AoPS Academy
of positive integers such that
be a circle inscribed inside a rhombus 
. Let 
and 
be variable points on 
and 
respectively, such as 
is always the tangent line to . and the area of triangle 
is the same.
, find the smallest integer 
such that any graph with 
vertices and at least edges must contain at least two disjoint cycles (cycles with no common vertices).
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.
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 and 
end in the same digit. Find the sum of the two smallest beautiful numbers.
satisfying 
Prove that 
Proposed by Phan Ngoc Chau
and verifying 
. 
where 
are primes, show that 
can never be a perfect square.
be a real number, and let ![$P(x) \in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/4/5/3/453a624c3b002c1b0e78e0023b24dd22ddd03557.png)
be a monic polynomial with degree 
, such that
.
has exactly positive real roots that are all less than or equal to 
.
.
be a real number, and let be a monic polynomial with degree 
, such that. has exactly positive real roots that are all less than or equal to .![\[\left| {P\left( 1 \right)} \right| \le \max \left\{ {{{\left[ {\frac{{n - \left\lfloor {n \cdot \frac{{W\left( {\frac{1}{e}} \right)}}{{W\left( {\frac{1}{e}} \right) + 1}}} \right\rfloor }}{{\left\lfloor {n \cdot \frac{{W\left( {\frac{1}{e}} \right)}}{{W\left( {\frac{1}{e}} \right) + 1}}} \right\rfloor }}} \right]}^{\left\lfloor {n \cdot \frac{{W\left( {\frac{1}{e}} \right)}}{{W\left( {\frac{1}{e}} \right) + 1}}} \right\rfloor }};{{\left[ {\frac{{n - \left\lceil {n \cdot \frac{{W\left( {\frac{1}{e}} \right)}}{{W\left( {\frac{1}{e}} \right) + 1}}} \right\rceil }}{{\left\lceil {n \cdot \frac{{W\left( {\frac{1}{e}} \right)}}{{W\left( {\frac{1}{e}} \right) + 1}}} \right\rceil }}} \right]}^{\left\lceil {n \cdot \frac{{W\left( {\frac{1}{e}} \right)}}{{W\left( {\frac{1}{e}} \right) + 1}}} \right\rceil }}} \right\}{\left( {\alpha - 1} \right)^n}\]](http://latex.artofproblemsolving.com/7/2/9/72930837fe3d4d4e2280353a89aa715011844320.png)
(Lambert 
function)
but 
...). 
.)
but ...). .)
satisfy that 
.
and 
Prove that
Let 
Prove that
be the Fibonacci numbers, defined by 
and 
.
,
![$\alpha_1,\dots,\alpha_{24} \in (0,\alpha]$](http://latex.artofproblemsolving.com/d/6/5/d655c0a7a1442ee2b3a4949fb85f1079bcf7fbf7.png)
be the roots, we have 
and we want to maximize 
.
,
and decreasing 
are equal to 
of the variables being equal to 
ones being equal to 
where 
and we need to prove that 
.
.![$t \in [0,1]$](http://latex.artofproblemsolving.com/6/7/3/6735b925696750e153b8d293780a7b620449b778.png)
and 
,
(we can then choose 
to finish). But this is equivalent to 
which immediately follows from the condition by weighted AM-GM.
over all choices of 
.
takes its maximum when 
is the unique root of 
,
,
is increasing for 
and decreasing for 
.
and 
i.e. that 
is bigger than both 
and 
which can be done by hand: It suffices to prove 
.
and the claim follows by noting that 
.
where we replace