of positive integers such that
be a circle inscribed inside a rhombus 
.
and 
be variable points on 
and 
respectively, such as 
is always the tangent line to 
is the same.
,
such that any graph with 
vertices and at least 
regions of a three-set Venn Diagram with 
colors such that each color is used at least once? Two colorings are considered the same if one can be reached from the other by rotation and reflection.
satisfy that 
.
and 
Prove that
Let 
Prove that
of them have the same color. I want to reshuffle the cupcakes such that each plate is replaced by a different color cupcake. Determine how many ways to reshuffle when:
.
.
is called beautiful if the are at least 
integers 
such that in base 
we have that 
end in the same digit. Find the sum of the two smallest beautiful numbers.
be the Fibonacci numbers, defined by 
and 
.
,
satisfying 
Prove that 
Proposed by Phan Ngoc Chau
and 
. 
where 
are primes, show that 
can never be a perfect square.
Y by Adventure10
An integer 
is called resistant, if it is coprime to the sum of all its divisors (including 
and 
).
Determine the maximum number of consecutive resistant numbers.
For instance:
*
has sum of divisors 
and hence is resistant.
*
has sum of divisors 
and hence is not resistant.
*
has sum of divisors 
and hence is resistant.
*
has sum of divisors 
and hence is not resistant.
is called resistant, if it is coprime to the sum of all its divisors (including 
and 
).Determine the maximum number of consecutive resistant numbers.
For instance:
*
has sum of divisors 
and hence is resistant.*
has sum of divisors 
and hence is not resistant.*
has sum of divisors 
and hence is resistant.*
has sum of divisors 
and hence is not resistant.
,
to 
.
denote the sum of the divisors of 
such that 
where 
is odd, then the sum of divisors of 
is 
,
,
,
and thus 
or 
.
,
and 
,
,
and 
for some 
,
,
,
for some 
.
