2024 AIME II Problems/Problem 3
Problem
Find the number of ways to place a digit in each cell of a 2x3 grid so that the sum of the two numbers formed by reading left to right is , and the sum of the three numbers formed by reading top to bottom is
. The grid below is an example of such an arrangement because
and
.
Solution 1
Consider this table:
We note that , because
, meaning it never achieves a unit's digit sum of
otherwise. Since no values are carried onto the next digit, this implies
and
. We can then simplify our table into this:
We want , or
, or
. Since zeroes are allowed, we just need to apply stars and bars on
, to get
. ~akliu
Solution 2
Let this be the table:
Then,
and
. We look at the first equation. Since
and
are digits,
and
, so
, which means
for the last digit to be
. Similarly,
and
.