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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 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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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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IGO 2022 advanced/free P2
Tafi_ak   16
N an hour ago by mcmp
Source: Iranian Geometry Olympiad 2022 P2 Advanced, Free
We are given an acute triangle $ABC$ with $AB\neq AC$. Let $D$ be a point of $BC$ such that $DA$ is tangent to the circumcircle of $ABC$. Let $E$ and $F$ be the circumcenters of triangles $ABD$ and $ACD$, respectively, and let $M$ be the midpoints $EF$. Prove that the line tangent to the circumcircle of $AMD$ through $D$ is also tangent to the circumcircle of $ABC$.

Proposed by Patrik Bak, Slovakia
16 replies
Tafi_ak
Dec 13, 2022
mcmp
an hour ago
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100 Points but the Contestants get 0
tastymath75025   29
N 2 hours ago by popop614
Source: USA Winter TST for IMO 2020, Problem 6, by Michael Ren
Let $P_1P_2\dotsb P_{100}$ be a cyclic $100$-gon and let $P_i = P_{i+100}$ for all $i$. Define $Q_i$ as the intersection of diagonals $\overline{P_{i-2}P_{i+1}}$ and $\overline{P_{i-1}P_{i+2}}$ for all integers $i$.

Suppose there exists a point $P$ satisfying $\overline{PP_i}\perp\overline{P_{i-1}P_{i+1}}$ for all integers $i$. Prove that the points $Q_1,Q_2,\dots, Q_{100}$ are concyclic.

Michael Ren
29 replies
tastymath75025
Jan 27, 2020
popop614
2 hours ago
J
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GOTEEM #5: Circumcircle passes through fixed point
tworigami   22
N 2 hours ago by ohiorizzler1434
Source: GOTEEM: Mock Geometry Contest
Let $ABC$ be a triangle and let $B_1$ and $C_1$ be variable points on sides $\overline{BA}$ and $\overline{CA}$, respectively, such that $BB_1 = CC_1$. Let $B_2 \neq B_1$ denote the point on $\odot(ACB_1)$ such that $BC_1$ is parallel to $B_1B_2$, and let $C_2 \neq C_1$ denote the point on $\odot(ABC_1)$ such that $CB_1$ is parallel to $C_1C_2$. Prove that as $B_1, C_1$ vary, the circumcircle of $\triangle AB_2C_2$ passes through a fixed point, other than $A$.

Proposed by tworigami
22 replies
tworigami
Jan 2, 2020
ohiorizzler1434
2 hours ago
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Easy Geometry
pokmui9909   3
N 3 hours ago by whwlqkd
Source: FKMO 2025 P4
Triangle $ABC$ satisfies $\overline{CA} > \overline{AB}$. Let the incenter of triangle $ABC$ be $\omega$, which touches $BC, CA, AB$ at $D, E, F$, respectively. Let $M$ be the midpoint of $BC$. Let the circle centered at $M$ passing through $D$ intersect $DE, DF$ at $P(\neq D), Q(\neq D)$, respecively. Let line $AP$ meet $BC$ at $N$, line $BP$ meet $CA$ at $L$. Prove that the three lines $EQ, FP, NL$ are concurrent.
3 replies
pokmui9909
3 hours ago
whwlqkd
3 hours ago
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Rubik's cube problem
ilikejam   18
N 3 hours ago by mpcnotnpc
If I have a solved Rubik's cube, and I make a finite sequence of (legal) moves repeatedly, prove that I will eventually resolve the puzzle.

(this wording is kinda goofy but i hope its sorta intuitive)
18 replies
ilikejam
Mar 28, 2025
mpcnotnpc
3 hours ago
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line AO passes through the midpoint of segment EF
toanrathay   0
3 hours ago
Given a triangle \( ABC \) with \( AB < AC \) and the angle bisector \( AD \).
The line passing through \( A \) and perpendicular to \( AC \) intersects the line passing through \( B \) and parallel to \( AD \) at point \( E \).
The line passing through \( A \) and perpendicular to \( AB \) intersects the line passing through \( C \) and parallel to \( AD \) at point \( F \).
Let \( O \) be the intersection of the three perpendicular bisectors of triangle \( ABC \).
Prove that line \( AO \) passes through the midpoint of segment \( EF \).
0 replies
toanrathay
3 hours ago
0 replies
J
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Inequalities
sqing   1
N 4 hours ago by sqing
Let $ a,b\geq 0 $ and $a^2+b^2+ab+a+b=1. $ Prove that
$$\frac{179657}{450000}\geq a^2+b^2+3ab(a+ b-0.13562)\geq \frac{3-\sqrt 5}{2}$$
1 reply
sqing
4 hours ago
sqing
4 hours ago
J
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Weird fractions
wangyanliluke   1
N 5 hours ago by Facejo
While I was doing a question I made this really weird observation:

So first, we suppose $S$ is the infinite sum $\frac{1}{1}-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...$. Then $S$ is more than $0$ since $\frac{1}{1}>\frac{1}{2}$, $\frac{1}{3}>\frac{1}{4}$, and so on. But we can rewrite it as $\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...-2(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+...)=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...-(\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...)=0.$ So is $S$ more than $0$ or equal to $0$? Help is much appreciated
1 reply
wangyanliluke
5 hours ago
Facejo
5 hours ago
J
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Probably appeared before
steven_zhang123   1
N 5 hours ago by lyllyl
In the plane, there are two line segments $AB$ and $CD$, with $AB \neq CD$. Prove that there exists and only exists one point $P$ such that $\triangle PAB \sim \triangle PCD$.($P$ corresponds to $P$, $A$ corresponds to $C$)
Click to reveal hidden text
1 reply
steven_zhang123
6 hours ago
lyllyl
5 hours ago
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Hard geometry
jannatiar   2
N 5 hours ago by sami1618
Source: 2024 AlborzMO P4
In triangle \( ABC \), let \( I \) be the \( A \)-excenter. Points \( X \) and \( Y \) are placed on line \( BC \) such that \( B \) is between \( X \) and \( C \), and \( C \) is between \( Y \) and \( B \). Moreover, \( B \) and \( C \) are the contact points of \( BC \) with the \( A \)-excircle of triangles \( BAY \) and \( AXC \), respectively. Let \( J \) be the \( A \)-excenter of triangle \( AXY \), and let \( H' \) be the reflection of the orthocenter of triangle \( ABC \) with respect to its circumcenter. Prove that \( I \), \( J \), and \( H' \) are collinear.

Proposed by Ali Nazarboland
2 replies
jannatiar
Mar 4, 2025
sami1618
5 hours ago
J
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Perpendicular following tangent circles
buzzychaoz   19
N Today at 1:44 AM by cursed_tangent1434
Source: China Team Selection Test 2016 Test 2 Day 2 Q6
The diagonals of a cyclic quadrilateral $ABCD$ intersect at $P$, and there exist a circle $\Gamma$ tangent to the extensions of $AB,BC,AD,DC$ at $X,Y,Z,T$ respectively. Circle $\Omega$ passes through points $A,B$, and is externally tangent to circle $\Gamma$ at $S$. Prove that $SP\perp ST$.
19 replies
buzzychaoz
Mar 21, 2016
cursed_tangent1434
Today at 1:44 AM
J
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A projectional vision in IGO
Shayan-TayefehIR   15
N Today at 1:34 AM by mcmp
Source: IGO 2024 Advanced Level - Problem 3
In the triangle $\bigtriangleup ABC$ let $D$ be the foot of the altitude from $A$ to the side $BC$ and $I$, $I_A$, $I_C$ be the incenter, $A$-excenter, and $C$-excenter, respectively. Denote by $P\neq B$ and $Q\neq D$ the other intersection points of the circle $\bigtriangleup BDI_C$ with the lines $BI$ and $DI_A$, respectively. Prove that $AP=AQ$.

Proposed Michal Jan'ik - Czech Republic
15 replies
Shayan-TayefehIR
Nov 14, 2024
mcmp
Today at 1:34 AM
J
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Circles tangent to BC at B and C
MarkBcc168   9
N Today at 1:20 AM by channing421
Source: ELMO Shortlist 2024 G3
Let $ABC$ be a triangle, and let $\omega_1,\omega_2$ be centered at $O_1$, $O_2$ and tangent to line $BC$ at $B$, $C$ respectively. Let line $AB$ intersect $\omega_1$ again at $X$ and let line $AC$ intersect $\omega_2$ again at $Y$. If $Q$ is the other intersection of the circumcircles of triangles $ABC$ and $AXY$, then prove that lines $AQ$, $BC$, and $O_1O_2$ either concur or are all parallel.

Advaith Avadhanam
9 replies
MarkBcc168
Jun 22, 2024
channing421
Today at 1:20 AM
J
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Iran TST 2009-Day3-P3
khashi70   66
N Today at 12:50 AM by ihategeo_1969
In triangle $ABC$, $D$, $E$ and $F$ are the points of tangency of incircle with the center of $I$ to $BC$, $CA$ and $AB$ respectively. Let $M$ be the foot of the perpendicular from $D$ to $EF$. $P$ is on $DM$ such that $DP = MP$. If $H$ is the orthocenter of $BIC$, prove that $PH$ bisects $ EF$.
66 replies
khashi70
May 16, 2009
ihategeo_1969
Today at 12:50 AM
J
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J
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Inequalities
sqing   11
N Yesterday at 12:25 PM by sqing
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that
$$a(b+c+ 5bc +1)\leq\frac{676}{675}$$$$a(b+c+6bc +1)\leq\frac{245}{243}$$
11 replies
sqing
Mar 26, 2025
sqing
Yesterday at 12:25 PM
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Inequalities
G H J
Reveal topic tags
inequalities
L
G H BBookmark kLocked kLocked NReply
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sqing
41315 posts
sqing
#1 h  Mar 26, 2025, 11:34 AM
Y by
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that
$$a(b+c+ 5bc +1)\leq\frac{676}{675}$$$$a(b+c+6bc +1)\leq\frac{245}{243}$$
Z K Y
The post below has been deleted. Click to close.
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sqing
41315 posts
sqing
#2 h  Mar 26, 2025, 11:58 AM
Y by
Let $ a,b,c\geq 0 $ and $ a(b+c+ 7bc +1)\geq\frac{50}{49}$. Prove that
$$ a+b+c \geq 1$$
Z K Y
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jasperE3
11133 posts
jasperE3
#3 h  Mar 27, 2025, 1:40 AM
Y by
I also had $a,b,c\ge0$, $a(b+c+2bc+1)\ge1$ implies $a+b+c\ge1$.
Z K Y
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sqing
41315 posts
sqing
#4 h  Mar 27, 2025, 2:39 AM
Y by
Thanks.
Let $ a,b,c\geq 0 $ and $ a(b+c +1)\geq 1.$ Prove that
$$ a+b+c \geq 1$$
Z K Y
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jasperE3
11133 posts
jasperE3
#5 h  Mar 27, 2025, 3:04 AM
Y by
sqing wrote:
Thanks.
Let $ a,b,c\geq 0 $ and $ a(b+c +1)\geq 1.$ Prove that
$$ a+b+c \geq 1$$

True.
Z K Y
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anduran
466 posts
anduran
#6 h  Mar 27, 2025, 3:29 AM
Y by
sqing wrote:
Thanks.
Let $ a,b,c\geq 0 $ and $ a(b+c +1)\geq 1.$ Prove that
$$ a+b+c \geq 1$$

$$b+c \geq \frac{1}{a} - 1$$And so $a+b+c \geq a + \frac{1}{a} -1 \geq 1,$ where the last inequality follows from $a+\frac{1}{a} \geq 2.$
Z K Y
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sqing
41315 posts
sqing
#7 h  Mar 27, 2025, 11:12 AM
Y by
Nice.Thanks.
$$ a>0,b+c \geq \frac{1}{a} - 1$$$$a+b+c \geq a + \frac{1}{a} -1 \geq 1$$
Z K Y
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DAVROS
1639 posts
DAVROS
#8 h  Yesterday at 10:28 AM
Y by
sqing wrote:
Let $ a,b,c\geq 0 $ and $ a(b+c+ 7bc +1)\geq\frac{50}{49}$. Prove that $ a+b+c \geq 1$
solution
Z K Y
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neeyakkid23
104 posts
neeyakkid23
#9 h  Yesterday at 11:40 AM
Y by
sqing wrote:
Let $ a,b,c\geq 0 $ and $ a(b+c +1)\geq 1.$ Prove that
$$ a+b+c \geq 1$$

Solution
Z K Y
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sqing
41315 posts
sqing
#10 h  Yesterday at 11:49 AM
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Thanks.
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sqing
#11 h  Yesterday at 11:51 AM
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Very very nice.Thank DAVROS.
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#12 h  Yesterday at 12:25 PM
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Let $ a, b, c>0 $ and $\frac{a^2}{a^2+b+c}+\frac{b^2}{b^2+c+a}+\frac{c^2}{c^2+a+b}\leq1$. Prove that
$$ab+bc+ca\leq 3$$
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