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k a January Highlights and 2025 AoPS Online Class Information
jlacosta   0
Jan 1, 2025
Happy New Year!!! Did you know, 2025 is the first perfect square year that any AoPS student has experienced? The last perfect square year was 1936 and the next one will be 2116! Let’s make it a perfect year all around by tackling new challenges, connecting with more problem-solvers, and staying curious!

We have some fun new things happening at AoPS in 2025 with new courses, such as self-paced Introduction to Algebra B, more coding, more physics, and, well, more!

There are a number of upcoming events, so be sure to mark your calendars for the following:

[list][*]Accelerated AIME Problem Series classes start on January 6th and 7th. These AIME classes will run three times a week throughout the month of January. With this accelerated track, you can fit three months of contest tips and training into four weeks finishing right in time for the AIME I on February 6th.
[*]Join our Math Jam on January 7th to learn about our Spring course options. We'll work through a few sample problems, discuss how the courses work, and answer your questions.
[*]RSVP for our New Year, New Challenges webinar on January 9th. We’ll discuss how you can meet your goals, useful strategies for your problem solving journey, and what classes and resources are available.
Have questions? Our Academic Success team is only an email away, write to us at success@aops.com.[/list]
AoPS Spring classes are open for enrollment. Get a jump on 2025 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! 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
Jan 1, 2025
0 replies
k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
Moving pebbles in n boxes
61plus   31
N 19 minutes ago by math004
Source: European Girls’ Mathematical Olympiad-2014 - DAY 2 - P5
Let $n$ be a positive integer. We have $n$ boxes where each box contains a non-negative number of pebbles. In each move we are allowed to take two pebbles from a box we choose, throw away one of the pebbles and put the other pebble in another box we choose. An initial configuration of pebbles is called solvable if it is possible to reach a configuration with no empty box, in a finite (possibly zero) number of moves. Determine all initial configurations of pebbles which are not solvable, but become solvable when an additional pebble is added to a box, no matter which box is chosen.
31 replies
2 viewing
61plus
Apr 13, 2014
math004
19 minutes ago
Sequence limit
Snoop76   1
N 2 hours ago by vanstraelen
Source: Own
The $x_n$ sequence satisfies the following relation : $2 cos(n x_n)  + cos(x_n - n x_n)= cos(2x_n + nx_n)$.$ $ Find $\lim_{n\to\infty} cos(x_n)$.
1 reply
Snoop76
Nov 25, 2024
vanstraelen
2 hours ago
Euler Line Madness
raxu   72
N 2 hours ago by cj13609517288
Source: TSTST 2015 Problem 2
Let ABC be a scalene triangle. Let $K_a$, $L_a$ and $M_a$ be the respective intersections with BC of the internal angle bisector, external angle bisector, and the median from A. The circumcircle of $AK_aL_a$ intersects $AM_a$ a second time at point $X_a$ different from A. Define $X_b$ and $X_c$ analogously. Prove that the circumcenter of $X_aX_bX_c$ lies on the Euler line of ABC.
(The Euler line of ABC is the line passing through the circumcenter, centroid, and orthocenter of ABC.)

Proposed by Ivan Borsenco
72 replies
raxu
Jun 26, 2015
cj13609517288
2 hours ago
What is maximum?
mihaig   8
N 2 hours ago by PaixiaoLover
Source: Nguyen Minh Tho
Let $a,b,c$ be reals such that
$$2(a^2+b^2+c^2)=15+ab+bc+ca.$$Find
$$\max(a-b)(a-c).$$
8 replies
mihaig
Jan 1, 2025
PaixiaoLover
2 hours ago
The Bank of Bath
TelMarin   96
N 2 hours ago by abeot
Source: IMO 2019, problem 5
The Bank of Bath issues coins with an $H$ on one side and a $T$ on the other. Harry has $n$ of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly $k>0$ coins showing $H$, then he turns over the $k$th coin from the left; otherwise, all coins show $T$ and he stops. For example, if $n=3$ the process starting with the configuration $THT$ would be $THT \to HHT  \to HTT \to TTT$, which stops after three operations.

(a) Show that, for each initial configuration, Harry stops after a finite number of operations.

(b) For each initial configuration $C$, let $L(C)$ be the number of operations before Harry stops. For example, $L(THT) = 3$ and $L(TTT) = 0$. Determine the average value of $L(C)$ over all $2^n$ possible initial configurations $C$.

Proposed by David Altizio, USA
96 replies
TelMarin
Jul 17, 2019
abeot
2 hours ago
Olympiad with Beginner Friendly Theory
WheatNeat   11
N 2 hours ago by BR1F1SZ
I want to practice on olympiad level problems, but I still don't have all the theory I need to do most national and such olympiads/TSTs (currently working through vol 2 and only a third of the way through). Are there any good olympiads that I should practice on that are relatively beginner friendly (theory-wise)?
11 replies
WheatNeat
Nov 19, 2024
BR1F1SZ
2 hours ago
xy+yz+xz=3xyz
shivangjindal   53
N 2 hours ago by PaixiaoLover
Source: Balkan Mathematics Olympiad 2014 - Problem-1
Let $x,y$ and $z$ be positive real numbers such that $xy+yz+xz=3xyz$. Prove that \[ x^2y+y^2z+z^2x \ge 2(x+y+z)-3 \] and determine when equality holds.

UK - David Monk
53 replies
shivangjindal
May 4, 2014
PaixiaoLover
2 hours ago
Similar to P:11.2
mathuz   13
N 2 hours ago by ehuseyinyigit
Source: All Russian 2014 Grade 9 Day 2 P2
Let $ABCD$ be a trapezoid with $AB\parallel CD$ and $ \Omega $ is a circle passing through $A,B,C,D$. Let $ \omega $ be the circle passing through $C,D$ and intersecting with $CA,CB$ at $A_1$, $B_1$ respectively. $A_2$ and $B_2$ are the points symmetric to $A_1$ and $B_1$ respectively, with respect to the midpoints of $CA$ and $CB$. Prove that the points $A,B,A_2,B_2$ are concyclic.


I. Bogdanov
13 replies
mathuz
May 3, 2014
ehuseyinyigit
2 hours ago
Functional Equations for starters
Sadece_Threv   4
N 2 hours ago by Mhremath
Source: Own
1) Find all functions $f: \mathbb{R} \to \mathbb{R}$ that satisfies
$$f(x+f(y))=f(x)+y$$for all $x,y\in \mathbb{R}$

2) Find all functions $f: \mathbb{R^+} \to \mathbb{R^+}$ that satisfies
$$f(x+f(y))=f(x)+y$$for all $x,y\in \mathbb{R^+}$
4 replies
Sadece_Threv
Today at 1:28 PM
Mhremath
2 hours ago
Diophantine Equation with Primes and Squares
djmathman   13
N 3 hours ago by MuradSafarli
Source: Indian RMO 2013 Mumbai Region Problem 2
Find all triples $(p,q,r)$ of primes such that $pq=r+1$ and $2(p^2+q^2)=r^2+1$.
13 replies
djmathman
Feb 1, 2014
MuradSafarli
3 hours ago
Vector units
youochange   2
N 3 hours ago by EaZ_Shadow
\[
\left( \frac{\hat{\jmath}}{\hat{k}} \right) \times \hat{k} = ?
\]
2 replies
youochange
6 hours ago
EaZ_Shadow
3 hours ago
Minimum of an expresion with squares and square roots
DensSv   3
N 3 hours ago by ehuseyinyigit
Source: 2023 Shortlist for NMO in Romania, for 8th grade
For $a$ and $b$ real numbers, define
$$E(a,b)=\sqrt{\bigg(\frac{1}{2}-a\bigg)^{2}+\bigg(\frac{1}{2}-b\bigg)^{2}}+\sqrt{\bigg(\frac{1}{2}+a\bigg)^{2}+\bigg(\frac{1}{2}+b\bigg)^{2}}$$a) Prove that $E(a,b)\geq \sqrt{2}$ for every real number $a,b$ and determine the equality case.
b) Find $min\{E(a,a+2)|a\in \mathbb{R}\}$.
3 replies
DensSv
Yesterday at 9:45 PM
ehuseyinyigit
3 hours ago
Junior Balkan Mathematical Olympiad 2020- P3
Lukaluce   12
N 4 hours ago by wizixez
Source: JBMO 2020
Alice and Bob play the following game: Alice picks a set $A = \{1, 2, ..., n \}$ for some natural number $n \ge 2$. Then, starting from Bob, they alternatively choose one number from the set $A$, according to the following conditions: initially Bob chooses any number he wants, afterwards the number chosen at each step should be distinct from all the already chosen numbers and should differ by $1$ from an already chosen number. The game ends when all numbers from the set $A$ are chosen. Alice wins if the sum of all the numbers that she has chosen is composite. Otherwise Bob wins. Decide which player has a winning strategy.

Proposed by Demetres Christofides, Cyprus
12 replies
Lukaluce
Sep 11, 2020
wizixez
4 hours ago
thanks u!
Ruji2018252   1
N 4 hours ago by arqady
$a,b,c>0$ and $a+b+c=3$. Prove:
$$\dfrac{\sqrt{a^3+b^3+8}}{ab+2}+\dfrac{\sqrt{b^3+c^3+8}}{bc+2}+\dfrac{\sqrt{a^3+c^3+8}}{ac+2}\ge 3$$
1 reply
Ruji2018252
Today at 4:14 PM
arqady
4 hours ago
2017 IGO Advanced P2
bgn   3
N Sep 25, 2023 by Assassino9931
Source: 4th Iranian Geometry Olympiad (Advanced) P2
We have six pairwise non-intersecting circles that the radius of each is at least one (no circle lies in the interior of any other circle). Prove that the radius of any circle intersecting all the six circles, is at least one.

Proposed by Mohammad Ali Abam - Morteza Saghafian
3 replies
bgn
Sep 15, 2017
Assassino9931
Sep 25, 2023
2017 IGO Advanced P2
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G H BBookmark kLocked kLocked NReply
Source: 4th Iranian Geometry Olympiad (Advanced) P2
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bgn
178 posts
#1 • 2 Y
Y by Adventure10, Mango247
We have six pairwise non-intersecting circles that the radius of each is at least one (no circle lies in the interior of any other circle). Prove that the radius of any circle intersecting all the six circles, is at least one.

Proposed by Mohammad Ali Abam - Morteza Saghafian
This post has been edited 1 time. Last edited by bgn, Sep 15, 2017, 6:58 AM
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ThE-dArK-lOrD
4071 posts
#2 • 2 Y
Y by leru007, Adventure10
Denote the center of six circles by $O_i$ and their radii by $r_i$ where $i\in \{ 1,2,...,6\}$.
Let $O$ and $r$ be the center and radius of the circle intersecting all six circles. Consider angle formed by rays $\overrightarrow{OO_i}$ where $i\in \{ 1,2,...,6\}$.
Suppose that $r<1$.
Not hard to show that there exist $i,j\in \{ 1,2,...,6\}$ that $i\neq j$ and $\angle{O_iOO_j}\leq 60^{\circ}$.
Law of cosine gives us $O_iO_j^2=OO_i^2+OO_j^2-2\times OO_i\times OO_j\times \cos( \angle{O_iOO_j})\leq OO_i^2+OO_j^2-OO_i\times OO_j$.
WLOG $OO_i\geq OO_j$, we get $OO_i\times OO_j\geq OO_j^2\Rightarrow 0\geq OO_j^2-OO_i\times OO_j$.
But we also have $O_iO_j>r_i+r_j\geq r_i+1>r_i+r\geq OO_i$, so $O_iO_j^2> OO_i^2$.
Hence $O_iO_j^2> OO_i^2+OO_j^2-OO_i\times OO_j$, contradiction. So $r\geq 1$, done.
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VeeEmKay
20 posts
#3 • 2 Y
Y by Adventure10, Mango247
Same notation as ThE-dArK-lOrD used, but more synthetic argument:
After showing that there exists such $\angle O_iOO_j \le 60^\circ$ we could say that then (WLOG) $r+r_i \ge OO_i \ge O_iO_j > r_i+r_j$, which implies $r > r_j \ge 1\square$
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Assassino9931
1019 posts
#4
Y by
Isn't this problem flavour/main argument suspiciously similar to IMO Shortlist 2020 G4?

(EDIT: I just saw this has been mentioned in the early posts in the IMO SL thread)
This post has been edited 1 time. Last edited by Assassino9931, Sep 25, 2023, 2:16 PM
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