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 Mathzeus1024
For a non-empty finite complex number set 
, define the "Tao root" of as 
. Given the integer 
, let the set 
Let 
be the number of non-empty subsets in which the Tao root of 
is 
, 
is the number of non-empty subsets of whose Tao root is 
. Compare the sizes of 
and 
.
, define the "Tao root" of as 
. Given the integer 
, let the set 
Let 
be the number of non-empty subsets in which the Tao root of 
is 
, 
is the number of non-empty subsets of whose Tao root is 
. Compare the sizes of 
and 
.This post has been edited 1 time. Last edited by parmenides51, Apr 6, 2024, 6:36 AM
