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k a My Retirement & New Leadership at AoPS
rrusczyk   1571
N Mar 26, 2025 by SmartGroot
I write today to announce my retirement as CEO from Art of Problem Solving. When I founded AoPS 22 years ago, I never imagined that we would reach so many students and families, or that we would find so many channels through which we discover, inspire, and train the great problem solvers of the next generation. I am very proud of all we have accomplished and I’m thankful for the many supporters who provided inspiration and encouragement along the way. I'm particularly grateful to all of the wonderful members of the AoPS Community!

I’m delighted to introduce our new leaders - Ben Kornell and Andrew Sutherland. Ben has extensive experience in education and edtech prior to joining AoPS as my successor as CEO, including starting like I did as a classroom teacher. He has a deep understanding of the value of our work because he’s an AoPS parent! Meanwhile, Andrew and I have common roots as founders of education companies; he launched Quizlet at age 15! His journey from founder to MIT to technology and product leader as our Chief Product Officer traces a pathway many of our students will follow in the years to come.

Thank you again for your support for Art of Problem Solving and we look forward to working with millions more wonderful problem solvers in the years to come.

And special thanks to all of the amazing AoPS team members who have helped build AoPS. We’ve come a long way from here:IMAGE
1571 replies
rrusczyk
Mar 24, 2025
SmartGroot
Mar 26, 2025
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H
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.

Are you ready to level up with Olympiad training? Registration is open with early bird pricing available for our WOOT programs: MathWOOT (Levels 1 and 2), CodeWOOT, PhysicsWOOT, and ChemWOOT. What is WOOT? WOOT stands for Worldwide Online Olympiad Training and is a 7-month high school math Olympiad preparation and testing program that brings together many of the best students from around the world to learn Olympiad problem solving skills. Classes begin in September!

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
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Problem 07 OIMU
KyloRen   3
N a minute ago by emi3.141592
Source: OIMU 2024
Show that the equacion $x^{3}+2y^{3}+3z^{3}=4$ has infinitely many solutions with $x,y,z$ rational numbers.
3 replies
KyloRen
Dec 21, 2024
emi3.141592
a minute ago
J
H
An Integral Inequality from the Chinese Internet
Blast_S1   2
N 21 minutes ago by Blast_S1
Source: Xiaohongshu
Let $f(x)\in C[0,3]$ satisfy $f(x) \ge 0$ for all $x$ and
$$\int_0^3 \frac{1}{1 + f(x)}\,dx = 1.$$Show that
$$\int_0^3\frac{f(x)}{2 + f(x)^2}\,dx \le 1.$$
2 replies
Blast_S1
Yesterday at 2:39 AM
Blast_S1
21 minutes ago
J
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What functions check these conditions?
TheBlackPuzzle913   2
N 2 hours ago by Filipjack
Source: RMO shortlist, Mihai Bunget and Dragoș Gabriel Borugă
Find all functions $ f : \mathbb{R} \rightarrow (0, \infty) $ that are twice differentiable and satisfy $ 3(f'(x))^2 \le 2f(x)f''(x) , \forall x \in \mathbb{R} $
2 replies
TheBlackPuzzle913
4 hours ago
Filipjack
2 hours ago
J
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Limit serie
Moubinool   0
3 hours ago
Source: Oral examination Ecole Polytechnique France
A(n) is a sequence given by
$$A(n)=\frac{1}{n} \sum_{ k , integer, \sqrt{2}< k/n < \sqrt{2} +1} \frac{1}{\sqrt{k/n - \sqrt{2}}}$$Find limit of A(n) when n tend to +oo
0 replies
Moubinool
3 hours ago
0 replies
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Newton's Third Law of Motion
aoum   0
Mar 18, 2025
Newton's Third Law of Motion: The Law of Action and Reaction

Newton's Third Law of Motion is one of the most profound and intuitive principles in classical mechanics. It describes the reciprocal nature of forces between interacting objects and is fundamental to understanding motion in physical systems.


[center]IMAGE[/center]


[center]Rockets work by creating unbalanced high pressure that pushes the rocket upwards while exhaust gas exits through an open nozzle.[/center]

1. Statement of Newton's Third Law

Newton's Third Law can be stated mathematically as:

\[ \mathbf{F}_{A \to B} = -\mathbf{F}_{B \to A}, \]
which means:

For every action, there is an equal and opposite reaction.

In other words, if object $A$ exerts a force on object $B$, then object $B$ simultaneously exerts a force of equal magnitude and opposite direction on object $A$.

[list]
[*] $\mathbf{F}_{A \to B}$ represents the force exerted by object $A$ on object $B$.
[*] $\mathbf{F}_{B \to A}$ represents the force exerted by object $B$ on object $A$.
[/list]

This law applies to all interactions—whether they involve gravity, electromagnetism, or physical contact forces.

2. Understanding Action-Reaction Pairs

Action-reaction pairs are forces that two bodies exert on each other. These pairs:

[list]
[*] Always have equal magnitudes: $|\mathbf{F}_{A \to B}| = |\mathbf{F}_{B \to A}|$.
[*] Always point in opposite directions: $\mathbf{F}_{A \to B} = -\mathbf{F}_{B \to A}$.
[*] Always act on different objects: Each force in the pair is applied to a separate body.
[/list]

It is crucial to remember that these forces do not cancel out because they act on different objects.

3. Examples of Newton's Third Law

[list]
[*] Walking: When you push backward on the ground, the ground pushes you forward with an equal and opposite force.
[*] Rocket Propulsion: Exhaust gases are expelled backward, and the rocket moves forward due to the reaction force.
[*] Collisions: When two objects collide, each exerts an equal and opposite force on the other, regardless of their masses.
[*] Magnetic Attraction and Repulsion: If a magnet pulls on a piece of iron, the iron pulls back on the magnet with equal strength.
[/list]

4. Mathematical Formulation and Proof

Let two objects $A$ and $B$ interact. According to Newton’s Second Law:

\[ \mathbf{F}_{A \to B} = m_B \mathbf{a}_B, \quad \mathbf{F}_{B \to A} = m_A \mathbf{a}_A. \]
Since these forces are equal and opposite:

\[ m_B \mathbf{a}_B = -m_A \mathbf{a}_A, \]
This implies that the total momentum of the system remains constant:

\[ \frac{d}{dt} (\mathbf{p}_A + \mathbf{p}_B) = 0, \]
where $\mathbf{p} = m \mathbf{v}$ represents linear momentum.

This conservation of momentum is a direct consequence of Newton’s Third Law.

5. Applications of Newton's Third Law

Newton’s Third Law has widespread applications in physics and engineering:

[list]
[*] Aviation: The thrust of jet engines is countered by the expulsion of exhaust gases.
[*] Mechanical Systems: In gears and pulleys, forces between components are always paired and balanced.
[*] Space Exploration: Spacecraft maneuver using reaction forces from expelled propellants (even in the vacuum of space).
[*] Engineering Structures: When a structure pushes on the ground, the ground pushes back to maintain equilibrium.
[/list]

6. Newton’s Third Law in Different Force Types

Newton’s Third Law applies universally across different physical forces:

[list]
[*] Gravitational Force: The Earth pulls on the Moon, and the Moon pulls back with equal and opposite force.
[*] Electromagnetic Force: Charges exert equal and opposite forces on each other.
[*] Contact Forces: When a book rests on a table, the table provides an equal and opposite normal force.
[*] Tension Forces: When pulling a rope, the rope pulls back with an equal and opposite tension.
[/list]

7. Newton's Third Law and Non-Inertial Frames

In non-inertial (accelerating) frames of reference, fictitious forces arise, but Newton’s Third Law still holds within the proper frame by accounting for these pseudo-forces.

8. Limitations and Extensions of Newton’s Third Law

While the law holds in most classical systems, it requires modification in certain advanced contexts:

[list]
[*] Relativity Theory: At relativistic speeds (close to the speed of light), forces become dependent on the frame of reference, and momentum conservation is governed by Einstein’s equations.
[*] Electromagnetic Fields: In time-varying electromagnetic fields, forces are mediated by field energy and may not be strictly equal and opposite at every instant.
[*] Quantum Mechanics: On microscopic scales, interactions are described by quantum fields where Newton’s classical laws must be reinterpreted probabilistically.
[/list]

9. Experimental Evidence for Newton's Third Law

Newton’s Third Law has been confirmed through numerous experiments:

[list]
[*] Tension and Compression Tests: Measurements of stress in materials show equal and opposite internal forces.
[*] Collision Studies: High-speed cameras record equal and opposite forces during elastic and inelastic collisions.
[*] Rocket Testing: Exhaust mass flow and thrust measurements match the predictions of Newton’s Third Law.
[/list]

10. Newton's Third Law and Conservation Laws

Newton’s Third Law is closely tied to the law of conservation of linear momentum. For any closed system:

\[ \mathbf{p}_{\text{total}} = \sum m \mathbf{v} = \text{constant}. \]
The conservation of angular momentum is also a consequence of equal and opposite torque interactions.

11. Summary

Newton’s Third Law of Motion establishes the reciprocity of forces between interacting bodies. It underpins the conservation of momentum and has universal applications across all areas of classical physics and beyond.

12. References

[list]
[*] Newton, I. Philosophiæ Naturalis Principia Mathematica (1687).
[*] Halliday, D., Resnick, R., & Walker, J. Fundamentals of Physics (2013).
[*] Wikipedia: Newton’s Laws of Motion
[/list]
0 replies
aoum
Mar 18, 2025
0 replies
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No more topics!
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Ahlfors 3.3.1.2
centslordm   4
N Mar 26, 2025 by Safal
If \[T_1 z = \frac{z + 2}{z + 3}, \qquad T_2 z = \frac z{z + 1},\]find $T_1 T_2z, \,T_2 T_1z$ and ${T_1}^{-1} T_2 z.$
4 replies
centslordm
Jan 8, 2025
Safal
Mar 26, 2025
J
Ahlfors 3.3.1.2
G H J
Reveal topic tags
complex analysis
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centslordm
4736 posts
centslordm
#1 h  Jan 8, 2025, 12:07 AM
Y by
If \[T_1 z = \frac{z + 2}{z + 3}, \qquad T_2 z = \frac z{z + 1},\]find $T_1 T_2z, \,T_2 T_1z$ and ${T_1}^{-1} T_2 z.$
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Mathzeus1024
768 posts
Mathzeus1024
#2 h  Mar 23, 2025, 8:38 PM • 1 Y
Y by centslordm
$T_{1}(T_{2}(z)) = \textcolor{red}{\frac{3z+2}{4z+3}}$.

$T_{2}(T_{1}(z)) = \textcolor{red}{\frac{z+2}{2z+5}}$.

$T_{1}^{-1}(T_{2}(z)) = \textcolor{red}{z-2}$.
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Safal
151 posts
Safal
#3 h  Mar 23, 2025, 9:52 PM • 1 Y
Y by centslordm
Hope this helps
This post has been edited 5 times. Last edited by Safal, Mar 23, 2025, 9:56 PM
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Mathzeus1024
768 posts
Mathzeus1024
#4 h  Mar 25, 2025, 9:14 AM
Y by
Which course would these covered under? Abstract algebra?

Can't say I've heard of this topic before.
This post has been edited 1 time. Last edited by Mathzeus1024, Mar 25, 2025, 9:17 AM
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Safal
151 posts
Safal
#5 h  Mar 26, 2025, 6:51 PM
Y by
Mathzeus1024 wrote:
Which course would these covered under? Abstract algebra?

Can't say I've heard of this topic before.

Well , not abstract algebra but complex analysis, these are basic stuffs of complex analysis. Well these things are actually interesting as it's application is on Modular Forms in Number Theory.
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