Solution APMO 2017 ( Number 3)

Solution APMO 2017 ( Number 3)

Problem 3.

Let A(n) denote the number of sequences a_1\geq a_2\geq ...\geq a_k of positive integers for which a_1+a_2+...+a_k=n and each a_i+1 is a power of two (i = 1, 2, . . . , k). Let B(n) denote the number of sequences b_1\geq b_2\geq ...\geq b_m of positive integers for which b_1+...+b_m=n and each inequality b_i\geq 2b_{i+1} holds (j = 1, 2, . . . , m − 1). Prove that A(n) = B(n) for every positive integer n.

Also Read:

Also Read :

Solution :

We say that a sequence a_1\geq a_2\geq ...\geq a_k of positive integers has type A if a_i+1 is a power of two for i = 1, 2, . . . , k. We say that a sequence b_1\geq b_2\geq ...\geq b_m of positive integer has type B if b_i\geq 2b_{i+1} for j = 1, 2, . . . , m − 1.

Recall that the binary representation of a positive integer expresses it as a sum of distinct powers of two in a unique way. Furthermore, we have the following formula for every positive integer N

2^N-1=2^{N-1}+2^{N-2}+...+2^1+2^0.

Given a sequence a_1\geq a_2\geq ...\geq a_k of type A, use the preceding formula to express each term as a sum of powers of two. Write these powers of two in left-aligned rows, in decreasing order of size. By construction, ai is the sum of the numbers in the ith row. For example, we obtain the following array when we start with the type A sequence 15, 15, 7, 3, 3, 3, 1.

apmo3.PNG

Define the sequence b_1,b_2,...,b_n by setting b_i to be the sum of the numbers in the jth column of the array. For example, we obtain the sequence 27, 13, 5, 2 from the array above. We now show that this new sequence has type B. This is clear from the fact that each column in the array contains at least as many entries as the column to the right of it and that each number larger than 1 in the array is twice the number to the right of it. Furthermore, it is clear that a_1+a_2+...+a_k=b_1+b_2+...+b_m , since both are equal to the sum of all the entries in the array.

We now show that we can do this operation backwards. Suppose that we are given a type B sequence b_1\geq b_2\geq ...\geq b_m . We construct an array inductively as follows:

  • We fill b_m left-aligned rows with the numbers 2^{m-1},2^{m-2},...,2^1,2^0
  • Then we fill b_{m-1}-2b_m left aligned rows with the numbers 2^{m-2},2^{m-3},...2^1,2^0
  • Then we fill b_{m-2}-2b_{m-1} left aligned rows with the numbers 2^{m-3},2^{m-4},...2^1,2^0 and the so on.
  • In the last step we fill b_1-2b_2 left aligned rows with the number 1.

For example, if we start with the type B sequence 27, 13, 5, 2, we obtain once again the array above. We define the sequence a_1,a_2,...,a_k by setting a_i to be the sum of the numbers in the ith row of the array. By construction, this sequence has type A. Furthermore, it is clear that a_1+...+a_k=b_1+...+b_m, since once again both sums are equal to the sum of all the entries in the array.

We have defined an operation that starts with a sequence of type A, produces an array whose row sums are given by the sequence, and outputs a sequence of type B corresponding to the column sums. We have also defined an operation that starts with a sequence of type B, produces an array whose column sums are given by the sequence, and outputs a sequence of type A corresponding to the row sums. The arrays produced in both cases comprise left- aligned rows of the form 2^{N-1},2{N-2},...,2^1,2^0,with non-increasing lengths. Let us refer to arrays obeying these properties as marvelous.

To show that these two operations are inverses of each other, it then suffices to prove that marvelous arrays are uniquely defined by either their row sums or their column sums. The former is obviously true and the latter arises from the observation that each step in the above inductive algorithm was forced in order to create a marvelous array with the prescribed column sums.

Thus, we have produced a bijection between the sequences of type A with sum n and the sequences of type B with sum n. So we can conclude that A(n) = B(n) for every positive integer n.

 

Remark The solution above provides a bijection between type A and type B sequences via an algorithm. There are alternative ways to provide such a bijection. For example, given the numbers a_1\geq ...\geq a_k we may define the b_i{}'s as

b_{j}=\sum_{i}\left \lfloor \frac{a_i+1}{2^j} \right \rfloor

Conversely, given the numbers b_1\geq ...\geq b_m, one may define the a_i{}'s by taking, as in the solution, b_m numbers equal to 2^m-1,b_{m-1}-2b_m numbers equal to 2^{m-1}-1,…, and b_1-2b_2 number equal to 2^1-1. One now needs to verify that these maps are mutually inverse.

%d bloggers like this: