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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
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0 replies
jlacosta
May 1, 2025
0 replies
d+2 pts in R^d can partition
EthanWYX2009   0
28 minutes ago
Source: Radon's Theorem
Show that: any set of $d + 2$ points in $\mathbb R^d$ can be partitioned into two sets whose convex hulls intersect.
0 replies
EthanWYX2009
28 minutes ago
0 replies
hard inequality omg
tokitaohma   4
N an hour ago by arqady
1. Given $a, b, c > 0$ and $abc=1$
Prove that: $ \sqrt{a^2+1} + \sqrt{b^2+1} + \sqrt{c^2+1} \leq \sqrt{2}(a+b+c) $

2. Given $a, b, c > 0$ and $a+b+c=1 $
Prove that: $ \dfrac{\sqrt{a^2+2ab}}{\sqrt{b^2+2c^2}} + \dfrac{\sqrt{b^2+2bc}}{\sqrt{c^2+2a^2}} + \dfrac{\sqrt{c^2+2ca}}{\sqrt{a^2+2b^2}} \geq \dfrac{1}{a^2+b^2+c^2} $
4 replies
tokitaohma
Yesterday at 5:24 PM
arqady
an hour ago
ISI UGB 2025 P4
SomeonecoolLovesMaths   6
N an hour ago by Atmadeep
Source: ISI UGB 2025 P4
Let $S^1 = \{ z \in \mathbb{C} \mid |z| =1 \}$ be the unit circle in the complex plane. Let $f \colon S^1 \longrightarrow S^2$ be the map given by $f(z) = z^2$. We define $f^{(1)} \colon = f$ and $f^{(k+1)} \colon = f \circ f^{(k)}$ for $k \geq 1$. The smallest positive integer $n$ such that $f^{(n)}(z) = z$ is called the period of $z$. Determine the total number of points in $S^1$ of period $2025$.
(Hint : $2025 = 3^4 \times 5^2$)
6 replies
SomeonecoolLovesMaths
Yesterday at 11:24 AM
Atmadeep
an hour ago
An innocent-looking inequality
Bryan0224   0
an hour ago
Source: Idk
If $\{a_i\}_{1\le i\le n }$ and $\{b_i\}_{1\le i\le n}$ are two sequences between $1$ and $2$ and they satisfy $\sum_{i=1}^n a_i^2=\sum_{i=1}^n b_i^2$, prove that $\sum_{i=1}^n\frac{a_i^3}{b_i}\leq 1.7\sum_{i=1}^{n} a_i^2$, and determine when does equality hold
Please answer this @sqing :trampoline:
0 replies
+1 w
Bryan0224
an hour ago
0 replies
No more topics!
Can this sequence be bounded?
darij grinberg   70
N May 8, 2025 by ezpotd
Source: German pre-TST 2005, problem 4, ISL 2004, algebra problem 2
Let $a_0$, $a_1$, $a_2$, ... be an infinite sequence of real numbers satisfying the equation $a_n=\left|a_{n+1}-a_{n+2}\right|$ for all $n\geq 0$, where $a_0$ and $a_1$ are two different positive reals.

Can this sequence $a_0$, $a_1$, $a_2$, ... be bounded?

Proposed by Mihai Bălună, Romania
70 replies
darij grinberg
Jan 19, 2005
ezpotd
May 8, 2025
Can this sequence be bounded?
G H J
G H BBookmark kLocked kLocked NReply
Source: German pre-TST 2005, problem 4, ISL 2004, algebra problem 2
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darij grinberg
6555 posts
#1 • 8 Y
Y by Davi-8191, Adventure10, jhu08, megarnie, HWenslawski, ImSh95, Mango247, and 1 other user
Let $a_0$, $a_1$, $a_2$, ... be an infinite sequence of real numbers satisfying the equation $a_n=\left|a_{n+1}-a_{n+2}\right|$ for all $n\geq 0$, where $a_0$ and $a_1$ are two different positive reals.

Can this sequence $a_0$, $a_1$, $a_2$, ... be bounded?

Proposed by Mihai Bălună, Romania
This post has been edited 1 time. Last edited by djmathman, Sep 27, 2015, 2:12 PM
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Valentin Vornicu
7301 posts
#2 • 6 Y
Y by jhu08, megarnie, Adventure10, AlphaBetaGammaOmega, ImSh95, Mango247
Yes, actually this is a Romanian proposal by Mihai Bălună. Have fun with this nice problem.
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Peter Scholze
644 posts
#3 • 5 Y
Y by Adventure10, jhu08, megarnie, ImSh95, Mango247
? yes?
there are thousands of proofs given(in particular, by darij and me), that this sequence is always unbounded.
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darij grinberg
6555 posts
#4 • 4 Y
Y by jhu08, Adventure10, ImSh95, Mango247
Well, it's not exactly that I didn't give a proof, but it wasn't quite correct either... So, the problem is not too trivial...

darij
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zhaobin
2382 posts
#5 • 4 Y
Y by Adventure10, jhu08, ImSh95, Mango247
I think a_i must be position.

BTW,I am intrested in this problem.can you post the solution.
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Xell
40 posts
#6 • 4 Y
Y by Adventure10, jhu08, ImSh95, Mango247
First, some basic facts: all the terms of the sequence are positive, two consecutive terms are always different, we have $u_{n+2}=u_{n+1}-u_{n}$ or $u_{n+2}=u_{n+1}+u_{n}$, but if $u_{n}>u_{n+1}$, then necessarily, $u_{n+2}=u_{n+1}+u_{n}$.

I will then prove the following result: if $u_{n}<u_{n+1}$ for some integer n, then for some $m>n$ we have $u_{n+1}<u_{m}$ and $u_{n}+u_{n+1} < u_{m+1}$.

Property 1: let a,b be two positive real numbers with $a<b$ and suppose that $u_{n}=b-a$,$u_{n+1}=2kb-(2k-1)a$ for some integer k>0. If $u_{n+2}=u_{n+1}-u_{n}$ then we have $u_{n+2}<u_{n+1}$ so, from a previous remark, we have necessarily, $u_{n+3}=u_{n+1}+u_{n+2}$. We check that in that case, $u_{n+2}>b$ and $u_{n+3}>a+b$.

Property 2: let a,b be two positive real numbers with $a<b$ and suppose that $u_{n}=2kb-(2k-1)a$,$u_{n+1}=(2k+1)b-2ka$ for some integer k>0. If $u_{n+2}=u_{n+1}+u_{n}$, then We check that $u_{n+1}> b$ and $u_{n+2}> a+b$.

Suppose now that $a=u_{n}<u_{n+1}=b$. If $u_{n+2}=a+b$, the result is proved. So suppose that $u_{n+2}=b-a$. Then necessarily, $u_{n+3}=u_{n+2}+u_{n+1}=2b-a$. Property 1 (with k=1) shows that the result is proved if $u_{n+4}=u_{n+3}-u_{n+2}$, so suppose that $u_{n+4}=u_{n+3}+u_{n+2}=3b-2a$. Property 2 (with k=1) proves the result if $u_{n+5}=u_{n+4}+u_{n+3}$ so suppose that $u_{n+5}=u_{n+4}-u_{n+3}=b-a$.$u_{n+4}>u_{n+5}$, so $u_{n+6}=u_{n+5}+u_{n+4}=4b-3a$. And so on (= induction) we show that the result is proved by using property 1 and 2 unless we have the following sequence:
a,b,b-a,2b-a,3b-2a,b-a,4b-3a,5b-4a,...,b-a,2kb-(2k-1)a,(2k+1)b-2ka,b-a,...
But in that last case, 2kb-(2k-1)a, can be as high as we want, for example higher than b, and (2k+1)b-2ka higher than a+b.
The result is therefore proved.

Now, there are always consecutive terms such that $u_{n}<u_{n+1}$ (if $u_{n}>u_{n+1}$ for some n, then $u_{n+2}=u_{n}+u_{n+1}>u_{n+1}$). Name $a=u_{n}<u_{n+1}=b$. Then an easy induction using the result shows that one can find terms higher than $k(a+b)$ for every integer k, which shows that such a sequence can not be bounded.
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k2c901_1
146 posts
#7 • 3 Y
Y by Adventure10, jhu08, ImSh95
This problem was used as problem 1 of the final exam of the 3rd TST of Taiwan 2005.
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indybar
398 posts
#8 • 5 Y
Y by ultralako, Adventure10, jhu08, ImSh95, Mango247
I'm not really clear with Xell's solution. Can anybody elaborate?
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bomb
365 posts
#9 • 3 Y
Y by Adventure10, jhu08, ImSh95
Wasnt this posted before???

Bomb
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Philip_Leszczynski
327 posts
#10 • 4 Y
Y by Adventure10, jhu08, ImSh95, Mango247
Since $a_n = |a_{n+1}-a_{n+2}|$, we see that all elements of $\{a_n\}$ are nonnegative.
We first show that all elements of $\{a_n\}$ are positive.

Suppose for the sake of contradiction that there exists an $i \ge 0$ such that $a_{i+2}=0$.
(Obviously $a_0,a_1 \ne 0$ since this is given.)
$a_i = |a_{i+1} - a_{i+2}|$, so $a_i = a_{i+1}$. Let $x=a_i=a_{i+1}$.
We will show that for any $j \ge 0$, the set $\{a_k|j \le k \le i_1 \}$ has only two distinct elements: $x$ and $0$.
For the base case we have $j=i$. We will induct backwards on $j$, from $i$ to $0$.
By the inductive hypothesis, $a_{j+1} \epsilon \{0,x\}$, $a_{j+2} \epsilon \{0,x\}$.
Since $a_j = |a_{j+1} - a_{j+2}|$, $a_j \epsilon \{|0-0|,|0-x|,|x-0|,|x-x|\} = \{0,x\}$.
(Remember that $x$ is nonnegative.)
This completes the induction. So $a_0 \epsilon \{0,x\}$ and $a_1 \epsilon \{0,x\}$.
Since we are given that $a_0$ and $a_1$ are positive, $a_0=a_1=x$. (If $x=0$, then we have our contradiction.)
But we are also given that $a_0$ and $a_1$ are different, so we have our contradiction.
Therefore, all elements of $\{a_n\}$ are positive, and as a corollary, no two consecutive elements of $\{a_n\}$ are equal.

We will now show that $a_0$ cannot be the largest element of $\{a_n\}$.
Suppose for the sake of contradiction that it is.
Let $a_0 = x/k$, and let $a_1 = 1/k$. Then $k>0$ and $x>1$.
$ka_0 = x$
$ka_1 = 1$
$ka_2 = b$
$a_0 = |a_1 - a_2|$, so $ka_0 = k|a_1 - a_2| = |ka_1 - ka_2| = |1-b|$.
Then $x=|1-b|$. So $x=1-b$ or $x=b-1$. The former is impossible since it would imply that $b=1-x$, which would make $b$ negative.
So $x=b-1$, and $b=x+1$. $b>x$. So $a_2 = b/k > x/k = a_0$. This contradicts the assumption that $a_0$ is the largest element of $\{a_n\}$.

We will now suppose, for the sake of contradiction, that $a_{i+1}$, for $i \ge 0$, is the largest element of $\{a_n\}$.
Let $a_0 = 1/k$, and let $a_1 = x/k$. Then $k>0$ and $x>1$.
$ka_i = 1$
$ka_{i+1} = x$
$ka_{i+2} = x-1$ (It can't be $1-x$, since this is negative.)
$ka_{i+3} = b$
$ka_{i+1} = k|a_{i+2} - a_{i+3}| = |ka_{i+2} - ka_{i+3}| = |x-1-b|$
$x=|x-1-b|$
The option $x=x-1-b$ is impossible, since it implies $b=-1$.
Then $x=b+1-x$, and $2x=b+1$, and $b=2x-1$.
Suppose $2x-1<x$. Then $x-1<0$, so $x<1$. Contradiction.
So $a_{i+3} = b/k > x/k = a_{i+1}$. This contradicts our assumption that $a_{i+1}$ is the largest of $\{a_n\}$.

We conclude that $\{a_n\}$ does not have a largest element, and it is therefore not bounded.

QED.
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darktreb
732 posts
#11 • 4 Y
Y by Adventure10, jhu08, ImSh95, Mango247
This was also the first problem of the first MOP test for the Black group (USAMO winners) in 2005.
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Valentin Vornicu
7301 posts
#12 • 4 Y
Y by Adventure10, jhu08, ImSh95, Mango247
darktreb wrote:
This was also the first problem of the first MOP test for the Black group (USAMO winners) in 2005.
Also in the Romanian TST in 2005 :) Spread problem :D
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nipolo
7 posts
#13 • 4 Y
Y by Adventure10, jhu08, ImSh95, Mango247
What a easy problem????

There is no bounded.It is unlimited or it has at least 2 limits
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ZetaX
7579 posts
#14 • 4 Y
Y by Adventure10, jhu08, ImSh95, Assassino9931
Philip_Leszczynski wrote:
[...]
We conclude that $\{a_n\}$ does not have a largest element, and it is therefore not bounded.

QED.
You can't make that assumption, e.g the sequence $a_i = 1- \frac{1}{i}$ doesn't have a largest element and is bounded.
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me@home
2349 posts
#15 • 4 Y
Y by Adventure10, jhu08, ImSh95, Mango247
I'm guessing nipolo's brief "proof" is incorrect? Or if it is correct, can someone elaborate?
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