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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
Thursday at 11:16 PM
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.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
Thursday at 11:16 PM
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
Infinitely many numbers of a given form
Stefan4024   19
N 9 minutes ago by cursed_tangent1434
Source: EGMO 2016 Day 2 Problem 6
Let $S$ be the set of all positive integers $n$ such that $n^4$ has a divisor in the range $n^2 +1, n^2 + 2,...,n^2 + 2n$. Prove that there are infinitely many elements of $S$ of each of the forms $7m, 7m+1, 7m+2, 7m+5, 7m+6$ and no elements of $S$ of the form $7m+3$ and $7m+4$, where $m$ is an integer.
19 replies
Stefan4024
Apr 13, 2016
cursed_tangent1434
9 minutes ago
Very easy case of a folklore polynomial equation
Assassino9931   1
N 11 minutes ago by iamnotgentle
Source: Bulgaria EGMO TST 2025 P6
Determine all polynomials $P(x)$ of odd degree with real coefficients such that $P(x^2 + 2025) = P(x)^2 + 2025$.
1 reply
+2 w
Assassino9931
an hour ago
iamnotgentle
11 minutes ago
Process on scalar products and permutations
Assassino9931   2
N 18 minutes ago by Assassino9931
Source: RMM Shortlist 2024 C1
Fix an integer $n\geq 2$. Consider $2n$ real numbers $a_1,\ldots,a_n$ and $b_1,\ldots, b_n$. Let $S$ be the set of all pairs $(x, y)$ of real numbers for which $M_i = a_ix + b_iy$, $i=1,2,\ldots,n$ are pairwise distinct. For every such pair sort the corresponding values $M_1, M_2, \ldots, M_n$ increasingly and let $M(i)$ be the $i$-th term in the list thus sorted. This denes an index permutation of $1,2,\ldots,n$. Let $N$ be the number of all such permutations, as the pairs run through all of $S$. In terms of $n$, determine the largest value $N$ may achieve over all possible choices of $a_1,\ldots,a_n,b_1,\ldots,b_n$.
2 replies
Assassino9931
2 hours ago
Assassino9931
18 minutes ago
Square problem
Jackson0423   2
N 24 minutes ago by Jackson0423
Construct a square such that the distances from an interior point to the vertices (in clockwise order) are
1,7,8,4 respectively.
2 replies
Jackson0423
Yesterday at 4:08 PM
Jackson0423
24 minutes ago
IMO Shortlist Problems
ABCD1728   2
N 33 minutes ago by ABCD1728
Source: IMO official website
Where can I get the official solution for ISL before 2005? The official website only has solutions after 2006. Thanks :)
2 replies
ABCD1728
Yesterday at 12:44 PM
ABCD1728
33 minutes ago
Estimate on number of progressions
Assassino9931   1
N 33 minutes ago by BlizzardWizard
Source: RMM Shortlist 2024 C4
Let $n$ be a positive integer. For a set $S$ of $n$ real numbers, let $f(S)$ denote the number of increasing arithmetic progressions of length at least two all of whose terms are in $S$. Prove that, if $S$ is a set of $n$ real numbers, then
\[ f(S) \leq \frac{n^2}{4} + f(\{1,2,\ldots,n\})\]
1 reply
Assassino9931
2 hours ago
BlizzardWizard
33 minutes ago
2^x+3^x = yx^2
truongphatt2668   10
N 36 minutes ago by MittenpunktpointX9
Prove that the following equation has infinite integer solutions:
$$2^x+3^x = yx^2$$
10 replies
truongphatt2668
Apr 22, 2025
MittenpunktpointX9
36 minutes ago
find the radius of circumcircle!
jennifreind   1
N 37 minutes ago by ricarlos
In $\triangle \rm ABC$, $  \angle \rm B$ is acute, $\rm{\overline{BC}} = 8$, and $\rm{\overline{AC}} = 3\rm{\overline{AB}}$. Let point $\rm D$ be the intersection of the tangent to the circumcircle of $\triangle \rm ABC$ at point $\rm A$ and the perpendicular bisector of segment $\rm{\overline{BC}}$. Given that $\rm{\overline{AD}} = 6$, find the radius of the circumcircle of $\triangle \rm BCD$.
IMAGE
1 reply
jennifreind
Yesterday at 2:12 PM
ricarlos
37 minutes ago
A folklore polynomial game
Assassino9931   1
N an hour ago by YaoAOPS
Source: RMM Shortlist 2024 A1, also Bulgaria Regional Round 2016, Grade 12
Fix a positive integer $d$. Yael and Ziad play a game as follows, involving a monic polynomial of degree $2d$. With Yael going first, they take turns to choose a strictly positive real number as the value of one of the coecients of the polynomial. Once a coefficient is assigned a value, it cannot be chosen again later in the game. So the game
lasts for $2d$ rounds, until Ziad assigns the final coefficient. Yael wins if $P(x) = 0$ for some real
number $x$. Otherwise, Ziad wins. Decide who has the winning strategy.
1 reply
Assassino9931
2 hours ago
YaoAOPS
an hour ago
Game on board with gcd and lcm
Assassino9931   0
an hour ago
Source: Bulgaria EGMO TST 2025 P3
On the board are written $n \geq 2$ positive integers with least common multiple $K$ and greatest common divisor $1$. It is known that $K$ is not a perfect square and is not among the initially written numbers. Two players $A$ and $B$ play the following game, taking turns alternatingly, with $A$ starting first. In a move the player has to write a number which has not been written so far, by taking two distinct integers $a$ and $b$ from the board and write LCM$(a,b)$ or LCM$(a,b)$/$a$. The player who writes $1$ or $K$ loses. Who has a winning strategy?
0 replies
Assassino9931
an hour ago
0 replies
Popular children at camp with algebra and geometry
Assassino9931   0
2 hours ago
Source: RMM Shortlist 2024 C3
Fix an odd integer $n\geq 3$. At a maths camp, there are $n^2$ children, each of whom selects
either algebra or geometry as their favourite topic. At lunch, they sit at $n$ tables, with $n$ children
on each table, and start talking about mathematics. A child is said to be popular if their favourite
topic has a majority at their table. For dinner, the students again sit at $n$ tables, with $n$ children
on each table, such that no two children share a table at both lunch and dinner. Determine the
minimal number of young mathematicians who are popular at both mealtimes. (The minimum is across all sets of topic preferences and seating arrangements.)
0 replies
Assassino9931
2 hours ago
0 replies
Triangles in dissections
Assassino9931   0
2 hours ago
Source: RMM Shortlist 2024 C2
Fix an integer $n\geq 3$ and let $A_1A_2\ldots A_n$ be a convex polygon in the plane. Let $\mathcal{M}$ be the set of all midpoints $M_{i,j}$ of segments $A_iA_j$ where $i\neq j$. Assume that all of these midpoints are distinct, i.e. $\mathcal{M}$ consists of $\frac{n(n-1)}{2}$ elements. Dissect the polygon $M_{1,2}M_{2,3}\ldots M_{n,1}$ into triangles so that the following hold:

(1) The intersection of every two triangles (interior and boundary) is either empty or a common
vertex or a common side.
(2) The vertices of all triangles lie in M (not all points in M are necessarily used).
(3) Each side of every triangle is of the form $M_{i,j}M_{i,k}$ for some pairwise distinct indices $i,j,k$.

Prove that the total number of triangles in such a dissection is $3n-8$.
0 replies
Assassino9931
2 hours ago
0 replies
Tangency geo
Assassino9931   0
2 hours ago
Source: RMM Shortlist 2024 G1
Let $ABC$ be an acute triangle with $\angle ABC > 45^{\circ}$ and $\angle ACB > 45^{\circ}$. Let $M$ be the midpoint of the side $BC$. The circumcircle of triangle $ABM$ intersects the side $AC$ again at $X\neq A$ and the circumcircle of triangle $ACM$ intersects the side $AB$ again at $Y\neq A$. The point $P$ lies on the perpendicular bisector of the segment $BC$ so that the points $P$ and $A$ lie on the same side of $XY$ and $\angle XPY = 90^{\circ} + \angle BAC$. Prove that the circumcircles of triangles $BPY$ and $CPX$ are tangent.
0 replies
1 viewing
Assassino9931
2 hours ago
0 replies
Inequalities in real math research
Assassino9931   0
2 hours ago
Source: RMM Shortlist 2024 A3
For a positive integer $n$ denote $F_n(x_1,x_2,\ldots,x_n) = 1 + x_1 + x_1x_2 + \cdots +x_1x_2\ldots x_n$. For any real numbers $x_1\geq x_2 \geq \ldots \geq x_k \geq 0$ prove that
\[ \prod_{i=1}^k F_i(x_{k-i+1},x_{k-i+2},\ldots,x_k) \geq \prod_{i=1}^k F_i(x_i,x_i,\ldots,x_i)\]
0 replies
Assassino9931
2 hours ago
0 replies
Sequence with condition on million consecutive terms
jbaca   13
N Apr 26, 2025 by Ilikeminecraft
Source: 2021 Iberoamerican Mathematical Olympiad, P3
Let $a_1,a_2,a_3, \ldots$ be a sequence of positive integers and let $b_1,b_2,b_3,\ldots$ be the sequence of real numbers given by
$$b_n = \dfrac{a_1a_2\cdots a_n}{a_1+a_2+\cdots + a_n},\ \mbox{for}\ n\geq 1$$Show that, if there exists at least one term among every million consecutive terms of the sequence $b_1,b_2,b_3,\ldots$ that is an integer, then there exists some $k$ such that $b_k > 2021^{2021}$.
13 replies
jbaca
Oct 20, 2021
Ilikeminecraft
Apr 26, 2025
Sequence with condition on million consecutive terms
G H J
G H BBookmark kLocked kLocked NReply
Source: 2021 Iberoamerican Mathematical Olympiad, P3
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jbaca
225 posts
#1
Y by
Let $a_1,a_2,a_3, \ldots$ be a sequence of positive integers and let $b_1,b_2,b_3,\ldots$ be the sequence of real numbers given by
$$b_n = \dfrac{a_1a_2\cdots a_n}{a_1+a_2+\cdots + a_n},\ \mbox{for}\ n\geq 1$$Show that, if there exists at least one term among every million consecutive terms of the sequence $b_1,b_2,b_3,\ldots$ that is an integer, then there exists some $k$ such that $b_k > 2021^{2021}$.
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blackbluecar
302 posts
#2 • 1 Y
Y by PRMOisTheHardestExam
Notice that if $b_i$ and $b_j$ are consecutive integer terms then $i-j$ is bounded. This implies that there exists some integer $k$ where if $b_i=k=b_j$ and $b_i$ and $b_j$ are consecutive terms with this property, then $i-j$ is bounded. Now, notice that the only thing that decreases the denominator is $a_i=1$ and it does so linearly. And, if $a_i>1$ then the numerator increases exponentially while the denominator grows linearly. Thus, we require an arbitrarily large number of $a_i=1$'s to drag $b_i$ down to $k$, contradicting $i-j$ bounded.
This post has been edited 2 times. Last edited by blackbluecar, Dec 19, 2021, 4:54 PM
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L567
1184 posts
#3 • 1 Y
Y by PRMOisTheHardestExam
Suppose not, and let the $b_i$ be bounded above by $M$. Ignore the first $10000000M$ terms and only consider the rest, so the denominator of the $b_i$'s now only has numbers $> 10000000M$.

Let $N$ be a sufficiently large number and consider the first $N$ terms. Every million terms, there is at least one integer, so there are at least $\frac{N}{10^6}$ integers. Since there are only $M$ possibilities for these, some integer appears at least $\frac{N}{10^6M}$ times, so by PHP, we can find two of these that are at most $10^6M$ apart.

Say for the first number, we had the $b_i$ equal to $\frac{x}{y}$. The numbers in between cant be all $1$'s since then the integers cannot be equal. Let $P$ be the product of the numbers in between and $S$ be the sum.

We have $\frac{xP}{y+S} = \frac{x}{y} \implies yP = y+S$. Note that we cant have $P = 1$ because then LHS is too small, so $P \ge 2$. I claim the LHS is now too big. Note that if an $a_i$ in between is $> 1$, then replacing it with $a_i - 1$ only decreases the difference between RHS and LHS.

So, the worst case is when one number is $2$ and others are $1$, but then we want to show $2y > y+S \iff y > S$, which is true since $y$ was at least $10000000M$ and $S$ is at most $1000000M$ since there are those many numbers, so it is indeed impossible for the $b_i$ to be bounded, as claimed. $\blacksquare$

Remark

oops @below, I meant "un"bounded
This post has been edited 1 time. Last edited by L567, Oct 21, 2021, 4:40 PM
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juckter
323 posts
#4 • 3 Y
Y by PRMOisTheHardestExam, guptaamitu1, MatBoy-123
L567's remark wrote:
We can also prove $a_i$ is bounded but this is useless for the problem.

Wrong. Assume FTSOC that $b_i$ is bounded above.

Claim. $a$ is bounded.

Proof. Assume otherwise and choose a large $k$ such that $a_k > \max(a_1, a_2, \dots, a_{k - 1})$. Let $r \ge k - 10^6$ be such that $b_r$ is an integer and let $k = r + t$. Let $P_r = a_1\dots a_r$ and $S_r = a_1 + \dots + a_r$, so $P_r \ge S_r$. Then

\begin{align*}
b_k = \frac{a_1 \dots a_k}{a_1 + a_2 + \dots + a_k} > \frac{P_ra_k}{S_r + ta_k} \ge \frac{S_ra_k}{S_r + ta_k} \ge \frac{S_ra_k}{2\max(S_r, ta_k)} = \min \left(\frac{a_k}{2}, \frac{S_r}{2t}\right)
\end{align*}
Since $S_r \ge k - 10^6$ and both $k$ and $a_k$ can be arbitrarily large, this provides a contradiction. $\square$

Assume now that $a$ is bounded above by $M$. Let $c_1, c_2, \dots$ be the sequence of integers in the sequence $b_1, b_2, \dots$. Notice that the numerators of the $c_i$ are $M$-smooth, i.e. all of their prime factors are at most $M$. Since the $c_i$ are integers it follows that all the denominators must be $M$-smooth too.

On the other hand, by the problem condition the sequence of denominators is strictly increasing, and any two consecutive denominators differ by at most $10^6 M$. This means the sequence of denominators has a positive lower density, which is a contradiction as the $M$-smooth numbers have density $0$.
This post has been edited 1 time. Last edited by juckter, Oct 21, 2021, 4:04 PM
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v_Enhance
6877 posts
#5 • 4 Y
Y by PRMOisTheHardestExam, HamstPan38825, Mathematicsislovely, pog
Let $K = 10^{10^{100}}$ be a huge absolute constant. We consider two cases. The first case is captured in the following claim.

Claim: Suppose that after the first $K$ terms, at least one of every $K$ terms $a_i$ is greater than $1$. Then $b_i$ is unbounded.
Proof. In that case, consider the first $n$ terms for some large $n$. Let $M = \max(a_1, \dots, a_n)$. We have that \[ b_n \ge \frac{ 		2^{\left\lfloor 10^{-6} n - 1 \right\rfloor} M} 	{M+M+\dots+M} 	\ge \frac{1}{n} 		2^{\left\lfloor 10^{-6} n - 1 \right\rfloor} \]since the product of every million terms is at least $2$ and one of the products is actually equal to $M$. The right-hand side is unbounded in $n$, as desired. $\blacksquare$

Thus, we may focus on the case where $a_N = a_{N+1} = \dots = a_{N+K-1} = 1$ for some $N \ge K$. Look at two indices $b_{N+i}$ and $b_{N+j}$ with $|j-i| \le 10^6$ and such that $b_i$ and $b_j$ are integers, as promised by the problem condition. We may write \[ b_i = \frac{X}{Y+i}, \qquad b_j = \frac{X}{Y+j} \]where $Y = a_1 + \dots + a_N \ge N \ge K$. Then \[ X \ge \operatorname{lcm} (Y+i, Y+j) 	= \frac{(Y+i)(Y+j)}{\gcd(Y+i, Y+j)} 	\ge \frac{1}{10^6} (Y+i)(Y+j) \]which means that \[ b_j \ge \frac{Y+j}{10^6} > \frac{K}{10^6} \]as needed.
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SnowPanda
186 posts
#6
Y by
Solution
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guptaamitu1
656 posts
#7
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Assume contrary. Let $\text{one million} = x$ and $2021^{2021} = y$. Note
$$ \frac{a_{n+k}}{a_n} = a_{n+1} \cdots a_{n+k} \cdot \frac{a_1 + a_2 + \cdots + a_n}{a_1 + \cdots + a_{n+k}} $$Observe decreasing any of $a_1,\ldots,a_{k+n}$ would decrease the above value and increasing them would increase it (this fact will be used several times). By assumption the sequence $\{b_i\}_{i \ge 1}$ is bounded above by $y$. Fix a large $M$. Pick a $N > M$ such that $b_N$ is the max possible integer among $b_M,b_{M+1},\ldots$. We have two cases:

Case 1: All of $a_{N+1},\ldots,a_{N+x}$ equal $1$. Pick a $1 \le r \le x$ such that $a_{N+r} \in \mathbb Z$. Then both
$$ b_N = \frac{a_1a_2 \cdots a_N}{a_1 + \cdots + a_N} ~~,~~ b_{N+r} = \frac{a_1 \cdots a_N}{a_1 + \cdots + a_N + r} $$are integers.
In particular $a_1a_2 \cdots a_N$ is divisible by both $a_1 + \cdots + a_N, a_1 + \cdots + a_N + x$. Thus,
$$ b_N \ge \frac{\text{lcm}(a_1 + \cdots + a_N, a_1 + \cdots + a_N+r)}{a_1 + a_2 + \cdots + a_N} \ge \frac{a_1 + a_2 + \cdots + a_N + r}{r} \ge \frac{M}{x}  $$But as $M$ is large, so $b_N > y$, contradiction. $\square$

Case 2: $a_{N+s} > 1$ for some $1 \le s \le x$. Then,
$$ \frac{b_{N+s}}{b_N} = a_{N+1} \cdots a_{N+s} \cdot \frac{a_1 + \cdots + a_N}{a_1 + \cdots + a_{N+s}} \ge 2 \cdot \frac{N}{N+s+1} \ge \frac{2M}{M+s+1} \ge \frac{2M}{M+x+1} $$Now pick a $1 \le t \le x$ such that $b_{N + s + t} \in \mathbb Z$. Then,
$$ \frac{b_{N+s+t}}{b_{N+s}} = a_{N+s+1} \cdots a_{N+s+t} \cdot \frac{a_1 + \cdots + a_{N+s}}{a_1 + \cdots + a_{N+s+t} } \ge \frac{N+s}{N+s+t} \ge \frac{M+s}{M+s+t} \ge \frac{M}{M+t} \ge \frac{M}{M+x} $$Thus it follows that
$$ \frac{b_{N+s+t}}{b_N} \ge \frac{2M}{M+x+1} \cdot \frac{M}{M+x} $$As $M$ is large, so $b_{N+s+t} > b_N$, which contradicts the maximality assumption on $b_N$. $\blacksquare$
This post has been edited 1 time. Last edited by guptaamitu1, Feb 7, 2022, 6:55 PM
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megarnie
5603 posts
#8
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Suppose that $b_i\le 2021^{2021}$ for all $i$.

Claim: There exist arbitrarily large groups of consecutive $a_i$ all equal to $1$.
Proof: Suppose this was not the case. Then for any arbitrarily large $k$, we have that at least one of $k$ consecutive terms in the sequence $(a_i)$ is greater than $1$.

Fix a $k=C> 69^{420^{665^{1434}}}$ satisfying that condition. Notice that for any $n$ at least $n$ of $a_1, a_2, \ldots, a_{n\cdot C}$ are greater than $1$, which implies that \[b_{n\cdot C} \ge \frac{2^n}{a_1 + a_2 + \cdots + a_{n\cdot C}}\]Notice that the function $\frac{kx}{k+x}$ for fixed $k$ is increasing over the interval $(1, \infty)$, which implies that \[b_{n\cdot C} \ge \frac{2^n}{n\cdot 2 + (C-n)\cdot 1} = \frac{2^n}{n(C+1)},\]which exceeds $2021^{2021}$ for large $n$. $\square$


Now we have for any $M$, there exist $a_i, a_{i+1}, \ldots, a_{i+M-1}$ all equal to $1$.

Fix a large $M > 665^{1434^{7776}}$ for which $a_i = a_{i+1} = \cdots = a_{i+M-1} = 1$, where $i> 665^{1434^{7776}}$ is a positive integer.

Notice that there exist $x<y$ strictly between $i$ and $i+M - 1$ such that $y-x\le 10^6$ and $b_x, b_y\in \mathbb{Z}$.

Let $x+d = y, a_1a_2 \cdots a_x = P, a_1 + a_2 \cdots + a_x = S$. We have \[b_x = \frac{P}{S}, b_y = \frac{P}{S+d}\]This implies that $\mathrm{lcm}(S, S+d)\mid P$, so \[P\ge \frac{S(S+d)}{\gcd(S, S+d)} \ge \frac{S(S+d)}{10^6},\]which implies that $b_x \ge \frac{S+d}{10^6}$, which exceeds $2021^{2021}$ because $S\ge x$ is a sufficiently large positive integer, contradiction.
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IAmTheHazard
5001 posts
#9 • 1 Y
Y by megarnie
We prove that $b_i$ is unbounded which clearly finishes. Suppose for the sake of contradiction that the $b_i$ are all bounded above by some positive integer $M$.
First note that $S_n:=a_1+\cdots+a_n$ is clearly unbounded, so $P_n:=a_1\ldots a_n$ must be as well, since it's bounded below by $S_n$ at least once in every million consecutive terms. We now prove the central claim.

Claim: There cannot exist one trillion consecutive terms $a_{n+1}=\cdots=a_{n+10^{12}}=1$ for $n$ sufficiently large.
Proof: We will first prove that there cannot exist one billion consecutive terms $a_{n+1}=\cdots=a_{n+10^9}=1$ if $b_n$ is an integer, if $n$ is sufficiently large, so $S_n\geq GM$, where $G$ is Graham's number. Suppose that we have $P_n=kS_n$, so $k\leq M$. Then if $a_{n+1}=\cdots=a_{n+10^9}=1$, we have
$$b_{n+10^9}=\frac{kS_n}{S_n+10^9}>k-1 \iff kS_n>kS_n-S_n+10^9k-10^9 \iff S_n>10^9k-10^9,$$which is true. Thus $b_i \in (k-1,k)$ for all $n+1 \leq i \leq n+10^9$, so we have $10^9$ consecutive noninteger $b_i$: illegal. To finish, note that if $a_{n+1}=\cdots=a_{n+10^{12}}=1$ we can find some $1 \leq i \leq 10^6$ such that $b_i$ is an integer, so applying this lemma implies that there are at most $10^6+10^9$ consecutive one terms at a time, which clearly implies the desired claim.

Now for large $n$ such that $S_n \geq \mathrm{TREE}(3)M$ (note that $\mathrm{TREE}(3)\gg G$), if we have $a_{n+1}=\cdots=a_{n+k}=1$ and $a_{n+k+1}=c>1$, we have
$$b_{n+k+1}=\frac{cP_n}{S_n+c+k}\geq \frac{1.001P_n}{S_n}=1.001b_n \iff cS_n\geq 1.001(S_n+c+k) \iff S_n \geq \frac{1.001c+1.001k}{c-1.001}\geq 1.001+\frac{1.001k+1.001^2}{c-1.001}.$$By our claim, $k\leq M$, hence the last inequality is certainly true as $c\geq 2$. Thus we can find a subsequence of the $b_i$ which grows exponentially, which contradicts our initial assumption. This clearly gives us some $b_k>2021^{2021}$, so we're done. $\blacksquare$
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mathlogician
1051 posts
#10
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The goal is to prove that $(b_i)$ is unbounded; suppose otherwise. Now we first prove that we can never have an infinitely long string of $1$s, since otherwise for infinitely many consecutive $b_i$ we have $b_i < 1$ as $a_1a_2\cdots a_i$ is bounded whilst $a_1+a_2+\cdots + a_i$ is unbounded. Now take a very large $n$ and $k$ such that some term between $a_n$ and $a_{n+k}$ is greater than $1$ and with $b_n \geq b_k$; we can do this since $(b_i)$ is bounded. We get that $S \left( a_{n+1}a_{n+2}\dots a_{n+k} \right) \leq a_{n+1} + a_{n+2} + \dots + a_{n+k}$ where $S = a_1 + a_2 + \dots + a_n$. Suppose that $t$ of the $a_i$ between $a_n$ and $a_{n+k}$ are greater than $1$. Then if $t = 1$ then with sufficiently high $S$ we note that $S(a_i -1) > 10^6-1 + a_i$ which is a contradiction. Otherwise if $t>1$ we can just induct down on $t$ by noticing $a_i > 1$ contributes a lot more to the left-hand side than the right-hand side, done.
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Bigtaitus
75 posts
#11 • 1 Y
Y by Vahe_Arsenyan
We assume FTSOC that $b_k\leq 2021^{2021}$ $\forall k.$ Now the key claim is the following:

Claim. For all $a\in\mathbb{N}$ $\exists b s.t. a_{b+i}=1 \forall i\in \{1,2,\dots, a\}.$

Proof. For some $a_i$ with $i$ arbitrarely big, let $c_1,c_2,...,c_k$ be the $a_j\geq 2$ for $j\leq i.$ Now: $$b_i=\frac{c_1\cdots c_k}{c_1+\cdots + c_k+(i-k)}.$$If we can show that $(i-k)>ck$ has a solution $k$ $\forall c,$ then the claim is proven by $PHP.$
In order to show this, we claim that $b_i$ would dicrease if we swap some $a_l\to a_l-1f.$ For the sake of simplicity we'll replace $a_1\to a_1-1.$ This is equivalent to: $$\frac{(c_1-1)c_2\cdots c_k}{c_1-1+c_2+\cdots c_k}\leq \frac{c_1\cdots c_k}{c_1+c_2+\cdots c_k}$$which is trivially true by expanding. So, iterating this procces until all $c_i's$ are $2$ we get that $$b_i\leq \frac{2^{k}}{2k+(i-k)}\leq 2021^{2021},$$which ends the claim.
\blacksquare

This means that we can get a $d$ such that there exist $d_1,d_2,...,d_m<10^6$ satisfying that all $b_{d}, b_{d+d_1}, b_{d+d_1+d_2}, \dots, b_{d+d_1+\cdots +d_m}$ are all integers, and $a_d,a_{d+1}, \dots a_{d+d_1+\cdots +d_k}=1.$ Let $f(i)=a_1+a_2+\cdots a_i.$ Now we have that $lcm(f(d),f(d+d_1),\dots f(d+d_1+\cdots +d_m))\mid a_1\cdot a_2\cdots a_d.$ Bounding the quotient of this $lcm$ and $f(d)$ is trivial, and ends the problem immediately.
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HamstPan38825
8857 posts
#12
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Let $\{x_i\}$ be the maximal subsequence of $\{b_i\}$ that consists of only integers. (For convenience, the indices of $\{x_i\}$ will match those of $\{b_i\}$ and thus not be always consecutive.) I claim that for every $x_i$, there exists an $r$ such that $x_{i+r} > x_i$, which will imply the result.

Suppose for the sake of contradiction that there exists an $n$ such that $\{x_i\}$ is constant at $c$ for all $i \geq n$. Then, for $k > \ell > n$, we have
\begin{align*}
a_1a_2\cdots a_k &= c(a_1+a_2+\cdots+a_k) \\
a_1a_2\cdots a_\ell &= c(a_1+a_2+\cdots+a_\ell) \\
\implies a_{\ell+1}a_{\ell+2}\cdots a_k(a_1+a_2+\cdots+a_\ell) &= a_1+a_2+\cdots + a_k.
\end{align*}Let $N = a_1+a_2+\cdots + a_\ell$. Then suppose $a_{\ell+1}+a_{\ell+2}+\cdots + a_k = m \cdot N$, such that $a_{\ell+1}a_{\ell+2}\cdots a_n > m \cdot N - 10^6$ as $k - \ell < 10^6$. It follows that $$Na_{\ell + 1}a_{\ell+2} \cdots a_k > N(m \cdot N - 10^6) > (m+1)N = a_1+a_2+\cdots+ a_k,$$which is an obvious contradiction.
This post has been edited 1 time. Last edited by HamstPan38825, Dec 21, 2023, 11:47 PM
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bin_sherlo
716 posts
#13
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Suppose that $\{b_i\}_{i=1}^{\infty}$ is bounded above by $C$. Since there are finitely many positive integers $b_i$ can take, there are infinitely many pairs where $b_i=b_j$. Assume that the value $T$ is taken infinitely many times. Let $T=b_{m_1}=\dots=b_{m_r}=\dots$
\[\frac{a_1\dots a_k}{a_1+\dots+a_k}=T=\frac{a_1\dots a_k\dots a_{k+l}}{a_1+\dots+a_{k+l}}\implies a_{k+1}\dots a_{k+l}=1+\frac{a_{k+1}+\dots+a_{k+l}}{a_1+\dots+a_k}\]Note that $a_1+\dots+a_k|a_{k+1}+\dots+a_{k+l}$ hence $k\leq a_{k+1}+\dots+a_{k+l}$. We see that
\[\frac{a_{k+1}+\dots+a_{k+l}}{l}\leq a_{k+1}\dots a_{k+l}=1+\frac{a_{k+1}+\dots+a_{k+l}}{a_1+\dots+a_k}\leq 1+\frac{a_{k+1}+\dots+a_{k+l}}{k}\]We pick $l\leq 10^6$ and $k$ sufficiently large (since there are infinitely many $b_i\in \mathbb{Z}^+$ and consecutive ones have difference at most $10^6$ we can apply this).
\[1\geq (a_{k+1}+\dots+a_{k+l})(\frac{1}{l}-\frac{1}{k})\geq k(\frac{1}{l}-\frac{1}{k})\geq \frac{k}{l}-1>1\]Which gives a contradiction as desired.$\blacksquare$
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Ilikeminecraft
611 posts
#15
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Suppose $b_n$ is bounded above. Let $A = 143414341434$ be a fixed constant.
Claim: Let $N$ be any integer. Then, there exists a value $k$ such that $a_k = a_{k + 1} = \dots = a_{k + N - 1} = 1.$
Proof: Suppose $c_1, \dots, c_k$ are the values greater than 1 in the set $a_1, a_2, \dots, a_m,$ where $m$ is some very big number. Then, \[b_m = \frac{c_1\dots c_k}{c_1 + \dots + c_k + (m - k)}.\]Observe that $\frac{x\cdot j}{x + j}$ is an increasing function over $x \in (1, \infty)$ for a fixed $j > 0.$ Thus, if we were to smooth the value of $b_m$ by forcing all of the $c_i = 2,$ we get \[b_m \geq  \frac{2^m}{2m + k - m} \geq 2021^{2021}\]for sufficiently large $m.$ Thus, the claim must be true.

We now prove that this is a contradiction. Assume $b_n$ is bounded above by $B.$ We know there exists $k$ such that $a_{k + 1} = a_{k + 2} = \dots = a_{AB + k} = 1.$ Clearly, at least $B + 1$ values in the set $\{b_{k + 1}, b_{k + 2}, \dots, b_{k + AB + k}\}$ must be integers. However, clearly, the values $b_{k + i} > b_{k + j}$ for $i > j.$ Thus, this is impossible.
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