[/quote]
,
(where 
are the digits and 
)
For example the permutation sum of 20253 is 
Let 
and 
.
be a triangle. Let 
be a points on the segment 
such that 
.
be the mid-point of 
.
intersect 
in 
and 
in 
respectively. Determine 
.
,
and 
e
are tangent to 
and 
respectively. 
intersects line 
.
.
intersects 
.
.
that satisfy the following condition for any real numbers 
and 
such that:
Prove that
.
be the midpoints of 
and 
respectively and let 
be the projection of 
.
on 
.
intersect 
at points 
and 
.
intersct 
.
intersect 
at 
and 
intersect 
.
are concurrent.
and 
respectively be the incentre, circumcentre and circumcircle of triangle 
are chosen on 
,
is the foot of the altitude from 
to 
.
are similarly defined, prove that lines 
and 
concurr on 
.
and let 
and 
and 
respectively. Let the circle with centre 
and 
.
,
and 
,
.
and 
be feet of perpendiculars from 
and 
respectively. Prove that 
,
and 
are concurrent.
be a sequence that consists of an initial block of 
positive distinct integers that then repeat periodically. This means that 
are 
for every positive integer 
prime numbers, find for which values of 
terms of the sequence.
such that, for all real numbers 
with 
,
s
of positive integers for which there exist positive integers 
and 
such that
holds for all integers 
(Note that 
denotes the greatest common divisor of integers 
and 
)
Y by AwesomeYRY, Adventure10
Let 
be a positive integer. Ana and Banana are playing the following game:
First, Ana arranges
cups in a row on a table, each facing upside-down. She then places a ball under a cup
and makes a hole in the table under some other cup. Banana then gives a finite sequence of commands to Ana,
where each command consists of swapping two adjacent cups in the row.
Her goal is to achieve that the ball has fallen into the hole during the game. Assuming Banana has no information
about the position of the hole and the position of the ball at any point, what is the smallest number of commands
she has to give in order to achieve her goal?
be a positive integer. Ana and Banana are playing the following game:First, Ana arranges
cups in a row on a table, each facing upside-down. She then places a ball under a cupand makes a hole in the table under some other cup. Banana then gives a finite sequence of commands to Ana,
where each command consists of swapping two adjacent cups in the row.
Her goal is to achieve that the ball has fallen into the hole during the game. Assuming Banana has no information
about the position of the hole and the position of the ball at any point, what is the smallest number of commands
she has to give in order to achieve her goal?
swaps to "determine" where the hole is. We then take one additional move to put the ball in the hole, making a total of 
.
.
be the total number of L balls and 
be the total number of R balls. Then we have shown that at least![\[ \binom{m_L}{2} + \binom{m_R}{2} + \binom{2n}{2} \]](http://latex.artofproblemsolving.com/b/8/e/b8eae8ccbe1a30891eafb4694bc01b1a08aedf63.png)
moves are required. By convexity and 
,
.
in order, take ball 
,
times. After this, the configuration is 
.
to 
in order, take ball 
times. After all of this, the configuration is 
.
,
moves, and step 2 needed 
moves. Adding these up gives 
.
.
pairs of cups must be swapped at some point (can't jump over). Next, we claim that each cup 
or 
.![\[\text{(A,B) where A is closer to endpoint} \geq \sum_{i=1}^{2n} \min(i-1,2n-i) = 1+\cdots + (n-1)+(n-1) + \cdots+1 = n^2-n\]](http://latex.artofproblemsolving.com/9/d/6/9d6e38920fd606461732aca3c8b868439953021d.png)
Thus, in total at least 
cups are needed. 
.
moves we flip 
into 
.![$[1,n]$](http://latex.artofproblemsolving.com/2/2/4/2240bb122bdadbe086c0a6291dcd5a3eb82ad5cf.png)
case, the other one is symmetric. For 
,
moves, and means that each cup 
will only need to go rightwards. 
.
to the endpoints 
.
moves for each iteration, for a total of 
,
