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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
21 minutes ago
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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[*]April 16th (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
21 minutes ago
0 replies
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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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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
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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
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Geo Mock #5
Bluesoul   1
N 32 minutes ago by Sedro
Consider triangle $ABC$ with $AB=13, BC=14, AC=15$. Denote the orthocenter of $\triangle{ABC}$ as $H$, the intersection of $(BHC)$ and $AC$ as $P\neq C$. Compute the length of $AP$.
1 reply
Bluesoul
Yesterday at 7:03 AM
Sedro
32 minutes ago
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inequalities hard
Cobedangiu   3
N an hour ago by sqing
problem
3 replies
Cobedangiu
Mar 31, 2025
sqing
an hour ago
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Find the probability
ali3985   0
an hour ago
Let $A$ be a set of Natural numbers from $1$ to $N$.
Now choose $k$ ($k \geq 3$) distinct elements from this set.

What is the probability of these numbers to be an increasing geometric progression ?
0 replies
ali3985
an hour ago
0 replies
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IMOC 2017 G2 , (ABC) <= (DEF) . perpendiculars related
parmenides51   7
N an hour ago by AshAuktober
Source: https://artofproblemsolving.com/community/c6h1740077p11309077
Given two acute triangles $\vartriangle ABC, \vartriangle DEF$. If $AB \ge DE, BC \ge EF$ and $CA \ge FD$, show that the area of $\vartriangle ABC$ is not less than the area of $\vartriangle DEF$
7 replies
parmenides51
Mar 20, 2020
AshAuktober
an hour ago
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The Sums of Elements in Subsets
bobaboby1   3
N an hour ago by bobaboby1
Given a finite set \( X = \{x_1, x_2, \ldots, x_n\} \), and the pairwise comparison of the sums of elements of all its subsets (with the empty set defined as having a sum of 0), which amounts to \( \binom{2}{2^n} \) inequalities, these given comparisons satisfy the following three constraints:

1. The sum of elements of any non-empty subset is greater than 0.
2. For any two subsets, removing or adding the same elements does not change their comparison of the sums of elements.
3. For any two disjoint subsets \( A \) and \( B \), if the sums of elements of \( A \) and \( B \) are greater than those of subsets \( C \) and \( D \) respectively, then the sum of elements of the union \( A \cup B \) is greater than that of \( C \cup D \).

The question is: Does there necessarily exist a positive solution \( (x_1, x_2, \ldots, x_n) \) that satisfies all these conditions?
3 replies
bobaboby1
Mar 12, 2025
bobaboby1
an hour ago
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3 var inquality
sqing   1
N 2 hours ago by sqing
Source: Own
Let $a,b,c>0$. Prove that
$$ \dfrac{ab}{a^2-ab+3b^2} + \dfrac{bc}{b^2-bc+3c^2} + \dfrac{ca}{c^2-ca+3a^2} \le1$$$$ \dfrac{ab}{a^2-ab+ 2b^2} + \dfrac{bc}{b^2-bc+2 c^2} + \dfrac{ca}{c^2-ca+ 2a^2}\le \dfrac{3}{2}$$$$ \dfrac{ab}{2a^2-ab+3b^2} + \dfrac{bc}{2b^2-bc+3c^2} + \dfrac{ca}{2c^2-ca+3a^2} \le \dfrac{3}{4}$$
1 reply
sqing
2 hours ago
sqing
2 hours ago
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Show that AB/AC=BF/FC
syk0526   75
N 2 hours ago by AshAuktober
Source: APMO 2012 #4
Let $ ABC $ be an acute triangle. Denote by $ D $ the foot of the perpendicular line drawn from the point $ A $ to the side $ BC $, by $M$ the midpoint of $ BC $, and by $ H $ the orthocenter of $ ABC $. Let $ E $ be the point of intersection of the circumcircle $ \Gamma $ of the triangle $ ABC $ and the half line $ MH $, and $ F $ be the point of intersection (other than $E$) of the line $ ED $ and the circle $ \Gamma $. Prove that $ \tfrac{BF}{CF} = \tfrac{AB}{AC} $ must hold.

(Here we denote $XY$ the length of the line segment $XY$.)
75 replies
syk0526
Apr 2, 2012
AshAuktober
2 hours ago
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Very hard FE problem
steven_zhang123   1
N 2 hours ago by GreekIdiot
Source: 0
Given a real number \(C\) such that \(x + y + z = C\) (where \(x, y, z \in \mathbb{R}\)), and a functional equation \(f: \mathbb{R} \rightarrow \mathbb{R}\) that satisfies \((f^x(y) + f^y(z) + f^z(x))((f(x))^y + (f(y))^z + (f(z))^x) \geq 2025\) for all \(x, y, z \in \mathbb{R}\), has a finite number of solutions. Find such \(C\).
(Here, $f^{n}(x)$ is the function obtained by composing $f(x)$ $n$ times, that is, $(\underbrace{f \circ f \circ \cdots \circ f}_{n \ \text{times}})(x)$)
1 reply
steven_zhang123
Mar 30, 2025
GreekIdiot
2 hours ago
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inequalities
Cobedangiu   8
N 2 hours ago by xytunghoanh
problem
8 replies
Cobedangiu
Mar 31, 2025
xytunghoanh
2 hours ago
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Any nice way to do this?
NamelyOrange   1
N 2 hours ago by HockeyMaster85
Source: Taichung P.S.1 math program tryouts

How many ordered pairs $(a,b,c)\in\mathbb{N}^3$ are there such that $c=ab$ and $1\le a\le b\le c\le60$?
1 reply
NamelyOrange
2 hours ago
HockeyMaster85
2 hours ago
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April Fools Geometry
awesomeming327.   4
N 2 hours ago by avinashp
Let $ABC$ be an acute triangle with $AB<AC$, and let $D$ be the projection from $A$ onto $BC$. Let $E$ be a point on the extension of $AD$ past $D$ such that $\angle BAC+\angle BEC=90^\circ$. Let $L$ be on the perpendicular bisector of $AE$ such that $L$ and $C$ are on the same side of $AE$ and
\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]Let the reflection of $D$ across $AB$ and $AC$ be $W$ and $Y$, respectively. Let $X\in AW$ and $Z\in AY$ such that $\angle XBE=\angle ZCE=90^\circ$. Let $EX$ and $EZ$ intersect the circumcircles of $EBD$ and $ECD$ at $J$ and $K$, respectively. Let $LB$ and $LC$ intersect $WJ$ and $YK$ at $P$ and $Q$. Let $PQ$ intersect $BC$ at $F$. Prove that $FB/FC=DB/DC$.
4 replies
awesomeming327.
Yesterday at 2:52 PM
avinashp
2 hours ago
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Olympiad Geometry problem-second time posting
kjhgyuio   4
N 2 hours ago by ND_
Source: smo problem
In trapezium ABCD,AD is parallel to BC and points E and F are midpoints of AB and DC respectively. If
Area of AEFD/Area of EBCF =√3 + 1/3-√3 and the area of triangle ABD is √3 .find the area of trapezium ABCD
4 replies
kjhgyuio
Today at 1:03 AM
ND_
2 hours ago
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Polynomial optimization problem
ReticulatedPython   2
N 3 hours ago by Mathzeus1024
Let $$p(x)=-ax^4+x^3$$, where $a$ is a real number. Prove that for all positive $a$, $$p(x) \le \frac{27}{256a^3}.$$
2 replies
ReticulatedPython
Mar 31, 2025
Mathzeus1024
3 hours ago
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a+b+c=3 inequality
JK1603JK   1
N 3 hours ago by KhuongTrang
Let $a,b,c\ge 0: a+b+c=3$ then prove
$$\color{black}{\sqrt{a+b+2c^{2}}+\sqrt{b+c+2a^{2}}+\sqrt{c+a+2b^{2}}\le 3\sqrt{\frac{a^2+b^2+c^2}{ab+bc+ca}+3}.}$$
1 reply
JK1603JK
4 hours ago
KhuongTrang
3 hours ago
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line bisecting area and perimeter, incenter - 1993 Cono Sur Shortlist G9
parmenides51   0
Oct 24, 2021
Prove that a line that divides a triangle into two polygons of equal area and equal perimeter passes through the center of the circle inscribed in the triangle. Prove an analogous property for a polygon that has an inscribed circle.
0 replies
parmenides51
Oct 24, 2021
0 replies
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line bisecting area and perimeter, incenter - 1993 Cono Sur Shortlist G9
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Reveal topic tags
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parmenides51
30629 posts
parmenides51
#1 h  Oct 24, 2021, 7:38 AM
Y by
Prove that a line that divides a triangle into two polygons of equal area and equal perimeter passes through the center of the circle inscribed in the triangle. Prove an analogous property for a polygon that has an inscribed circle.
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