Start the New Year strong with our problem-based courses! Enroll today!

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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
non trivial subset problem?
iStud   1
N 10 minutes ago by iStud
Source: KTOM Mock INAMO 2023 P6
Given a set $S=\{1,2,\dots,2023\}$ which is a subset of first $2023$ consecutive integers. A subset $\digamma$ contains some non-empty subsets from $S$ with this following trait: for every two sets $A,B\in\digamma$, if $A\cap B$ isn't empty, then the sum of all the numbers in $A\cap B$ and the sum of all the numbers in $A\cup B$ have the same parity. Determine the biggest value possible of $|\digamma|$.
1 reply
iStud
Saturday at 3:15 PM
iStud
10 minutes ago
IMO ShortList 2008, Number Theory problem 1
April   63
N 11 minutes ago by Maximilian113
Source: IMO ShortList 2008, Number Theory problem 1
Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$.

Proposed by Angelo Di Pasquale, Australia
63 replies
April
Jul 9, 2009
Maximilian113
11 minutes ago
A non-symmetric inequality with a bizarre nested radical as a coefficient
MyLifeMyChoice   6
N 19 minutes ago by MyLifeMyChoice
Source: from Facebook
Happy New Year 2025 everyone(!) Here, I just want to share my solution for this somewhat beast yet noteworthy problem :P

Now, my little curiosity is: could we show $f\left(r\right)\ge0$ in a shorter way, or perhaps there's an entirely different approach to the desired inequality from the very first that's more elegant ? :idea:

Important Note: In my proof, I have used Schur, Click to reveal hidden text and I denoted $q_0=\frac{54-9\sqrt{10}}{13}\approx1.96\ $ ;)

Thank you for reading and I really hope to see some nice things soon !
6 replies
MyLifeMyChoice
Jan 3, 2025
MyLifeMyChoice
19 minutes ago
Show that XD and AM meet on Gamma
MathStudent2002   87
N 30 minutes ago by cursed_tangent1434
Source: IMO Shortlist 2016, Geometry 2
Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$. The points $D$, $E$, $F$ are selected on sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$, $\overline{IE}\perp \overline{AI}$, and $\overline{IF}\perp \overline{AI}$. Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $\Gamma$.

Proposed by Evan Chen, Taiwan
87 replies
MathStudent2002
Jul 19, 2017
cursed_tangent1434
30 minutes ago
Geometry with point P inside triangle
Asyrafr09   6
N 32 minutes ago by teomihai
Given a triangle $ABC$ with $AB=13$, $BC=14$, $CA=15$. A point P lies inside a triangle $ABC$ such that
$$\angle PAB = \angle PBC=\angle PCA =\alpha$$
Find the value of tan $\alpha$ so it can be define in the form of $\frac{m}{n}$ so that gcd($m,n$)= 1. Determine the value of $m\ +\ n.$
6 replies
Asyrafr09
Dec 27, 2024
teomihai
32 minutes ago
Euclidean geom
JetFire008   1
N 36 minutes ago by TestX01
Source: Internet
Consider four given points $A, B, C, D$. There are four circles determined by the four points taken three at a time. If two of these four circles are orthogonal, show that the remaining two circles are also orthogonal.
1 reply
JetFire008
Saturday at 1:02 PM
TestX01
36 minutes ago
Inequalities with two variables
sqing   0
an hour ago
Source: Own
Let $ a,b \geq \frac{1}{2} . $ Prove that
$$ \left(2-\frac{a}{b^2}\right)\left(2-\frac{b}{a^2}\right)  \leq ab$$$$ \left(2-\frac{a^2}{b^3}\right)\left(2-\frac{b^2}{a^3}\right)  \leq ab$$$$ \left(2-\frac{a}{b^2}\right)\left(3-\frac{b}{a^2}\right)  \leq \frac{9}{4}ab$$$$ \left(2-\frac{a^2}{b^3}\right)\left(3-\frac{b^2}{a^3}\right)  \leq \frac{9}{4}ab$$
0 replies
sqing
an hour ago
0 replies
Inequalities with two variables
sqing   4
N 2 hours ago by lbh_qys
Source: Own
Let $a,b>0 . $ Prove that
$$ \frac{(a+b)^2}{4}+ \frac{2}{a+1}+ \frac{2}{b+1}- \sqrt{ab}  \ge2$$$$ \frac{a^2+ab+b^2}{3}+ \frac{2}{a+1}+ \frac{2}{b+1}-  \sqrt{ab}  \ge2$$$$ \frac{a^2+ab+b^2}{4}+ \frac{1}{a+1}+ \frac{1}{b+1}- \sqrt{ab}   \ge \frac{3}{4}$$$$ \frac{3(a+b)^2}{16}+ \frac{1}{a+1}+ \frac{1}{b+1}- \sqrt{ab}   \ge \frac{3}{4}$$
4 replies
sqing
Jan 3, 2025
lbh_qys
2 hours ago
Mathematical Reflections
JustPostTaiwanTST   14
N 2 hours ago by bjump
Source: 2019 Taiwan TST Round 1
Given a triangle $ \triangle ABC $. Denote its incenter and orthocenter by $ I, H $, respectively. If there is a point $ K $ with $$ AH+AK = BH+BK = CH+CK $$Show that $ H, I, K $ are collinear.

Proposed by Evan Chen
14 replies
JustPostTaiwanTST
Mar 31, 2020
bjump
2 hours ago
Easy
Omid Hatami   17
N 2 hours ago by RedFireTruck
Source: Iranian selection test for IMO 2004
$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$
17 replies
Omid Hatami
Jun 27, 2004
RedFireTruck
2 hours ago
IMO ShortList 2001, combinatorics problem 7
orl   27
N 3 hours ago by shendrew7
Source: IMO ShortList 2001, combinatorics problem 7
A pile of $n$ pebbles is placed in a vertical column. This configuration is modified according to the following rules. A pebble can be moved if it is at the top of a column which contains at least two more pebbles than the column immediately to its right. (If there are no pebbles to the right, think of this as a column with 0 pebbles.) At each stage, choose a pebble from among those that can be moved (if there are any) and place it at the top of the column to its right. If no pebbles can be moved, the configuration is called a final configuration. For each $n$, show that, no matter what choices are made at each stage, the final configuration obtained is unique. Describe that configuration in terms of $n$.

IMO ShortList 2001, combinatorics problem 7, alternative
27 replies
orl
Sep 30, 2004
shendrew7
3 hours ago
Very incredible inequality
AlexCenteno2007   0
3 hours ago
Source: Vasc...
Let a, b, c, r be positive reals. Prove that
$$  \frac{ab+(r-3)bc+ca}{(b-c)^2 +rbc}+\frac{bc+(r-3)ca+ab}{(c-a)^2 +rca}+\frac{ca+(r-3)ab+bc}{(a-b)^2 +rab}\geq\frac{3(r-1) }{r}$$
0 replies
AlexCenteno2007
3 hours ago
0 replies
Points and lines
bamboozle   1
N 4 hours ago by HoshimiyaMukuro
Source: AMC
"Prove that for any arrangement of 6 points in the plane, there is no set of lines L such that for any 3-element subset of the 6 points, say S, there exists at least one line in L that is incident to 2 points in S, and there exist two points in S such that there is no line from L that passes through them both."

How do I solve this question? What techniques are used to solve it? It involves only points and lines in the plane, so I assume Euclidean geometry.
1 reply
bamboozle
Yesterday at 7:52 AM
HoshimiyaMukuro
4 hours ago
IMO 2008, Question 4
orl   115
N 4 hours ago by Maximilian113
Source: IMO Shortlist 2008, A1
Find all functions $ f: (0, \infty) \mapsto (0, \infty)$ (so $ f$ is a function from the positive real numbers) such that
\[ \frac {\left( f(w) \right)^2 + \left( f(x) \right)^2}{f(y^2) + f(z^2) } = \frac {w^2 + x^2}{y^2 + z^2}
\]
for all positive real numbers $ w,x,y,z,$ satisfying $ wx = yz.$


Author: Hojoo Lee, South Korea
115 replies
orl
Jul 17, 2008
Maximilian113
4 hours ago
midpoints wanted, 4 equal inradii in an equilateral triangle
parmenides51   0
Jul 30, 2021
Source: 1995 Bulgaria NMO, Round 4, p4
Points $A_1,B_1,C_1$ are selected on the sides $BC$,$CA$,$AB$ respectively of an equilateral triangle $ABC$ in such a way that the inradii of the triangles $C_1AB_1$, $A_1BC_1$, $B_1CA_1$ and $A_1B_1C_1$ are equal. Prove that $A_1,B_1,C_1$ are the midpoints of the corresponding sides.
0 replies
parmenides51
Jul 30, 2021
0 replies
midpoints wanted, 4 equal inradii in an equilateral triangle
G H J
Source: 1995 Bulgaria NMO, Round 4, p4
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parmenides51
30515 posts
#1
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Points $A_1,B_1,C_1$ are selected on the sides $BC$,$CA$,$AB$ respectively of an equilateral triangle $ABC$ in such a way that the inradii of the triangles $C_1AB_1$, $A_1BC_1$, $B_1CA_1$ and $A_1B_1C_1$ are equal. Prove that $A_1,B_1,C_1$ are the midpoints of the corresponding sides.
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