The time is now - Spring classes are filling up!

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k a AMC 10/12 A&B Coming up Soon!
jlacosta   0
Nov 1, 2024
There is still time to train for the November 6th and November 12th AMC 10A/12A and AMC 10B/12B, respectively! Enroll in our weekend seminars to be held on November 2nd and 3rd (listed below) and you will learn problem strategies, test taking techniques, and be able to take a full practice test! Note that the “B” seminars will have different material from the “A” seminars which were held in October.

[list][*]Special AMC 10 Problem Seminar B
[*]Special AMC 12 Problem Seminar B[/list]
For those who want to take a free practice test before the AMC 10/12 competitions, you can simulate a real competition experience by following this link. As you assess your performance on these exams, be sure to gather data!

[list][*]Which problems did you get right?
[list][*]Was the topic a strength (e.g. number theory, geometry, counting/probability, algebra)?
[*]How did you prepare?
[*]What was your confidence level?[/list]
[*]Which problems did you get wrong?
[list][list][*]Did you make an arithmetic error?
[*]Did you misread the problem?
[*]Did you have the foundational knowledge for the problem?
[*]Which topics require more fluency through practice (e.g. number theory, geometry, counting/probability, algebra)?
[*]Did you run out of time?[/list][/list]
Once you have analyzed the results with the above questions, you will have a plan of attack for future contests! BEST OF LUCK to all competitors at this year’s AMC 10 and AMC 12!

Did you know that the day after both the AMC 10A/12A and AMC 10B/12B you can join a free math jam where our AoPS team will go over the most interesting problems? Find the schedule below under “Mark your calendars”.

Mark your calendars for these upcoming free math jams!
[list][*]November 20th: Amherst College Info Session, 7:30 pm ET: Matt McGann, Dean of Admission and Financial Aid at Amherst College, and Nathan Pflueger, math professor at Amherst College, will host an info session exploring both Amherst College specifically and liberal arts colleges generally. Topics include opportunities in math, the admission process, and financial aid for both US and international students.
[*]November 7th: 2024 AMC 10/12 A Discussion, Thursday, 7:30 pm ET:
[*]AoPS instructors will discuss problems from the AMC 10/12 A, administered November 6. We will discuss some of the most interesting problems from each test!
[*]November 13th: 2024 AMC 10/12 B Discussion, Wednesday, 7:30 pm ET:
[*]AoPS instructors will discuss problems from the AMC 10/12 B, administered November 12. We will discuss some of the most interesting problems from each test![/list]
AoPS Spring classes are open for enrollment. Get a jump on the New Year and enroll in our math, contest prep, coding, and science classes today! Need help finding the right plan for your goals? Check out our recommendations page!

Don’t forget: Highlight your AoPS Education on LinkedIn!
Many of you are beginning to build your education and achievements history on LinkedIn. Now, you can showcase your courses from Art of Problem Solving (AoPS) directly on your LinkedIn profile!

Whether you've taken our classes at AoPS Online or AoPS Academies or reached the top echelons of our competition training with our Worldwide Online Olympiad Training (WOOT) program, you can now add your AoPS experience to the education section on your LinkedIn profile.

Don't miss this opportunity to stand out and connect with fellow problem-solvers in the professional world and be sure to follow us at: https://www.linkedin.com/school/art-of-problem-solving/mycompany/ Check out our job postings, too, if you are interested in either full-time, part-time, or internship opportunities!

Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

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0 replies
jlacosta
Nov 1, 2024
0 replies
easy substitutions for a functional in reals
Circumcircle   3
N 15 minutes ago by megarnie
Source: Kosovo Math Olympiad 2025, Grade 11, Problem 2
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ with the property that for every real numbers $x$ and $y$ it holds that
$$f(x+yf(x+y))=f(x)+f(xy)+y^2.$$
3 replies
Circumcircle
2 hours ago
megarnie
15 minutes ago
Simple system of equations
topologicalsort   2
N 21 minutes ago by rchokler
Source: Bulgarian Autumn Tournament 2024, 10.1
Find all real solutions to the system of equations: $$\begin{cases} (x^2+xy+y^2)\sqrt{x^2+y^2} = 88 \\ (x^2-xy+y^2)\sqrt{x^2+y^2} = 40  \end{cases}$$
2 replies
topologicalsort
3 hours ago
rchokler
21 minutes ago
Perfect powers of functions on a finite set
Tintarn   2
N an hour ago by eddiekopp
Source: Baltic Way 2024, Problem 9
Let $S$ be a finite set. For a positive integer $n$, we say that a function $f\colon S\to S$ is an $n$-th power if there exists some function $g\colon S\to S$ such that
\[
    f(x) = \underbrace{g(g(\ldots g(x)\ldots))}_{\mbox{\scriptsize $g$ applied $n$ times}}
\]for each $x\in S$.

Suppose that a function $f\colon S\to S$ is an $n$-th power for each positive integer $n$. Is it necessarily true that $f(f(x)) = f(x)$ for each $x\in S$?
2 replies
Tintarn
6 hours ago
eddiekopp
an hour ago
A 5 y old result
mihaig   0
an hour ago
Source: Possibly Own
Let $a,b,c,d\geq0$ be reals satisfying $a^2+b^2+c^2+d^2=4.$
Prove
$$\sqrt3\left(a^3+b^3+c^3+d^3\right)+4\left(2-\sqrt3\right)\sqrt[4]{a^3b^3c^3d^3}\geq8.$$Study the equality cases.
0 replies
mihaig
an hour ago
0 replies
No more topics!
Fixed points of composition of inversions
Miquel-point   1
N Yesterday at 12:15 PM by EthanWYX2009
Source: KoMaL A. 714
Consider $n \ge 2$ pairwise disjoint disks $D_1,D_2,\ldots,D_n$ on the Euclidean plane. For each $k=1,2,\ldots,n$, denote by $f_k$ the inversion with respect to the boundary circle of $D_k$. (Here, $f_k$ is defined at every point of the plane, except for the center of $D_k$.) How many fixed points can the transformation $f_n\circ f_{n-1}\circ\ldots\circ f_1$ have, if it is defined on the largest possible subset of the plane?
1 reply
Miquel-point
May 21, 2023
EthanWYX2009
Yesterday at 12:15 PM
Fixed points of composition of inversions
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G H BBookmark kLocked kLocked NReply
Source: KoMaL A. 714
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Miquel-point
417 posts
#1 • 2 Y
Y by anantmudgal09, kiyoras_2001
Consider $n \ge 2$ pairwise disjoint disks $D_1,D_2,\ldots,D_n$ on the Euclidean plane. For each $k=1,2,\ldots,n$, denote by $f_k$ the inversion with respect to the boundary circle of $D_k$. (Here, $f_k$ is defined at every point of the plane, except for the center of $D_k$.) How many fixed points can the transformation $f_n\circ f_{n-1}\circ\ldots\circ f_1$ have, if it is defined on the largest possible subset of the plane?
Z K Y
The post below has been deleted. Click to close.
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EthanWYX2009
756 posts
#2 • 1 Y
Y by LLL2019
I really like this problem but don't have enough ability to solve. Here is the Chinese Translation of official solution(thanks GPT)
Attachments:
KoMaL A 714.pdf (408kb)
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