[/quote]
,
satisfying
and 
such that 
is divisible by 
?
be a triangle with incenter 
,
and 
at 
and 
.
and 
the reflections of 
be the intersection of 
with 
,
be the midpoint of 
and 
are perpendicular.
be a convex quadrilateral with perpendicular diagonals intersecting at 
.
be three diameters of the circumcircle of an acute triangle 
be an arbitrary point in the interior of 
,
be the orthogonal projection of 
,
be the point such that 
is the midpoint of 
,
be the point such that 
,
be the point such that 
.
is similar to triangle 
such that for any 
,
real numbers 
there exists real numbers 
such that 
for at least 
indices 
.
and 
denotes the fractional part of 
.
be an acute-angled scalene triangle. Let 
and 
be points on the rays 
and 
,
.
and 
be points on the rays 
and 
,
.
and 
intersect at 
,
is parallel to 
be the circumcircle of a triangle 
the midpoints of the sides 
,
the midpoint of the arc 
.
and 
intersect the perpendicular bisectors of 
and 
intersect at 
.
.
is equal to the largest prime divisor of 
.
be the set of positive real numbers. Find all functions 
such that, for all positive real numbers 
,
and 
,
,
,
.
of positive integers that satisfy the equation: 
.
be the set of real numbers. Determine all functions 
such that ![\[ f(x^2 - y^2) = x f(x) - y f(y) \]](http://latex.artofproblemsolving.com/6/8/f/68f78e0df627f6ac04052433c01627bf16f0af8e.png)
for all pairs of real numbers 
Y by Adventure10, Mango247
The Judgment of the Council of Sages proceeds as follows: the king arranges the sages in a line and places either a white hat or a black hat on each sage's head. Each sage can see the color of the hats of the sages in front of him, but not of his own hat or of the hats of the sages behind him. Then one by one (in an order of their choosing), each sage guesses a color. Afterward, the king executes those sages who did not correctly guess the color of their own hat. The day before, the Council meets and decides to minimize the number of executions. What is the smallest number of sages guaranteed to survive in this case?
See also http://www.artofproblemsolving.com/Forum/viewtopic.php?f=42&t=530553
See also http://www.artofproblemsolving.com/Forum/viewtopic.php?f=42&t=530553
