a/0/da063aa6968862dd21f797dd965aa160dd1b3522.png)
as 
is obtuse-angled (so ![$\angle A_{i}A_{1}A_{2} \not\in [89^{\circ}, 91^{\circ}]$](http://latex.artofproblemsolving.com/d/6/3/d63167c6a97acb493e000390546a7095e3d2d4ec.png)
)
is also obtuse-angled, which completes the construction.
replaced by 
for any 
.
,
