Min Number of Subsets of Strictly Increasing

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Source: India EGMO TST 2025 Day 1 P1

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#1 h Dec 13, 2024, 8:24 AM • 3 Y

Y by Supercali, GeoKing, NO_SQUARES

Let $n$ be a positive integer. Initially the sequence $0,0,\cdots,0$ ( $n$ times) is written on the board. In each round, Ananya choses an integer $t$ and a subset of the numbers written on the board and adds $t$ to all of them. What is the minimum number of rounds in which Ananya can make the sequence on the board strictly increasing?

Proposed by Shantanu Nene

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#2 h Dec 13, 2024, 8:57 AM • 1 Y

Y by GeoKing

Answer is $\lceil log_2 n\rceil$ .
Construction: Consider the binary representations of numbers' rows' (Like the first number is $0\dots 01$ ). In the $i.$ turn, add $2^{\lceil log_2 n\rceil-i+1}$ to the numbers whose $i.$ digit is $1$ . At the end of this process, $i.$ number on the row will be $i$ .
Example for the Construction: $0,0,0,0,0,0,0\rightarrow 0,0,0,4,4,4,4\rightarrow 0,2,2,4,4,6,6\rightarrow 1,2,3,4,5,6,7$ .
Lower Bound: Suppose that $2^{k-1}<n\leq 2^k$ and one can make the sequence increasing in $k-1$ turns (if it can be done in less than $k-1$ moves, then one can add $1$ to the last number for several times). For each number among $\{1,2,\dots,2^{k-1}+a\}$ , consider the binary strings where $i.$ number for $x$ is $1$ iff a number is added to $x$ in $i.$ turn. Note that each binary string must be different. However there cannot be more than $2^{k-1}$ distinct binary strings with $k-1$ digits which results in a contradiction as desired. $\blacksquare$

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#3 h Dec 14, 2024, 5:56 AM • 2 Y

Y by bin_sherlo, GeoKing

My problem! This one went through many iterations in a short amount of time.

We claim that the minimum number of moves needed is $\lceil \log_2(n) \rceil$ .

To show that $\lceil \log_2(n) \rceil$ moves are enough, it is enough to prove that $0,0, \dots, 0$ (of length $2^k$ ) can be made strictly increasing in $k$ moves, and then restrict our attention to the first $n$ positions, where $2^{k-1}<n \leq 2^k$ . Indeed, write the positions in binary from $0$ to $2^k-1$ , and on move $i$ , increment the positions that have a $2^{k-i}$ in their binary expansions by $2^{k-i}$ . At the end we will end up with $0,1,2, \dots, 2^k-1$ .

Now we show that at least $\lceil \log_2(n) \rceil$ moves are needed. For any sequence $\mathbf{a}$ , let $f(\mathbf{a})$ be the length of the longest non-increasing (i.e., weakly decreasing) subsequence in $\mathbf{a}$ . Suppose after applying the move once, we get the sequence $\mathbf{b}$ . We claim that $f(\mathbf{b}) \geq \frac{f(\mathbf{a})}{2}$ . Indeed, look at the longest non-increasing subsequence in $\mathbf{a}$ . Then there is a subsequence $\mathbf{a'}$ of this subsequence, having length at least $\frac{f(\mathbf{a})}{2}$ , such that either all elements of $\mathbf{a'}$ were selected in the move or none of the elements were selected (by PHP). In any case, $\mathbf{a'}$ remains non-increasing after the move, which proves the claim.

Now, if $k$ moves turn the sequence of all zeros strictly increasing, then $1 \geq \frac{n}{2^k}$ (since longest non-increasing subsequence in any strictly increasing sequence has length $1$ ). Therefore $k \geq \log_2(n)$ , as required.

Bonus:
Suppose instead of all zeros, the initial sequence is some $a_1 \geq a_2 \geq \cdots \geq a_n$ . Now, in terms of $n$ and the $a_i$ , what is the minimum number of moves needed to make the sequence strictly increasing?

This post has been edited 2 times. Last edited by Supercali, Dec 16, 2024, 1:21 AM

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#4 h Dec 14, 2024, 1:17 PM • 1 Y

Y by GeoKing

Does this work? (for the original problem)

Claim: If answer for $n$ is $k$ then $2^k >= n$ , i. e. $k \ge \lceil log_2(n)\rceil$
Proof: Let's say we add $t_1, t_2, ..., t_k$ in some order. These when summed in some order can give us at most $2^k$ values, and since all values above are to be distinct, $2^k >= n$

Claim: $k = \lceil \log_2(n) \rceil$ works.
Proof: It suffices to show this for $n = 2^a$ , for which we give an inductive constructon.

$a = 0$ is obvious.
If you have a construction for $n = 2^a$ , split $n = 2^{a+1}$ into a left side and right side of $2^a$ each.
Do the required operations on both the left and right side simultaneously to make them increasing; then do an operation on the entire right half to make its smallest element larger than the left side's largest, and we're done.

This post has been edited 1 time. Last edited by AshAuktober, Dec 14, 2024, 1:17 PM

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#5 h Dec 15, 2024, 12:34 PM • 1 Y

Y by GeoKing

Headsolved by me and blessed by Nimloth149131215208 since there is a little of Nimloth149131215208 in all of us

Solution Sketch
The minimum is $\lceil \log_2(n) \rceil$ .

Construction: Just simple binary works. $\blacksquare$

Proof of bound: For a sequence $\textbf{a}$ , denote $\textbf{a}'$ as the after effect of applying the operation in some manner and denote $f(\textbf{a})$ as the length of the longest subsequence of $\textbf{a}$ that is non-increasing(constant also works). The key realisation(by PHP) is that $f(\textbf{a}')\ge \frac{f(\textbf{a})}{2}$ but that kills, indeed, if $k$ is the number of moves, we get that $\frac{n}{2^k}\le 1$ , as desired

Alternate proof of bound due to Nimloth149131215208: Say we applied $k$ operations such that the end result is a strictly increasing sequence. For each of the $n$ elements in the list, consider the subset of the $k$ operations that acted on it. The key realisation is that no two different elements are associated to the same subset since that would contradict injectivity of the final sequence, and thus $2^k\ge n$ , just as we desired.

This post has been edited 1 time. Last edited by quantam13, Dec 16, 2024, 2:53 AM
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