[/quote]
,
and real 
,
.
such that 
such that![\[ f(xy+f(x)) + f(y) = xf(y) + f(x+y) \]](http://latex.artofproblemsolving.com/4/5/5/455594f4b2c4718152c23140e9faf4ab0edc3a1b.png)
for all real numbers 
and 
.
,
as the number of positive divisors of 
as the sum of those divisors. For example, 
and 
.![\[
\sigma(n) = \tau(n) \cdot \lceil \sqrt{n} \rceil
\]](http://latex.artofproblemsolving.com/d/b/2/db2028224f7109d21fff61c6e0575ec7699b433d.png)
of the triangle 
is extended beyond 
to 
so that 
.
is extended beyond 
to 
so that 
.
,
has no solutions in integers 
.
be positive real numbers so that 
.![\[ \left( a - 1 + \frac 1b \right) \left( b - 1 + \frac 1c \right) \left( c - 1 + \frac 1a \right) \leq 1.
\]](http://latex.artofproblemsolving.com/7/6/e/76e794a62090c8c4b5f3a70ad4d036669418ca69.png)
that satisfy 
and 
.
be the center of the square inscribed in acute triangle 
and the other is on 
.
are defined in a similar way for inscribed squares with two vertices on sides 
are concurrent.
There are no zeros in the representation of this number;
and 
are good.
be the circumcenter and 
the orthocenter of an acute triangle 
,
and 
is equal to the sum of the areas of the other two.
?
be a convex quadrilateral such that diagonals 
and 
intersect at right angles, and let 
be their intersection. Prove that the reflections of 
are concyclic.
Y by
Given a prime number 
, a set is said to be -good if the set contains exactly three elements 
and 
.
Find all prime number such that 
can be partitioned into several -good sets.
Proposed by capoouo
, a set is said to be -good if the set contains exactly three elements 
and 
.Find all prime number
such that 
can be partitioned into several -good sets.Proposed by capoouo
This post has been edited 1 time. Last edited by capoouo, Apr 21, 2024, 5:42 PM
Reason: remove awkward line-breaks; add "proposed by ..."
Reason: remove awkward line-breaks; add "proposed by ..."
.
rewrite into 
.
as 
,
.
runs through 
we achieve 
sets. The contrary is obvious.
,
,
?
.
doesn't satisfy as to partition 
into several 
.
.
.
.
be primitive root of 
as 
.
,
,
and we achieves 
