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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Wednesday at 3:18 PM
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

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[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Wednesday at 3:18 PM
0 replies
J
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Matrices and Determinants
Saucepan_man02   1
N 4 minutes ago by removablesingularity
Hello

Can anyone kindly share some problems/handouts on matrices & determinants (problems like Putnam 2004 A3, which are simple to state and doesnt involve heavy theory)?

Thank you..
1 reply
+1 w
Saucepan_man02
an hour ago
removablesingularity
4 minutes ago
J
H
Equivalent definition for C^1 functions
Ciobi_   2
N 3 hours ago by Alphaamss
Source: Romania NMO 2025 11.3
Prove that, for a function $f \colon \mathbb{R} \to \mathbb{R}$, the following $2$ statements are equivalent:
a) $f$ is differentiable, with continuous first derivative.
b) For any $a\in\mathbb{R}$ and for any two sequences $(x_n)_{n\geq 1},(y_n)_{n\geq 1}$, convergent to $a$, such that $x_n \neq y_n$ for any positive integer $n$, the sequence $\left(\frac{f(x_n)-f(y_n)}{x_n-y_n}\right)_{n\geq 1}$ is convergent.
2 replies
Ciobi_
Wednesday at 1:54 PM
Alphaamss
3 hours ago
J
H
Strange limit
Snoop76   7
N 3 hours ago by Alphaamss
Find: $\lim_{n \to \infty} n\cdot\sum_{k=1}^n \frac 1 {k(n-k)!}$
7 replies
Snoop76
Mar 29, 2025
Alphaamss
3 hours ago
J
H
(n^3+3n)^2/(n^6-64)
ThE-dArK-lOrD   11
N 6 hours ago by jolynefag
Source: IMC 2019 Day 1 P1
Evaluate the product
$$\prod_{n=3}^{\infty} \frac{(n^3+3n)^2}{n^6-64}.$$
Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan and Karen Keryan, Yerevan State University and American University of Armenia, Yerevan
11 replies
ThE-dArK-lOrD
Jul 31, 2019
jolynefag
6 hours ago
J
No more topics!
H
J
H
Show the existence of a neighborhhod
Alidq   0
Mar 28, 2025
Source: some derivative problem handout
Let $f:(a,b) \rightarrow \mathbb{R}$ a twice differentiable function with a continuous second derivative, for which there exists a unique $t \in (a,b)$ such that $f(t)=0$. If $f'(t) \neq 0$, show that there exists a neighborhood $V = (t- \delta, t+ \delta)$ of $t$ such that for any $x_0 \in V$ , the sequence defined by the recurrence relation $$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$converges to $t$.
0 replies
Alidq
Mar 28, 2025
0 replies
J
Show the existence of a neighborhhod
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Reveal topic tags
Convergence
Recurrence
neighborhood
derivative
calculus
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Source: some derivative problem handout
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Alidq
28 posts
Alidq
#1 h  Mar 28, 2025, 9:45 PM
Y by
Let $f:(a,b) \rightarrow \mathbb{R}$ a twice differentiable function with a continuous second derivative, for which there exists a unique $t \in (a,b)$ such that $f(t)=0$. If $f'(t) \neq 0$, show that there exists a neighborhood $V = (t- \delta, t+ \delta)$ of $t$ such that for any $x_0 \in V$ , the sequence defined by the recurrence relation $$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$converges to $t$.
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