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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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0 replies
jlacosta
May 1, 2025
0 replies
k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
Inspired by Kosovo 2010
sqing   0
2 minutes ago
Source: Own
Let $ a,b>0  , a+b\leq k $. Prove that
$$\left(1+\frac{1}{a(b+1)}\right)\left(1+\frac{1}{b(a+1)}\right)\geq\left(1+\frac{4}{k(k+2)}\right)^2$$Let $ a,b>0  , a+b\leq 2 $. Prove that
$$\left(1+\frac{1}{a(b+1)}\right)\left(1+\frac{1}{b(a+1)}\right)\geq \frac{9}{4} $$
0 replies
1 viewing
sqing
2 minutes ago
0 replies
Kosovo MO 2010 Problem 5
Com10atorics   20
N 15 minutes ago by sqing
Source: Kosovo MO 2010 Problem 5
Let $x,y$ be positive real numbers such that $x+y=1$. Prove that
$\left(1+\frac {1}{x}\right)\left(1+\frac {1}{y}\right)\geq 9$.
20 replies
+1 w
Com10atorics
Jun 7, 2021
sqing
15 minutes ago
Geometry Parallel Proof Problem
CatalanThinker   0
25 minutes ago
Source: No source found, just yet, please share if you find it though :)
Let M be the midpoint of the side BC of triangle ABC. The bisector of the exterior angle of point A intersects the side BC in D. Let the circumcircle of triangle ADM intersect the lines AB and AC in E and F respectively. If the midpoint of EF is N, prove that MN || AD.
I have done some constructions, but still did not quite get to the answer, see diagram attached below
0 replies
CatalanThinker
25 minutes ago
0 replies
Find all functions $f$ is strictly increasing : \(\mathbb{R^+}\) \(\rightarrow\)
guramuta   3
N 44 minutes ago by ja.
Find all functions $f$ is strictly increasing : \(\mathbb{R^+}\) \(\rightarrow\) \(\mathbb{R^+}\) such that:
i) $f(2x)$ \(\geq\) $2f(x)$
ii) $f(f(x)f(y)+x) = f(xf(y)) + f(x) $
3 replies
guramuta
Yesterday at 1:45 PM
ja.
44 minutes ago
Fourth powers and square roots
willwin4sure   39
N an hour ago by awesomeming327.
Source: USA TSTST 2020 Problem 4, by Yang Liu
Find all pairs of positive integers $(a,b)$ satisfying the following conditions:
[list]
[*] $a$ divides $b^4+1$,
[*] $b$ divides $a^4+1$,
[*] $\lfloor\sqrt{a}\rfloor=\lfloor \sqrt{b}\rfloor$.
[/list]

Yang Liu
39 replies
willwin4sure
Dec 14, 2020
awesomeming327.
an hour ago
Interesting inequalities
sqing   1
N an hour ago by sqing
Source: Own
Let $ a,b >0 $ and $ a^2-ab+b^2\leq 1 $ . Prove that
$$a^4 +b^4+\frac{a }{b +1}+ \frac{b }{a +1} \leq 3$$$$a^3 +b^3+\frac{a^2}{b^2+1}+ \frac{b^2}{a^2+1} \leq 3$$$$a^4 +b^4-\frac{a}{b+1}-\frac{b}{a+1} \leq 1$$$$a^4+b^4 -\frac{a^2}{b^2+1}- \frac{b^2}{a^2+1}\leq 1$$$$a^3+b^3 -\frac{a^3}{b^3+1}- \frac{b^3}{a^3+1}\leq 1$$
1 reply
sqing
an hour ago
sqing
an hour ago
Sum of 1/(a^5(b+2c))^2 at least 1/3 [USA TST 2010 2]
MellowMelon   42
N an hour ago by Adywastaken
Let $a, b, c$ be positive reals such that $abc=1$. Show that \[\frac{1}{a^5(b+2c)^2} + \frac{1}{b^5(c+2a)^2} + \frac{1}{c^5(a+2b)^2} \ge \frac{1}{3}.\]
42 replies
MellowMelon
Jul 26, 2010
Adywastaken
an hour ago
Weird function?
ItzsleepyXD   2
N an hour ago by ItzsleepyXD
Source: Own
Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \),
\[
f(x + f(2y)) + f(x^2 - y) = f(f(x)) f(x + 1) + 2y - f(y).
\]
2 replies
ItzsleepyXD
Apr 11, 2025
ItzsleepyXD
an hour ago
Almost similar one but more answer lol
ItzsleepyXD   0
an hour ago
Source: Own , Modified
Find all non decreasing functions or non increasing function $f \colon \mathbb{R} \to \mathbb{R}$ such that for all $x,y \in \mathbb{R}$

$$ f(x+f(y))=f(x)+f(y) \text{ or } f(f(f(x)))+y$$.
0 replies
ItzsleepyXD
an hour ago
0 replies
A lot of unexpected answer from non decreasing function
ItzsleepyXD   0
an hour ago
Source: Own
Find all non decreasing function $f : \mathbb{R} \to \mathbb{R}$ such that for all $x,y \in  \mathbb{R}$ and $m,n \in \mathbb{N}_0$ such that $m+n \neq 0$ there exist $m',n' \in \mathbb{N}_0$ such that $m'+n'=m+n+1$ and $$f(f^m(x)+f^n(y))=f^{m'}(x)+f^{n'}(y)$$. Note : $f^0(x)=x$ and $f^{n}(x)=f(f^{n-1}(x))$ for all $n \in \mathbb{N}$ . original
0 replies
ItzsleepyXD
an hour ago
0 replies
Cute Inequality
EthanWYX2009   0
2 hours ago
Let $a_1,\ldots ,a_n\in\mathbb R\backslash\{0\},$ determine the minimum and maximum value of
\[\frac{\sum_{i,j=1}^n|a_i+a_j|}{\sum_{i=1}^n|a_i|}.\]
0 replies
EthanWYX2009
2 hours ago
0 replies
Interesting inequality
sealight2107   3
N 2 hours ago by NguyenVanHoa29
Source: Own
Let $a,b,c>0$ such that $a+b+c=3$. Find the minimum value of:
$Q=\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{1}{a^3+b^3+abc}+\frac{1}{b^3+c^3+abc}+\frac{1}{c^3+a^3+abc}$
3 replies
sealight2107
May 6, 2025
NguyenVanHoa29
2 hours ago
Is this FE is solvable?
ItzsleepyXD   0
2 hours ago
Source: Own , If not appear somewhere before
Find all function $f : \mathbb{R} \to \mathbb{R}$ such that for all $x,y \in  \mathbb{R}$ . $$f(x+f(y))+f(x+y)=2x+f(y)+f(f(y))$$. Original
0 replies
ItzsleepyXD
2 hours ago
0 replies
Symmetry in Circumcircle Intersection
Mimii08   2
N 2 hours ago by mashumaro
Hi! Here's another geometry problem I'm thinking about, and I would appreciate any help with a proof. Thanks in advance!

Let AD and BE be the altitudes of an acute triangle ABC, with D on BC and E on AC. The line DE intersects the circumcircle of triangle ABC again at two points M and N. Prove that CM = CN.

Thanks for your time and help!
2 replies
Mimii08
4 hours ago
mashumaro
2 hours ago
ISL 2023 C2
OronSH   11
N Apr 24, 2025 by zRevenant
Source: ISL 2023 C2
Determine the maximal length $L$ of a sequence $a_1,\dots,a_L$ of positive integers satisfying both the following properties:
[list=disc]
[*]every term in the sequence is less than or equal to $2^{2023}$, and
[*]there does not exist a consecutive subsequence $a_i,a_{i+1},\dots,a_j$ (where $1\le i\le j\le L$) with a choice of signs $s_i,s_{i+1},\dots,s_j\in\{1,-1\}$ for which \[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\][/list]
11 replies
OronSH
Jul 17, 2024
zRevenant
Apr 24, 2025
ISL 2023 C2
G H J
G H BBookmark kLocked kLocked NReply
Source: ISL 2023 C2
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OronSH
1738 posts
#1 • 4 Y
Y by KevinYang2.71, peace09, nchuong1312, aidan0626
Determine the maximal length $L$ of a sequence $a_1,\dots,a_L$ of positive integers satisfying both the following properties:
  • every term in the sequence is less than or equal to $2^{2023}$, and
  • there does not exist a consecutive subsequence $a_i,a_{i+1},\dots,a_j$ (where $1\le i\le j\le L$) with a choice of signs $s_i,s_{i+1},\dots,s_j\in\{1,-1\}$ for which \[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\]
This post has been edited 1 time. Last edited by OronSH, Jul 17, 2024, 12:28 PM
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MarkBcc168
1595 posts
#2 • 5 Y
Y by OronSH, peace09, GrantStar, megarnie, aidan0626
Buffed version: replace $2^{2023}$ with any positive integer $n$.
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EeEeRUT
70 posts
#3 • 2 Y
Y by Quidditch, Want-to-study-in-NTU-MATH
Solved with Quidditch
The answer is \( L = 2^{2024} - 1 \). Next, we will demonstrate that there exists a sequence of length \( 2^{2024} - 1 \) that satisfies the conditions.

We will proceed with mathematical induction on \( n \), where \( a_i \leqslant 2^n \), showing that there exists a sequence of length \( 2^{n+1} - 1 \) that satisfies this condition.

1. Base case: Considering \( n = 1 \), a sequence of length \( 2^2 - 1 = 3 \) is possible. Clearly, the sequence \( 2, 1, 2 \) satisfies this.

2. Inductive step: Let \( a_1, a_2, \dots, a_{2^n-1} \) be a sequence that satisfies \( a_i \leqslant 2^{n-1} \). Consider the sequence:
- Double each element: \( 2a_1, 2a_2, \dots, 2a_{2^n-1} \) denoted as \( A \).
- Append \( 1 \) after \( A \).
- Reverse \( A \) and append it again after \( 1 \). Denote this sequence as \( B \).$$\underbrace{2a_1,2a_2,\dots,2a_{2^n-1}}_A,1,\underbrace{2a_{2^n-1},\dots,2a_2,2a_1}_B$$
It can be shown that every term in the sequence \( A \cup \{1\} \cup B \) is less than or equal to \( 2 \cdot 2^{n-1} = 2^n \). If there exists a subsequence \( k_i, k_{i+1}, \dots, k_j \) in \( A \) such that \( s_i k_i + s_{i+1} k_{i+1} + \dots + s_j k_j = 0 \), then:
\[
   s_i \left( \frac{k_i}{2} \right) + s_{i+1} \left( \frac{k_{i+1}}{2} \right) + \dots + s_j \left( \frac{k_j}{2} \right) = 0
   \]which contradicts the assumption from the induction hypothesis of case \( n-1 \). Similarly, if such a subsequence exists in \( B \), it contradicts the assumption from the induction hypothesis of case \( n-1 \). Therefore, the subsequence must include \( 1 \), which clearly contradicts the assumption because the sum of an odd number with an even number cannot be \( 0 \).

Thus, the sequence defined above has a length of \( 2^{n+1} - 1 \) and satisfies the condition.

3. Conclusion: Therefore, if \( a_i \leqslant 2^{2023} \), there exists a sequence of length \( 2^{2024} - 1 \).

Next, we will demonstrate that there is no sequence of length \( 2^{2024} \) that satisfies the above conditions.

Consider \( a_1, a_2, \dots, a_{2^{2024}} \). We will demonstrate by finding contradictions, assuming that there is no \( \pm a_i \pm a_{i+1} \dots \pm a_{j} = 0 \), when \( 1 \leqslant i \leqslant j \leqslant 2^{2024} \).$$b_{i+1}=\left\{\begin{array}{l}{b_i - a_{i+1}\text{ if }b_i>0}\\{b_i + a_{i+1}\text{ if }b_i<0}\end{array}\right.$$
By considering the sequence \( b_i \), which is defined by \( b_1 = a_1 \) and for any integer \( i \) where \( 1 \leqslant i \leqslant 2^{2024} \), it is easy to see that:
\[ b_i = s_1 a_1 + s_2 a_2 + s_3 a_3 + \dots + s_i a_i \]where \( s_1, s_2, \dots, s_i \in \{1, -1\} \), defined by \( s_k = \text{sgn}(b_{k-1}) \) for all integers \( k \) where \( 1 < k \leq i \), and \( s_1 = 1 \).

We can observe that for every \( i \), it holds that \( -(2^{2023} - 1) \leq b_i \leq 2^{2023} \).

Since:
\[ -(2^{2023} - 1) \leq b_i \leq 2^{2023} \]
The possible values of \( b_i \) are:
\[ -(2^{2023} - 1), -(2^{2023} - 2), \dots, -1, 1, 2, \dots, 2^{2023} \]
There are \( 2^{2024} \) possible values for \( b_i \). However, the sequence \( b_i \) consists of \( 2^{2024} \) elements.

By the pigeonhole principle, at least two numbers in the sequence \( b_i \) must be equal. Suppose \( b_s \) and \( b_t \) are equal, where \( s < t \). Then:
\[ b_t - b_s = 0 \Rightarrow s_{s+1} a_{s+1} + s_{s+2} a_{s+2} + \dots + s_t a_t = 0 \]
This contradicts our earlier assumption that there is no subsequence \( \pm a_i \pm a_{i+1} \dots \pm a_{j} = 0 \).

Therefore, we can conclude that there does not exist a sequence of length \( 2^{2024} \) that satisfies the conditions outlined.
This post has been edited 1 time. Last edited by EeEeRUT, Mar 20, 2025, 2:47 AM
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Marinchoo
407 posts
#4 • 3 Y
Y by lomta, isomoBela, TensorGuy666
The answer is $L = 2^{k+1}-1$ where $2023$ is replaced by $k$. Call a consecutive subsequence $a_i, a_{i+1}, \ldots, a_j$ bad if there exist $s_i, s_{i+1}, \ldots, s_j\in\{-1, 1\}$ such that
\[a_is_i+a_{i+1}s_{i+1}+\ldots+a_js_j=0.\]The construction for $L = 2^{k+1} - 1$ is by induction: start with the sequence $s_1 = 1$ for $k=1$ and construct the sequence $s_{k+1}$ as $2\cdot s_k, 1, 2\cdot s_k$ (that is, $s_{k+1}$ is two copies of the elements of $s_k$, multiplied by $2$, separated by a $1$). If $s_k$ has no bad subsequence, so does $s_{k+1}$ due to parity and the induction hypothesis on the two copies of $s_k$.

To prove $L = 2^k$ doesn't work, we fix signs for each $a_i$ by focusing on the prefixes. That is, we start by picking $s_1 = 1$ and then pick $s_{i+1}$ depending on the previously picked $s_1, s_2, \ldots, s_i$, adhering to the following rules:
  • If $a_1s_1+a_2s_2+\ldots+a_is_i < 2^{k}$, then $s_{i+1} = 1$.
  • Else, $s_{i+1} = -1$.
Note that at any point, the prefix sum $c_i = a_1s_1+a_2s_2+\ldots+a_is_i$ is non-negative and strictly less than $2^{k+1}$ as $1\leq a_i\leq 2^k$. Therefore, there exists an index $i$ with $c_i=0$, which corresponds to a bad subsequence, or two indices $i<j$, such that $c_i=c_j$. However, $c_j - c_i = 0$, so $a_i, a_{i+1}, \ldots, a_j$ is a bad subsequence, and we're done once again. This finishes.
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Tyx2019
22 posts
#5 • 1 Y
Y by NCbutAN
The answer is $2^{n+1}-1$ where we replace $2023$ with $n$. The construction is the same as the ones provided above.

Lemma
Proof
The rest
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MarkBcc168
1595 posts
#6
Y by
Since no one has posted the solution to the buffed version yet, I will.

If we replace $2^{2023}$ with general $n$, the answer is $L=2^{k+1}-1$ where $k=\lfloor\log_2 n\rfloor$.

Construction for $\boldsymbol{n=2^k}$ and $\boldsymbol{L=2^{k+1}-1}$

We take $a_i=2^{k-\nu_2(i)}$. For example, if $k=3$, then the sequence is
$$8\ 4\ 8\ 2\ 8\ 4\ 8\ \textbf 1\ 8\ 4\ 8\ 2\ 8\ 4\ 8.$$To see why this works, note that the subsequence cannot go through $1$ in the middle (otherwise, the sum is odd). Thus, it must be strictly within the left half or the right half. However, each half is the construction for $k-1$ multiplied by $2$, so we can repeat this argument to realize that no subsequence and signing work.
Bound for $\boldsymbol{n=2^{k+1}-1}$ and $\boldsymbol{L=2^{k+1}}$

We use induction on $k$. The base case $k=0$ is clear. For the inductive step, the idea is to consider the partial sums $s_i = a_1+a_2+\dots+a_i$ (and $s_0=0$). We have $2^{k+1}+1$ of these, so by Pigeonhole, at least $2^k+1$ of these have the same parity. This means that we can split group the numbers into $2^k$ blocks so that each block has even sum. For example, if if $L = 8$ and $s_1$, $s_3$, $s_4$, $s_5$, $s_8$ have the same parity, then the blocks are $(a_2\ a_3)\ (a_4)\ (a_5)\ (a_6\ a_7\ a_8)$.

Within each block, we may choose $\pm$ signs so that the results is in $[0, 2^{k+1}-2]$ (by choosing one by one and maintain that interval). If any of these were $0$, we are done. Otherwise, we divide the result of each block by $2$. We then get $2^k$ numbers in $[1, 2^k-1]$, and so we can use the induction hypothesis for $k-1$. Done.
This post has been edited 2 times. Last edited by MarkBcc168, Aug 17, 2024, 8:32 AM
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blackbluecar
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Replace $2023$ with $n$. We claim the answer is $L=2^{n+1}-1$.

First, we consider the following sequences of integers $A_0,A_1, \ldots$
\[ A_0=1 \]\[ A_1=2, 1, 2 \]\[ A_2=4, 2, 4, 1, 4, 2, 4\]\[ A_3=8, 4, 8, 2, 8, 4, 8, 1, 8, 4, 8, 2, 8, 4, 8 \]\[ \cdots \]To get from $A_k$ to $A_{k+1}$, we slot $2^{k+1}$ between any pair of numbers, as well as the two ends. We claim that $A_n$ is our desired equality case for $L=2^{n+1}-1$. First, we note that $A_n$ has length $2^{n+1}-1$ and has no entries exceeding $2^n$.

Notice that in this construction, given any consecutive instances of $2^m$, the entry at their midpoint is strictly less than $2^m$. Thus, given any contiguous block $B$ in $A_n$, the minimal element appears exactly once. So, we cannot partition $B$ into two multisets with equal sum, since the multiset with the unique minimum in $B$ will have a sum with a strictly lower $\nu_2$.

Now, all that is left to show is that $L \geq 2^{n+1}$ has such a contiguous block. AFTOC that no such block exists. Consider a sequence $e_1,e_2, \ldots, e_L \in \{-1,1\}$ with the following inductive definition
\[ e_k = 
    \begin{cases}
        1 & \text{if } e_1a_1+e_2a_2+ \cdots e_ka_k \leq 0\\
        -1 &  \text{otherwise}
    \end{cases} \]
It follows that $|e_1a_1+\cdots +e_ka_k| \leq 2^n$ for all $k \leq L$. By our assumption, $e_1a_1+\cdots +e_ka_k \not = 0$ for any $k$. Moreover, if $k \geq 2$ and $e_1a_1+\cdots +e_ka_k = \pm 2^n$ then $e_1a_1+\cdots +e_{k-1}a_{k-1} = 0$ which contradicts our assumption. Thus, if we consider the set $S$ denoting all possible values of $e_1a_1+\cdots +e_ka_k$ we see that $0 \not \in S$ and at least one of $2^n$ or $-2^n$ are not in $S$. Thus, $|S| \leq 2^{n+1}-1$ implying by PHP that some $d \in S$ appears at least \[ \left \lceil \frac{2^{n+1}}{|S|} \right \rceil \geq 2 \]times in $S$. Thus, there is some $i<j$ where \[ e_1a_1+\cdots +e_ia_i = e_1a_1+\cdots +e_ja_j = d \]but then, \[ e_{i+1}a_{i+1}+ \cdots +e_ja_j =0 \]Contradicting our assumption. $\blacksquare$
This post has been edited 4 times. Last edited by blackbluecar, Jul 29, 2024, 12:47 AM
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dkedu
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We claim the answer is $2^{n+1} - 1$ for general $2^n$.

The construction follows by making $\nu_2(a_i) = n - \nu_2(i)$. For example, when $n = 3$, we get construction $8\:4\:8\:2\:8\:4\:8\:1\:8\:4\:8\:2\:8\:4\:8$. One can see that either $i \le j < 8$ or $8 < i \le j$, since otherwise, the sum would have exactly one factor of $2$ which presents a contradiction. This split the construction into two halves. So divide by $2$ for the remaining halves and repeat the argument.

We will use induction to prove the bound, with the base case of $n = 1$ not being hard to see as the maximal sequence is $2, 1, 2$. We will now assume that for any $2^{n-1}$ positive integers less than $2^{n-2}$, there will be a consecutive subsequence that can be made into $0$ and prove that any sequence of length $2^{n}$ has a subsequence that can be made into $0$. Let the positions of the odd terms be $a_{n_1}, a_{n_2}, \ldots, a_{n_k}$ where $\{n_i\}_{i=1}^k$ is an increasing sequence. We can combine terms $a_{n_{2k-1}}, \ldots, a_{n_{2k}}$, picking signs to ensure that $|e_{n_{2k-1}}a_{n_{2k-1} + \cdots + e_{n_{2k}}a_{n_{2k}}}| \le 2^{n-1}$. If it is $0$, we have found a subsequence, so assume not. Now we get that the all terms are even. So divide by $2$ and apply the inductive hypothesis. If there are not enough terms, consider combining terms $a_{n_{2k}}, \ldots, a_{n_{2k+1}}$ instead and now this is guaranteed to have at least $2^{n-1}$ terms, so we are done by induction.
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SomeonesPenguin
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The answer is $2^{n+1}-1$ in general (numbers are less than $2^{n}$).

Proof

Construction
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cj13609517288
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The answer is $\boxed{2^{2024}-1}$, achieved by the sequence $a_k=2^{2023-\nu_2(k)}$.

To prove that $2^{2024}$ is impossible, consider starting from $0$, and for each new term in the sequence, either adding or subtracting it to stay within $[-(2^{2023}-1),2^{2023}]$. Note that if we ever repeat a number or get to $0$, we lose, because then we just found a substring that works. But there are only $2^{2024}-1$ nonzero numbers in this interval, contradiction. $\blacksquare$
This post has been edited 1 time. Last edited by cj13609517288, Aug 19, 2024, 1:17 PM
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Aiden-1089
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The answer is $2^{2024}-1$.
For any sequence $a_1, \dots, a_L$, define $b_i$ for $0 \leq i \leq L$ and $s_i$ for $1 \leq i \leq L$ as follows:
$b_0=0, s_i = \begin{cases} 1 & \text{if } b_{i-1}  \leq 0 \\ -1 &\text{if } b_{i-1}>0 \end{cases}, b_i=b_{i-1}+s_ia_i$ for $1 \leq i \leq L$.
Note that if $b_i=b_j$ for some $i<j$, then $s_{i+1}a_{i+1} + \cdots + s_ja_j =0$.

We claim that $-2^{2023} < b_i \leq 2^{2023}$ for all $i$. We prove this by induction.
Clearly this is true for $i=0$.
For $i>0$, if $b_{i-1} \leq 0$, that means $-2^{2023}<-2^{2023}+a_i<b_i=b_{i-1}+a_i \leq 0+2^{2023}=2^{2023}$. If $b_{i-1}>0$, $-2^{2023}=0-2^{2023}<b_i=b_{i-1}-a_i \leq 2^{2023}-a_i < 2^{2023}$, so we are done.
Then, if $L+1>2^{2023}+2^{2023}=2^{2024}$, there must exist $b_i=b_j$ where $i<j$ by pigeonhole principle. So $L \geq 2^{2024}-1$.

We now give a construction for $L=2^{2024}-1$.
Put $a_i=2^{2023-\nu_2(i)}$ for $1 \leq L \leq 2^{2024}$. Assume there exists $i \leq j$, $s_i, \dots, s_j$ such that the sum $S=s_ia_i+ \cdots s_ja_j=0$.
If $i \leq 2^{2023} \leq j$, then $S \equiv 1 \pmod{2}$, contradiction. Since $a_i=2^{2023-\nu_2(i)}=2^{2023-\nu_2(2^{2024}-i)}=a_{2^{2024}-i}$, we may WLOG assume $i,j <2^{2023}$.
Then if $i \leq 2^{2022} \leq j$, then $S \equiv 2 \pmod{4}$, which is again a contradiction. $a_i=2^{2023-\nu_2(i)}=2^{2023-\nu_2(2^{2023}-i)}=a_{2^{2023}-i}$, so WLOG assume $i,j < 2^{2022}$.
Inductively, we may show that $i,j<2^1$, so $i=j=1$, contradiction. So this construction works, so we are done. $\square$
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zRevenant
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Answer: $2^{2024}-1$.

Construction:

Let's define sequence $A_0$ as just number $1$. For convenience we will mean sequence $kB$ as sequence $B$, every term of which is increased by $k$ times. Now, define other sequences recurrently:

$A_{n+1}=2A_n, 1, 2A_n$

For example, $A_1=212$ and $A_2=4241424$.

Now, this construction works because there is only one $1$ meaning that by parity if it is included in the $s_i, ..., s_j$ the sum can't be zero. On the other hand, the parts on the $RHS$ and $LHS$ of $1$ are working by induction.

Bound:

We will prove by induction a stronger statement, that if we have a sequence bigger than that, we can find a sequence of '+' and '-' such that the amount of '+' equals to the amount of '-'. Suppose that we have a sequence of length greater or equal to $2^{n+1}-1$ if $a_i \le 2^n$. Then, we may assume that the sequence has exactly $2^{n+1}$ terms, say $a_1, a_2, ..., a_{2^{n+1}}$. We then define $b_i=\mid a_{2i} - a_{2i-1} \mid$. This gives us $2^n$ numbers that are between $0$ and $2^n$. Let $c_i=b_i-2^{n-1}$. By induction, we can assume that this sequence $c_i$ has a sequence of '+' and '-' that we searched for. Then the sequence $b_i$ can be put into $0$ yet we discard our statement about the amount of '+' and '-'. However, because of $b_i$'s definition, when we put signs next to $a_{2i}$ and $a_{2i-1}$ these signs will be different (that form $b_i$). Therefore the amount of '+' and '-' is again equal so induction is complete.
This post has been edited 1 time. Last edited by zRevenant, Apr 24, 2025, 9:47 AM
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