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 GeoKing
Suppose an acute scalene triangle 
has incentre 
and incircle touching 
at 
. Let 
be the antipode of 
in the circumcircle of . Point 
is chosen on the internal angle bisector of 
such that 
. Let 
be the midpoint of arc 
, and let 
be the midpoint of 
. Prove that 
has incentre 
and incircle touching 
at 
. Let 
be the antipode of 
in the circumcircle of . Point 
is chosen on the internal angle bisector of 
such that 
. Let 
be the midpoint of arc 
, and let 
be the midpoint of 
. Prove that 
which finishes because 
Let 
be the 
,
.
is a rectangle and the line perpendicular to 
through 
so 
.
and 
and 
and 
now write it in terms of 
and 
of 
with the help of 
and everything cancels out and you get the required ratio for similarity is true.
.
and 
,
.
,
which is known to be the power of 
,
be the Sharky Devil point (i.e., 
)
.![$$\left.
\begin{array}{l}
\measuredangle DIM = \measuredangle(\perp BC, AM) = \measuredangle MNZ \\[4pt]
\measuredangle IMD = \measuredangle AMS = \measuredangle AZS = \measuredangle AZI = \measuredangle PMN
\end{array}
\right\} \implies \triangle IMD\sim\triangle NMP$$](http://latex.artofproblemsolving.com/6/c/3/6c3f35256f9569173bc274195ec332eefd69c06f.png)
with corresponding midpoints 
at 
.
denote the midpoint of 
,
are collinear.
is a parallelogram
,
is a parallelogram
.
.
.
where 
,
These two facts imply the claim.
(also the 
is the internal bisector of 
be the reflection of 
,
.
be the second intersection of the 
,
,
.
is a rectangle, this is equivalent to 
where 
are the inradius and circumradius of 
.
the midpoint of segment 
a
.
denote the 
.
is the major 
must pass through 
and 
are similar.
point 
.
and 
it also follows that 
and 
.
is a rectangle.![\[\frac{MZ}{AM}=\frac{PX}{AP}\]](http://latex.artofproblemsolving.com/f/e/b/feb0887e1b63f9f327ea330531d81f177e8c9c94.png)
and since we know that 
and 
,![\[\frac{MZ\cdot AP}{AB \cdot AC}=\frac{PX}{AP}= \frac{PD}{AI}\]](http://latex.artofproblemsolving.com/3/2/d/32dfa9a555f7d353329a20bce6b0f19a6dd375f5.png)
Also we know 
.![\[\frac{UL}{UM} = \frac{MZ}{UM} = \frac{2MZ}{AI_a} = \frac{2 PD \cdot AB \cdot AC \cdot AI}{AI \cdot AP \cdot AB \cdot AC} = \frac{2PD}{AP}=\frac{PD}{NP}\]](http://latex.artofproblemsolving.com/9/9/9/99943daf444eacee5032acbe217cb9140b41bb76.png)
which in conjunction with 
is sufficient to imply the claim.![\[\measuredangle UML = \measuredangle DNP = \measuredangle I_aNP = \measuredangle I_aDI = \measuredangle MVI = \measuredangle MVD\]](http://latex.artofproblemsolving.com/f/c/3/fc3b55a750d04c4def18e86b1e94d8f99e0598eb.png)
which is the desired result.
the corresponding excenters, 
and 
the midpoint of 
the midpoint of segment 
by similarity between 
and 
.
,![\[ \cfrac{I_aK}{TK} = \cfrac{I_aA}{AN} \implies \cfrac{I_aK}{DK} = \cfrac{MJ}{AN} \]](http://latex.artofproblemsolving.com/c/d/2/cd2cb371683d4d88d818b6ffd74864245cb21144.png)
Then, as 
and 
are rectangles, it follows that 
,![\[ \angle IML = \angle JML = \angle KJD = 180^{\circ} - \angle IDJ = \angle DVM \]](http://latex.artofproblemsolving.com/9/2/6/92602a44b241c100a13e696cf333a55bf8dfc1b5.png)
