2017 AIME I Problems/Problem 11
We know that if is a median, then
will be the median of the medians.
WLOG, assume is in the upper left corner. One of the two other values in the top row needs to be below
, and the other needs to be above
. This can be done in
ways.
The other
can be arranged in
ways.
Finally, accounting for when
is in every other space, our answer is
. But we only need the last
digits, so
is our answer.
~Solution by SuperSaiyanOver9000, mathics42
Solution 2 (Complementary Counting with probability)
Notice that m can only equal 4, 5, or 6, and 4 and 6 are symmetric.
WLOG let
There is a chance that exactly one of 1, 2, 3 is in the same row.
There is a chance that the other two smaller numbers end up in the same row.
.