Difference between revisions of "2023 AIME II Problems/Problem 10"
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Number of divisors <cmath>(11+1) \cdot (5+1) \cdot (1 + 1) = \boxed{144}.</cmath> | Number of divisors <cmath>(11+1) \cdot (5+1) \cdot (1 + 1) = \boxed{144}.</cmath> | ||
'''vladimir.shelomovskii@gmail.com, vvsss''' | '''vladimir.shelomovskii@gmail.com, vvsss''' | ||
+ | |||
+ | ===Simpler Way to Finish=== | ||
+ | Note that there are 6!/(2!2!2!) ways for the possible column arrangements. Then, each of those columns have two numbers, which can be permuted either way, so we multiply by 2!. Finally, we have 4 of each of 0, 1, and 2 mod 3. Thus, let us put those 4 who are 0 mod 3, and we get 4! ways to do this. Similarly can be said for 1 and 2 mod 3, so we multiply by (4!)^3. Thus, the final answer is 6!/(2!2!2!)*2!*(4!)^3 which leads us to the same solution as above. | ||
== See also == | == See also == |
Revision as of 00:03, 31 December 2023
Problem
Let be the number of ways to place the integers
through
in the
cells of a
grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by
One way to do this is shown below. Find the number of positive integer divisors of
Solution 1
We replace the numbers which are 0 mod(3) to 0, 1 mod(3) to 1, and 2 mod(3) to 2.
Then, the problem is equivalent to arranging 4 0's,4 1's, and 4 2's into the grid (and then multiplying by to account for replacing the remainder numbers with actual numbers) such that no 2 of the same numbers are adjacent.
Then, the numbers which are vertically connected must be different. 2 1s must be connected with 2 2s, and 2 1s must be connected with 2 2s (vertically), as if there are less 1s connected with either, then there will be 2s or 3s which must be connected within the same number. For instance if we did did: - 12 x 3 - 13 x 1 Then we would be left with 23 x1, and 2 remaining 3s which aren't supposed to be connected, so it is impossible. Similar logic works for why all 1s can't be connected with all 2s.
Thus, we are left with the problem of re-arranging 2x 12 pairs, 2x 13 pairs, 2x 23 pairs. Notice that the pairs can be re-arranged horizontally in any configuration, but when 2 pairs are placed adjacent there is only 1 configuration for the rightmost pair to be set after the leftmost one has been placed.
We have ways to horizontally re-arrange the pairs, with 2 ways to set the initial leftmost column. Thus, there are 180 ways to arrange the pairs. Accounting for the initial simplification of the problem with 1-12 -> 0-3, we obtain the answer is:
The number of divisors is:
~SAHANWIJETUNGA
Solution 2 (Archaeological)
Let's carry out an archaeological study, that is, we will find the bones (the base) and "build up the meat."
1. Let "bones" of number be
Then the “skeleton” of the original table is
By condition, the table cannot have a column of two identical numbers (the difference of such numbers is a multiple of
There are zeros,
ones and
twos in the table (the order of the numbers in the columns is not important).
Therefore, there cannot be more than two identical columns (otherwise there will be four identical numbers in the remaining three, that is, the numbers in one column are the same).
Any such table has exactly columns
columns
and
columns
The number of possible tables of six elements of three types is
The number of possible tables of six elements, if the order of the digits is important, is
2. We are looking for the total number of options.
The column
can contain the following
pairs of numbers (the order is not important, it is already taken into account)
The second column
can contain
pairs of numbers (excluding the two that are in the first column).
Similarly with the other two columns, i.e. the total number of choices
Number of divisors
vladimir.shelomovskii@gmail.com, vvsss
Simpler Way to Finish
Note that there are 6!/(2!2!2!) ways for the possible column arrangements. Then, each of those columns have two numbers, which can be permuted either way, so we multiply by 2!. Finally, we have 4 of each of 0, 1, and 2 mod 3. Thus, let us put those 4 who are 0 mod 3, and we get 4! ways to do this. Similarly can be said for 1 and 2 mod 3, so we multiply by (4!)^3. Thus, the final answer is 6!/(2!2!2!)*2!*(4!)^3 which leads us to the same solution as above.
See also
2023 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.