[/quote]
,
be an acute non-isosceles triangle with circumcircle 
,
and orthocenter 
.
through 
through 
and 
lie on a circle, which is tangent to 
and intouch triangle 
.
be the foot of the perpendicular from 
onto 
.
,
intersect the sides 
respectively. Finally, let the rays 
,
meet the circumcircle of 
in 
,
respectively. Prove that the tangent from 
meet on the circumcircle of 
such that there exists exactly 
such that ![$f:[1,5]\rightarrow[1,5]$](http://latex.artofproblemsolving.com/6/2/0/620e1621905a4670627d46fcb367e1a8c461bb49.png)
satisfying 
.
such that 
.
May everyone not to upload solutions on this problem anymore until May 15/2025, this is an active problem on Mathematical Reflection! (Thank you Victoria_Discalceata1)
and 
for all 
be four distinct points on a line, in that order. The circles with diameters 
and 
intersect at 
and 
.
meets 
at 
.
be a point on the line 
intersects the circle with diameter 
and 
,
intersects the circle with diameter 
and 
.
are concurrent.
such that![\[
f(xy) = f(x)f(y) \;-\; f(x + y) \;+\; 1,
\quad \forall x, y \in \mathbb{R}.
\]](http://latex.artofproblemsolving.com/7/a/5/7a51a7be2306d18f4e80db1f5aac6e9bbf25c25c.png)
be a positive integer. For every positive integer 
the sequence 
is defined, where 
are integers. Among these 
be a sequence of real numbers with 
.
of a permutation 
of 
to be the minima of the sum 
over all real numbers 
.
,
,
be a nonempty set of positive integers. We say that a positive integer 
such that for any positive integer 
,
.
lie respectively on sides 
.
and 
and 
cross again at point 
.
Y by
There are 51 senators in a Senate. The Senate needs to be divided into 
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 such that it is always possible to arrange the committees so that no senator hates another senator on his or her committee.
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 such that it is always possible to arrange the committees so that no senator hates another senator on his or her committee.
