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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

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Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/list]
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0 replies
jlacosta
Mar 2, 2025
0 replies
Can this sequence be bounded?
darij grinberg   66
N a few seconds ago by shendrew7
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
66 replies
darij grinberg
Jan 19, 2005
shendrew7
a few seconds ago
Functional Equations Marathon March 2025
Levieee   5
N 9 minutes ago by mqoi_KOLA
1. before posting another problem please try your best to provide the solution to the previous solution because we don't want a backlog of many problems
2.one is welcome to send functional equations involving calculus (mainly basic real analysis type of proofs) as long it is of the form $\text{"find all functions:"}$
5 replies
Levieee
2 hours ago
mqoi_KOLA
9 minutes ago
Sequences and limit
lehungvietbao   14
N 17 minutes ago by eg4334
Source: Vietnam Mathematical OLympiad 2014
Let $({{x}_{n}}),({{y}_{n}})$ be two positive sequences defined by ${{x}_{1}}=1,{{y}_{1}}=\sqrt{3}$ and
\[ \begin{cases}  {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\   x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \] for all $n=1,2,3,\ldots$.
Prove that they are converges and find their limits.
14 replies
lehungvietbao
Jan 3, 2014
eg4334
17 minutes ago
IMO 2014 Problem 1
Amir Hossein   131
N 25 minutes ago by eg4334
Let $a_0 < a_1 < a_2 < \dots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n\geq 1$ such that
\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]
Proposed by Gerhard Wöginger, Austria.
131 replies
Amir Hossein
Jul 8, 2014
eg4334
25 minutes ago
No more topics!
A sequence related to the sigma function
Cookierookie   5
N Oct 30, 2024 by alba_tross1867
Source: 2024 Turkey TST P8
For an integer $n$, $\sigma(n)$ denotes the sum of postitive divisors of $n$. A sequence of positive integers $(a_i)_{i=0}^{\infty}$ with $a_0 =1$ is defined as follows: For each $n>1$, $a_n$ is the smallest integer greater than $1$ that satisfies
$$\sigma{(a_0a_1\dots a_{n-1})} \vert \sigma{(a_0a_1\dots a_{n})}.$$Determine the number of divisors of $2024^{2024}$ amongst the sequence.
5 replies
Cookierookie
Mar 18, 2024
alba_tross1867
Oct 30, 2024
A sequence related to the sigma function
G H J
G H BBookmark kLocked kLocked NReply
Source: 2024 Turkey TST P8
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Cookierookie
52 posts
#1 • 3 Y
Y by egxa, Amir Hossein, sami1618
For an integer $n$, $\sigma(n)$ denotes the sum of postitive divisors of $n$. A sequence of positive integers $(a_i)_{i=0}^{\infty}$ with $a_0 =1$ is defined as follows: For each $n>1$, $a_n$ is the smallest integer greater than $1$ that satisfies
$$\sigma{(a_0a_1\dots a_{n-1})} \vert \sigma{(a_0a_1\dots a_{n})}.$$Determine the number of divisors of $2024^{2024}$ amongst the sequence.
This post has been edited 3 times. Last edited by Cookierookie, Mar 18, 2024, 8:43 PM
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BarisKoyuncu
577 posts
#2
Y by
sketch: (will add the details later)

Consider all the numbers in the form $p^{2^n}$ where $p$ is a prime number and $n$ is a nonnegative integer. Rearrange these in increasing order and show that this sequence is the same as $a_0, a_1, a_2, \cdots$. After this, you can see that there are $1+13+11+11=36$ numbers in the sequence $a_i$ dividing the number $2^{6072}\cdot 11^{2024}\cdot 13^{2024}=2024^{2024}$.
This post has been edited 1 time. Last edited by BarisKoyuncu, Mar 19, 2024, 9:01 AM
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hakN
428 posts
#3 • 2 Y
Y by AnSoLiN, VicKmath7
Solution
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starchan
1601 posts
#4 • 4 Y
Y by Amir Hossein, Pranav1056, mxlcv, RM1729
very cute problem
solution
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math_comb01
659 posts
#5
Y by
cute!
Sketch
Remark
This post has been edited 3 times. Last edited by math_comb01, Mar 23, 2024, 2:17 PM
Reason: typos
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alba_tross1867
44 posts
#6
Y by
motivation
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N Quick Reply
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