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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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[*]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
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
Concurrent lines
MathChallenger101   4
N an hour ago by oVlad
Let $A B C D$ be an inscribed quadrilateral. Circles of diameters $A B$ and $C D$ intersect at points $X_1$ and $Y_1$, and circles of diameters $B C$ and $A D$ intersect at points $X_2$ and $Y_2$. The circles of diameters $A C$ and $B D$ intersect in two points $X_3$ and $Y_3$. Prove that the lines $X_1 Y_1, X_2 Y_2$ and $X_3 Y_3$ are concurrent.
4 replies
MathChallenger101
Feb 8, 2025
oVlad
an hour ago
Find the value
sqing   7
N an hour ago by giangtruong13
Source: Own
Let $a,b,c$ be distinct real numbers such that $ \frac{a^2}{(a-b)^2}+ \frac{b^2}{(b-c)^2}+ \frac{c^2}{(c-a)^2} =1. $ Find the value of $\frac{a}{a-b}+ \frac{b}{b-c}+ \frac{c}{c-a}.$
Let $a,b,c$ be distinct real numbers such that $\frac{a^2}{(b-c)^2}+ \frac{b^2}{(c-a)^2}+ \frac{c^2}{(a-b)^2}=2. $ Find the value of $\frac{a}{b-c}+ \frac{b}{c-a}+ \frac{c}{a-b}.$
Let $a,b,c$ be distinct real numbers such that $\frac{(a+b)^2}{(a-b)^2}+ \frac{(b+c)^2}{(b-c)^2}+ \frac{(c+a)^2}{(c-a)^2}=2. $ Find the value of $\frac{a+b}{a-b}+\frac{b+c}{b-c}+ \frac{c+a}{c-a}.$
7 replies
sqing
Mar 17, 2025
giangtruong13
an hour ago
2025 HMIC-5
EthanWYX2009   0
an hour ago
Source: 2025 HMIC-5
Compute the smallest positive integer $k > 45$ for which there exists a sequence $a_1, a_2, a_3, \ldots ,a_{k-1}$ of positive integers satisfying the following conditions:[list]
[*]$a_i = i$ for all integers $1 \le i \le 45;$
[*] $a_{k-i} = i$ for all integers $1 \le i \le 45;$
[*] for any odd integer $1 \le n \le k -45,$ the sequence $a_n, a_{n+1}, \ldots  , a_{n+44}$ is a permutation of
$\{1, 2, \ldots  , 45\}.$[/list]
Proposed by: Derek Liu
0 replies
EthanWYX2009
an hour ago
0 replies
I need the technique
DievilOnlyM   14
N an hour ago by Entei
Let a,b,c be real numbers such that: $ab+7bc+ca=188$.
FInd the minimum value of: $5a^2+11b^2+5c^2$
14 replies
DievilOnlyM
May 23, 2019
Entei
an hour ago
Cyclic Quads and Parallel Lines
gracemoon124   15
N an hour ago by Adywastaken
Source: 2015 British Mathematical Olympiad?
Let $ABCD$ be a cyclic quadrilateral. Let $F$ be the midpoint of the arc $AB$ of its circumcircle which does not contain $C$ or $D$. Let the lines $DF$ and $AC$ meet at $P$ and the lines $CF$ and $BD$ meet at $Q$. Prove that the lines $PQ$ and $AB$ are parallel.
15 replies
gracemoon124
Aug 16, 2023
Adywastaken
an hour ago
Something nice
KhuongTrang   33
N an hour ago by NguyenVanHoa29
Source: own
Problem. Given $a,b,c$ be non-negative real numbers such that $ab+bc+ca=1.$ Prove that

$$\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}\le 1+2\sqrt{a+b+c+abc}.$$
33 replies
KhuongTrang
Nov 1, 2023
NguyenVanHoa29
an hour ago
Geometry hard
Lukariman   2
N an hour ago by Primeniyazidayi
Given triangle ABC inscribed in circle (O). The bisector of angle A intersects (O) at D. Let M, N be the midpoints of AB, AC respectively. OD intersects BC at P and AD intersects MN at S. The circle circumscribed around triangle MPS intersects BC at Q different from P. Prove that QA is tangent to (O).
2 replies
Lukariman
2 hours ago
Primeniyazidayi
an hour ago
help!!!!!!!!!!!!
Cobedangiu   3
N 2 hours ago by sqing
help
3 replies
Cobedangiu
Mar 23, 2025
sqing
2 hours ago
Inequality
nguyentlauv   1
N 2 hours ago by NguyenVanHoa29
Source: Own
Let $a,b,c$ be positive real numbers such that $ab+bc+ca=3$ and $k\ge 0$, prove that
$$\frac{\sqrt{a+1}}{b+c+k}+\frac{\sqrt{b+1}}{c+a+k}+\frac{\sqrt{c+1}}{a+b+k} \geq \frac{3\sqrt{2}}{k+2}.$$
1 reply
1 viewing
nguyentlauv
Yesterday at 12:19 PM
NguyenVanHoa29
2 hours ago
Evan Chen Multivariable Calculus Book
Existing_Human1   4
N 2 hours ago by RedChameleon
What do you guys think of Evan Chen's multivariable calculus book (or whatever you want to call it), through MIT? Is it useful for learning multivariable calculus? Here is a link: book
4 replies
Existing_Human1
Today at 1:21 AM
RedChameleon
2 hours ago
Great similarity
steven_zhang123   3
N 2 hours ago by Lil_flip38
Source: a friend
As shown in the figure, there are two points $D$ and $E$ outside triangle $ABC$ such that $\angle DAB = \angle CAE$ and $\angle ABD + \angle ACE = 180^{\circ}$. Connect $BE$ and $DC$, which intersect at point $O$. Let $AO$ intersect $BC$ at point $F$. Prove that $\angle ACE = \angle AFC$.
3 replies
steven_zhang123
2 hours ago
Lil_flip38
2 hours ago
convex optimization
hakuj   0
3 hours ago
I got a problem that I cannot solve :
https://math.stackexchange.com/questions/5061495/property-of-the-optimal-solution-of-an-optimization-problem
0 replies
hakuj
3 hours ago
0 replies
Prove the statement
Butterfly   0
5 hours ago
Given an infinite sequence $\{x_n\} \subseteq  [0,1]$, there exists some constant $C$, for any $r>0$, among the sequence $x_n$ and $x_m$ could be chosen to satisfy $|n-m|\ge r $ and $|x_n-x_m|<\frac{C}{|n-m|}$.
0 replies
Butterfly
5 hours ago
0 replies
Morphism in a ring makes it a field
RobertRogo   1
N Today at 10:16 AM by ysharifi
Source: Daniel Jinga, Ionel Popescu, RNMO SHL, 2003
Let $A$ be a ring with unity in which $1+1 \neq 0$ and there is a morphism $f$ from the group $(A,+)$ to the monoid $(A,\cdot)$ such that for all $a\in A\setminus \{0\}$, there is a $b \in A$ such that $f(b)=a^2$. Prove that $A$ is a field.
1 reply
RobertRogo
Yesterday at 4:02 PM
ysharifi
Today at 10:16 AM
Putnam 2003 A6
btilm305   12
N Apr 25, 2025 by Ilikeminecraft
For a set $S$ of nonnegative integers, let $r_S(n)$ denote the number of ordered pairs $(s_1, s_2)$ such that $s_1 \in S$, $s_2 \in S$, $s_1 \neq s_2$, and $s_1 + s_2 = n$. Is it possible to partition the nonnegative integers into two sets $A$ and $B$ in such a way that $r_A(n) = r_B(n)$ for all $n$?
12 replies
btilm305
Jun 23, 2011
Ilikeminecraft
Apr 25, 2025
Putnam 2003 A6
G H J
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btilm305
439 posts
#1 • 2 Y
Y by Adventure10, Mango247
For a set $S$ of nonnegative integers, let $r_S(n)$ denote the number of ordered pairs $(s_1, s_2)$ such that $s_1 \in S$, $s_2 \in S$, $s_1 \neq s_2$, and $s_1 + s_2 = n$. Is it possible to partition the nonnegative integers into two sets $A$ and $B$ in such a way that $r_A(n) = r_B(n)$ for all $n$?
This post has been edited 1 time. Last edited by rrusczyk, May 8, 2018, 12:40 PM
Reason: LaTeX touch-up
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Kent Merryfield
18574 posts
#2 • 2 Y
Y by Adventure10, Mango247
The answer is yes. Once we make one arbitrary choice, such as $0 \in A,$ there are no further choices and the sets $A$ and $B$ are determined. We construct them as follows. Let $0 \in A$ and $1 \in B.$ After that, let $2n$ belong to the same set that $n$ belongs to and let $2n + 1$ belong to the other set. We won’t need it in the proof, but we could prove from this description the following characterization: write the number $n$ in binary. If the number of $1$s in the representation is even, then $n \in A,$ and if number of $1$s in the representation is odd, then $n \in B.$

For each $n,$ consider the set $\{0, 1, 2,\dots , n\}$ but delete from this set the number $n/2$ if $n$ is even. This set always has even cardinality and always contains $m$ members of $A$ and $m$ members of $B,$ where $m = n/2$ if $n$ is even and $(n + 1)/2$ if $n$ is odd. We must prove that the sizes of $A$ and $B$ balance as claimed. If $n$ is odd, the set consists of pairs $(2k, 2k + 1)$ in which each pair contains one element of each set. If $n$ is even, the set $\{0, 1, 2,\dots, n - 1\}$ has balanced membership from $A$ and $B.$ To get the set we want, we must delete $n/2$ and append $n,$ but these two numbers belong to the same set ($A$ or $B$) and hence the partition into $A$ and $B$ remains balanced.

Now consider a set which consists of $m$ members of set $A$ and $m$ members of set $B.$ We partition this set into $m$ disjoint pairs. (In practice, these pairs are chosen because they have the same sum, but this detail isn’t needed in the proof the follows.) Let $a$ be the number of such pairs in which both numbers belong to $A,$ $b$ be the number of such pairs in which both members belong to $B,$ and $c$ be the number of such pairs in which one member belongs to $A$ and one belongs to $B.$

Since the total number of pairs is $m,$ $a + b + c = m,$ but we don’t need this equation.

Accounting for all members of $A,$ $2a + c = m.$

Accounting for all members of $B,$ $2b + c = m.$

Combining the last two equations, we see that $a = b.$ Hence, for all $n,$ $r_A(n) = r_B(n).$
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peregrinefalcon88
299 posts
#3 • 6 Y
Y by Fermat_Theorem, Williamgolly, sqrtX, Ru83n05, Adventure10, Mango247
We will proceed with generating functions. Let $f(x) = \sum_{n = 0}^{\infty} a_nx^n$ and $g(x) = \sum_{n = 0}^{\infty} b_nx^n$ where $a_n = 1$ iff $n \in A$, otherwise $a_n = 0$, and define $b_n$ similarly. It is clear that $f(x)+g(x) = \frac{1}{1-x}$, and the problem statement desires that $f(x)^2-f(x^2) = g(x)^2-g(x^2) \rightarrow (f(x)-g(x))(f(x)+g(x)) = f(x^2)-g(x^2)$. Define $h(x) = f(x)-g(x)$ and the problem statement now desires that $h(x) = (1-x)h(x^2)$, and we have $h(x) = \sum_{n = 0}^{\infty} c_nx^n \rightarrow \sum_{n = 0}^{\infty} c_nx^n = \sum_{n = 0}^{\infty} c_nx^{2n}-c_nx^{2n+1} \rightarrow c_{2n+1} = -c_n, c_{2n} = c_n$, so it is clear that if we set $c_0 = 1$, then the sequence $c_n$ is well defined, and the reader can verify that $f(x)$ and $g(x)$ do exist with coefficients of only 1 and 0, so the sets do exist as desired.
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codyj
723 posts
#4 • 2 Y
Y by Adventure10, Mango247
$f_S(x):=\sum_{s_1\in S}x^{s_1}$. Then $f_A(x)^2-f_A(x^2)=f_B(x)^2-f_B(x^2)$ and $f_A(x)+f_B(x)=\frac1{1-x}$ is equivalent to the problem statement. Simplify to get $\frac{x}{1-x^2}=f_A(x)-(1-x)f_A(x^2)$. Expand to get $\sum_{\text{odd }n}x^n=\sum_{n\in A}(x^n-x^{2n}+x^{2n+1})$. thus such a set $A$ satisfies the problem statement iff the following two conditions are met: $n\in A\implies2\in A$ (a), $2n+1\in A\implies n\not\in A$ (b).

Now construct by induction. At each step, add the smallest element to $A$ such that each of these conditions is satisfied, and add all the powers of $2$ times it.
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anantmudgal09
1980 posts
#6 • 2 Y
Y by Adventure10, Mango247
The answer is yes. In fact, we can even characterize all such partitions of $\mathbb{N}$ where $\mathbb{N}$ is the set of all non-negative integers.

Indeed, consider a partition of $\mathbb{N}=A \cup B$ and consider the formal series $f(z),g(z)$ in the ring of formal series $R[[X]]$ (the usual operations of addition and multiplication) defined by the relations $$f(z) := \sum_{a \in A} z^a$$and $$g(z) := \sum_{b \in B} z^b$$and observe that $f(z)+g(z)=1+z+z^2+\dots=\frac{1}{1-z}$. Assume wlog that $1 \in A$ and note that $f(0)=1$ and $g(0)=0$. Notice that the number $f_n$ of pairs $(s_1,s_2)$ with $s_1 \not=s_2$ and $s_1,s_2 \in A$ such that $s_1+s_2=n$ is simply, the coefficient of $z^n$ in the formal series $f(z)^2-f(z^2)$. Using a similar argument for $g$, we conclude that $$f(z)^2-f(z^2)=g(z)^2-g(z^2).$$Thus, we have the relation $$\frac{f(z)-g(z)}{f(z^2)-g(z^2)}=1-z$$and successively replacing $z$ by $z^2$ in this relation and multiplying the equations obtained, we get $$f(z)-g(z)=\prod_{k \ge 0} \left(1-z^{2^k}\right)\cdot \lim_{k \rightarrow \infty} \left(f(z^{2^k})-g(z^{2^k})\right)$$and notice that by assumption, $f(0)=1$ and $g(0)=0$ and thus, we conclude that $$f(z)-g(z)=\prod_{k \ge 0} \left(1-z^{2^k}\right)=\sum_{j \ge 0} (-1)^{s_2(j)}\cdot x^j$$and thus, $A,B$ are respectively, the set of all non-negative integers with an odd (respectively, even) sum of digits in binary and they clearly satisfy the conditions by reversing the arguments.
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sqrtX
675 posts
#7
Y by
The answer is yes. Define a sequence $(c_{n})$ by $c_0 =1$ and
$$c_{2n} = c_{n},  \;\;\; c_{2n+1} =-c_{2n} $$Then we clearly have $c_{i} \in \{-1,1\}$. Let $A$ be the set of nonnegative integers $m$ for which $c_m =1$ and define $B$ analogously. We'll now work with power series in $\mathbb{Z}[[X]].$ Let
$$A(X) = \sum_{a\in A} X^{a} , \;\;\; B(X) = \sum_{b\in B} X^{b}$$and $R(X) =A(X)-B(X) = \sum_{i=0}^{\infty} c_{i} X^{i}.$ By definition of the sequence $(c_{n})$, we obtain
$$ R(X) = \sum_{i=0}^{\infty} c_{i} X^{i}= \sum_{j=0}^{\infty} c_{2j}X^{2j} +c_{2j+1}X^{2j+1} = \sum_{j=0}^{\infty} c_{j}X^{2j} -c_{j}X^{2j+1} =(1-X) R(X^{2}).$$Since $1-X$ has the inverse $\sum_{i=0}^{\infty} X^{i} =A(X) +B(X)$, we can write the last formula as
$$ \left(A(X) -B(X) \right)  \left(A(X)+B(X) \right) =A(X^{2}) -B(X^{2})$$or
$$ A(X)^{2} -A(X^{2}) = B(X)^{2} - B(X^{2}).$$Note, however, that
$$A(X)^{2} -A(X^{2}) = \sum_{n=0}^{\infty} r_{A}(n) X^{n}$$and so this partition works.
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HamstPan38825
8860 posts
#8
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WLOG $0 \in A$. Let $A(x)$ and $B(x)$ be the generating functions of $A$ and $B$ respectively. Then, we again have $$A^2(x) - A(x^2) = B^2(x) - B(x^2) \implies (A(x)+B(x))(A(x)-B(x)) = A(x^2)-B(x^2).$$However, $A(x) + B(x) = \frac 1{1-x}$, so $$A(x^2) - B(x^2) = \frac{A(x)-B(x)}{1-x}.$$Now, doing a similar trick as in 1996 USAMO 6, notice that $$\frac{A(x)-B(x)}{(1-x)(1-x^2)(1-x^4)\cdots} = A(x^{2^n}) - B(x^{2^n}) \to 1$$as we perform arbitrarily many iterations. Thus, $A(x) - B(x) = (1-x)(1-x^2)(1-x^4) \cdots$, so we can simply split the positive integers into two sets, one of which has an odd number of 1's in its binary representation, and one of which has an even number of 1's.
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YaoAOPS
1540 posts
#9
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I think it should also be explicitly possible to prove that Thue-Morse works by taking the involution $(s_1, s_2) \to (s_1', s_2')$ where we exchange the first power of $2$ at which $s_1 \ne s_2$. ie $(110, 101) \to (100, 111)$. This involution breaks down only if $s_1 = s_2$, at which point its not counted in $r_S$.
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Shreyasharma
682 posts
#10
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Solved with hints from YaoAOPS :love:

$\boxed{\text{Yes such a partition is possible}}$. Define the generating functions,
\begin{align*}
    F(x) &= \sum_{a \in A} x^a\\
    G(x) &= \sum_{b \in B} x^b
\end{align*}Note that we find that,
\begin{align*}
    F(x) + G(x) = \frac{1}{1-x}
\end{align*}as $A \cup B = \{0, 1, 2, \dots \}$. Then note that from the condition we find,
\begin{align*}
    F(x)^2 - F(x^2) = G(x)^2 - G(x^2)
\end{align*}This is due to the fact that $F(x)^2$ represents the number of partitions that $x^n$ can be partitioned over all $n$. We subtract $F(x^2)$ and $G(x^2)$ to account for $x^{2n} = x^n \cdot x^n$ which is not allowed by the problem condition. Then rearranging the second we have,
\begin{align*}
    [F(x) - G(x)] \cdot [F(x) + G(x)] &= F(x^2) - G(x^2)\\
    [F(x) - G(x)] \cdot \frac{1}{1-x} &= F(x^2) - G(x^2)\\
    \frac{1}{1-x} &= \frac{F(x^2) - G(x^2)}{F(x) - G(x)}
\end{align*}Then define $H(x) = F(x) - G(x)$. We find that,
\begin{align*}
    (1+x+x^2 + x^3 + \dots) &= \frac{H(x^2)}{H(x)}
\end{align*}Let $H(x) = \sum_{i \geq 0} c_ix^i$ for constants $c_i$. Then we require,
\begin{align*}
    (1+x+x^2+x^3+\dots)(c_0 + c_1x+c_2x^2+\dots) &= (c_0 + c_1x^2 + c_2x^4 + \dots)\\
    c_0 + x(c_0 + c_1) + x^2(c_0 + c_1 + c_2) + x^3(c_0 + c_0 + c_1 + c_2 + c_3) + \dots &= c_0 + c_1x^2 + c_2x^4 + \dots
\end{align*}Then we find the system,
\begin{align*}
    c_0 &= c_0\\
    c_0 + c_1 &= 0\\
    c_0 + c_1 + c_2 &= c_1\\
    c_0 + c_1 + c_2 + c_3 &= 0\\
    &\vdots
\end{align*}As such we conclude that $c_{2k+1} = -c_k$ and $c_{2k} = c_k$. Then set $c_0 = 1$. Now to define the sequence $F$ and $G$ we find that we have,
\begin{align*}
    F(x) - G(x) &= 1 - x + x^2 + x^3 + x^4 - x^5 + x^6 - x^7 + \dots
\end{align*}Then taking the partition of $F = 1+ x^2 + x^3 + x^4 + x^6 + \dots$ and $G(x) = x + x^5 + x^7 + \dots$ we are done. $\square$
This post has been edited 3 times. Last edited by Shreyasharma, Jan 13, 2024, 5:19 PM
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shendrew7
795 posts
#11
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Define the generating functions $A(x) = \sum_{n \ge 0} a_nx^n$ and $B(x) = \sum_{n \ge 0} b_nx^n$. The problem statement tells us
\[A(x)+B(x)=\sum_{n \ge 0} x^n = \frac{1}{1-x}, \quad A(x)^2-A(x^2) = B(x)^2-B(x^2).\]
If we let $P(x)=A(x)-B(x)$, the second condition can be rewritten as
\[P(x^2) = \frac{P(x)}{1-x} = P(x) \cdot (1+x)(1+x^2)(1+x^4)(1+x^8) \ldots\]\[\implies P(x) = (1-x)(1-x^2)(1-x^4)(1-x^8)\ldots\]
as a possible solution, plus each term is $\pm x^n$ as desired. Thus our answer is $\boxed{\text{yes}}$, with a possible partition being
\[\boxed{\begin{cases} \{A\} =& \text{Integers with an even number of 1's in binary.} \\ \{B\} =& \text{Integers with an odd number of 1's in binary.} \end{cases}}\]
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RedFireTruck
4223 posts
#12
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https://cdn.aops.com/images/9/d/8/9d831a1a535ee1fa0c8f2e9e0a9d7b1fba7c3262.png
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OronSH
1730 posts
#13
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yes partition based on the parity of the number of ones in its binary representation (morse thue sequence). to show this works let $A(x),B(x)$ be the generating functions for $A,B$ then $A(x)^2-A(x^2)=B(x)^2-B(x^2)$ so $A(x^2)-B(x^2)=(A(x)-B(x))(A(x)+B(x))$ but $A(x)+B(x)=\frac1{1-x}$ so $S(x)=A(x)-B(x)$ satisfies $(1-x)S(x^2)=S(x)$ which it does since clearly $2n$ and $2n+1$ are in different sets and clearly the even index terms form the same as the original sequence. thus this genfunc satisfies the equation
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Ilikeminecraft
616 posts
#14
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I claim the answer is $\boxed{\text{No}}.$ We have that $$\left(\sum_{k\in A} x^k\right)^2 - \sum_{k \in A}x^{2k} = \left(\sum_{k\in B} x^k\right)^2 - \sum_{k \in B}x^{2k}$$However, by rearrangement, we see that:
\begin{align*}
    \sum_{k\in B}x^{2k} - \sum_{k\in A}x^{2k} & = \left(\sum_{k\in B} x^k\right)^2 - \left(\sum_{k\in A} x^k\right)^2 \\
    & = \left(\sum_{k\in B} x^k - \sum_{k\in A} x^k\right)\left(\sum_{k\in B} x^k + \sum_{k\in A} x^k\right)
\end{align*}However, since $A\cup B,$ we see that
\begin{align*}
    \sum_{k\in B}x^{2k} - \sum_{k\in A}x^{2k} & = \left(\sum_{k\in B} x^k - \sum_{k\in A} x^k\right) \frac{1}{1 - x} \\
    & = \sum_{k\in B} x^k - \sum_{k\in A} x^k + \sum_{k\in B} x^{k + 1} - \sum_{k\in A} x^{k + 1} + \cdots
\end{align*}Now, define $f(x) = \sum_{k\in B} x^k - \sum_{k\in A}x^k.$ Hence, we have that $f(x^2) = \frac{f(x)}{1 - x}.$ Clearly, we have that $f(x) = (1 - x)(1 - x^2)(1 - x^4)(1 - x^8)\cdots$ work since $\frac1{1 - x} = (1 + x)(1 + x^2)(1 + x^4)\cdots$. Thus, we are done.
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