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a My Retirement & New Leadership at AoPS
rrusczyk   1345
N 2 hours ago by GoodGamer123
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
1345 replies
rrusczyk
Monday at 6:37 PM
GoodGamer123
2 hours ago
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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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Jury Meeting Lasting for Twenty Years
USJL   0
40 minutes ago
Source: 2025 Taiwan TST Round 2 Independent Study 1-C
2025 IMO leaders are discussing $100$ problems in a meeting. For each $i = 1, 2,\ldots , 100$, each leader will talk about the $i$-th problem for $i$-th minutes. The chair can assign one leader to talk about a problem of his choice, but he would have to wait for the leader to complete the entire talk of that problem before assigning the next leader and problem. The next leader can be the same leader. The next problem can be a different problem. Each leader’s longest idle time is the longest consecutive time that he is not talking.
Find the minimum positive integer $T$ so that the chair can ensure that the longest idle time for any leader does not exceed $T$.

Proposed by usjl
0 replies
USJL
40 minutes ago
0 replies
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Find the value
sqing   1
N an hour ago by lbh_qys
Source: Hunan changsha 2025
Let $ a,b,c $ be real numbers such that $ abc\neq 0,2a-b+c= 0 $ and $ a-2b-c=0. $ Find the value of $\frac{a^2+b^2+c^2}{ab+bc+ca}.$
Let $ a,b,c $ be real numbers such that $ abc\neq 0,a+2b+3c= 0 $ and $ 2a+3b+4c=0. $ Find the value of $\frac{ab+bc+ca}{a^2+b^2+c^2}.$
1 reply
sqing
2 hours ago
lbh_qys
an hour ago
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D1010 : How it is possible ?
Dattier   13
N an hour ago by Dattier
Source: les dattes à Dattier
Is it true that$$\forall n \in \mathbb N^*, (24^n \times B \mod A) \mod 2 = 0 $$?

A=1728400904217815186787639216753921417860004366580219212750904
024377969478249664644267971025952530803647043121025959018172048
336953969062151534282052863307398281681465366665810775710867856
720572225880311472925624694183944650261079955759251769111321319
421445397848518597584590900951222557860592579005088853698315463
815905425095325508106272375728975

B=2275643401548081847207782760491442295266487354750527085289354
965376765188468052271190172787064418854789322484305145310707614
546573398182642923893780527037224143380886260467760991228567577
953725945090125797351518670892779468968705801340068681556238850
340398780828104506916965606659768601942798676554332768254089685
307970609932846902
13 replies
Dattier
Mar 10, 2025
Dattier
an hour ago
J
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integral points
jhz   1
N an hour ago by gaussiemann144
Source: 2025 CTST P17
Prove: there exist integer $x_1,x_2,\cdots x_{10},y_1,y_2,\cdots y_{10}$ satisfying the following conditions:
$(1)$ $|x_i|,|y_i|\le 10^{10} $ for all $1\le i \le 10$
$(2)$ Define the set \[S = \left\{ \left( \sum_{i=1}^{10} a_i x_i, \sum_{i=1}^{10} a_i y_i \right) : a_1, a_2, \cdots, a_{10} \in \{0, 1\} \right\},\]then \(|S| = 1024\)，and any rectangular strip of width 1 covers at most two points of S.
1 reply
jhz
6 hours ago
gaussiemann144
an hour ago
J
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7 triangles in a square
gghx   2
N 2 hours ago by lightsynth123
Source: SMO junior 2024 Q3
Seven triangles of area $7$ lie in a square of area $27$. Prove that among the $7$ triangles there are $2$ that intersect in a region of area not less than $1$.
2 replies
gghx
Oct 12, 2024
lightsynth123
2 hours ago
J
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Practice AMC 10 Final Fives
freddyfazbear   1
N 2 hours ago by WannabeUSAMOkid
So someone pointed out to me that the last five problems on my previous practice AMC 10 test were rather low quality. Here are some problems that are (hopefully) better.

21.
A partition of a positive integer n is writing n as the sum of positive integer(s), where order does not matter. Find the number of partitions of 6.
A - 10, B - 11, C - 12, D - 13, E - 14

22.
Let n be the smallest positive integer that satisfies the following conditions:
- n is even
- The last digit of n is not 2 or 8
- n^2 + 1 is composite
Find the sum of the digits of n.
A - 3, B - 5, C - 8, D - 9, E - 10

23.
Find the sum of the coordinates of the reflection of the point (6, 9) over the line x + 2y + 3 = 0.
A - (-17.7), B - (-17.6), C - (-17.5), D - (-17.4), E - (-17.3)

24.
Find the number of ordered pairs of integers (a, b), where both a and b have absolute value less than 69, such that a^2 + 42b^2 = 13ab.
A - 21, B - 40, C - 41, D - 42, E - 69

25.
Let f(n) be the sum of the positive integer factors of n, where n is an integer. Find the sum of all positive integers n less than 1000 such that f(f(n) - n) = f(n).
A - 420, B - 530, C - 690, D - 911, E - 1034
1 reply
freddyfazbear
2 hours ago
WannabeUSAMOkid
2 hours ago
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What should I do
Jaxman8   0
2 hours ago
I recently mocked 2 AMC 10’s, and 2 AIME’s. My scores for the AMC 10 were both 123 and my AIME scores were 8 and 9 for 2010 I and II. What should I study for 2025-2026 AMCs? Goal is JMO.
0 replies
Jaxman8
2 hours ago
0 replies
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n-variable inequality
ABCDE   65
N 3 hours ago by LMat
Source: 2015 IMO Shortlist A1, Original 2015 IMO #5
Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\]for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.
65 replies
ABCDE
Jul 7, 2016
LMat
3 hours ago
J
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usamOOK geometry
KevinYang2.71   86
N 3 hours ago by deduck
Source: USAMO 2025/4, USAJMO 2025/5
Let $H$ be the orthocenter of acute triangle $ABC$, let $F$ be the foot of the altitude from $C$ to $AB$, and let $P$ be the reflection of $H$ across $BC$. Suppose that the circumcircle of triangle $AFP$ intersects line $BC$ at two distinct points $X$ and $Y$. Prove that $C$ is the midpoint of $XY$.
86 replies
KevinYang2.71
Mar 21, 2025
deduck
3 hours ago
J
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Scary Binomial Coefficient Sum
EpicBird08   38
N 3 hours ago by Mathandski
Source: USAMO 2025/5
Determine, with proof, all positive integers $k$ such that $$\frac{1}{n+1} \sum_{i=0}^n \binom{n}{i}^k$$is an integer for every positive integer $n.$
38 replies
EpicBird08
Mar 21, 2025
Mathandski
3 hours ago
J
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2 degree polynomial
PrimeSol   3
N 3 hours ago by PrimeSol
Let $P_{1}(x)= x^2 +b_{1}x +c_{1}, ... , P_{n}(x)=x^2+ b_{n}x+c_{n}$, $P_{i}(x)\in \mathbb{R}[x], \forall i=\overline{1,n}.$ $\forall i,j ,1 \leq i<j \leq n : P_{i}(x) \ne P_{j}(x)$.
$\forall i,j, 1\leq i<j \leq n : Q_{i,j}(x)= P_{i}(x) + P_{j}(x)$ polynomial with only one root.
$max(n)=?$
3 replies
PrimeSol
Mar 24, 2025
PrimeSol
3 hours ago
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Additive Combinatorics!
EthanWYX2009   3
N 3 hours ago by flower417477
Source: 2025 TST 15
Let \( X \) be a finite set of real numbers, \( d \) be a real number, and \(\lambda_1, \lambda_2, \cdots, \lambda_{2025}\) be 2025 non-zero real numbers. Define
\[A = \left\{ (x_1, x_2, \cdots, x_{2025}) : x_1, x_2, \cdots, x_{2025} \in X \text{ and } \sum_{i=1}^{2025} \lambda_i x_i = d \right\},\]\[B = \left\{ (x_1, x_2, \cdots, x_{2024}) : x_1, x_2, \cdots, x_{2024} \in X \text{ and } \sum_{i=1}^{2024} (-1)^i x_i = 0 \right\},\]\[C = \left\{ (x_1, x_2, \cdots, x_{2026}) : x_1, x_2, \cdots, x_{2026} \in X \text{ and } \sum_{i=1}^{2026} (-1)^i x_i = 0 \right\}.\]Show that \( |A|^2 \leq |B| \cdot |C| \).
3 replies
EthanWYX2009
Yesterday at 12:49 AM
flower417477
3 hours ago
J
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equal angles
jhz   2
N 3 hours ago by YaoAOPS
Source: 2025 CTST P16
In convex quadrilateral $ABCD, AB \perp AD, AD = DC$. Let $ E$ be a point on side $BC$, and $F$ be a point on the extension of $DE$ such that $\angle ABF = \angle DEC>90^{\circ}$. Let $O$ be the circumcenter of $\triangle CDE$, and $P$ be a point on the side extension of $FO$ satisfying $FB =FP$. Line BP intersects AC at point Q. Prove that $\angle AQB =\angle DPF.$
2 replies
jhz
Today at 12:56 AM
YaoAOPS
3 hours ago
J
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Flee Jumping on Number Line
utkarshgupta   23
N 4 hours ago by Ilikeminecraft
Source: All Russian Olympiad 2015 11.5
An immortal flea jumps on whole points of the number line, beginning with $0$. The length of the first jump is $3$, the second $5$, the third $9$, and so on. The length of $k^{\text{th}}$ jump is equal to $2^k + 1$. The flea decides whether to jump left or right on its own. Is it possible that sooner or later the flee will have been on every natural point, perhaps having visited some of the points more than once?
23 replies
utkarshgupta
Dec 11, 2015
Ilikeminecraft
4 hours ago
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2021 AMC10A Problem 1
Professor-Mom   54
N Mar 23, 2025 by Pengu14
What is the value of $$(2^2-2) - (3^2-3) + (4^2-4)?$$
$\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 8 \qquad \textbf{(E) } 12$
54 replies
Professor-Mom
Feb 5, 2021
Pengu14
Mar 23, 2025
J
2021 AMC10A Problem 1
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Reveal topic tags
AMC
AMC 10
AMC 10 A
2021 AMC 10A
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G H BBookmark kLocked kLocked NReply
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pog
4917 posts
pog
#43 h  Nov 24, 2021, 5:52 PM
Y by
mahaler wrote:
We need @pog to write an overly complicated solution for this.
i already wrote one
This post has been edited 2 times. Last edited by pog, Nov 24, 2021, 5:52 PM
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mahaler
3084 posts
mahaler
#44 h  Nov 24, 2021, 5:53 PM
Y by
pog wrote:
mahaler wrote:
We need @pog to write an overly complicated solution for this.
i already wrote one

Oh I thought I saw it somewhere...Thanks!
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TheSocialPi
5 posts
TheSocialPi
#45 h  Dec 8, 2021, 12:10 PM
Y by
Well yeah its 8.
Z K Y
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NTistrulove
183 posts
NTistrulove
#46 h  Dec 9, 2021, 2:19 PM
Y by
math9990 wrote:
Dr.Mathematics wrote:
Professor-Mom wrote:
What is the value of $$(2^2-2) - (3^2-3) + (4^2-4)?$$
$\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 8 \qquad \textbf{(E) } 12$

Answer is , 8!

Be careful with using exclamation marks after a number, it's fine here though cause it's obvious this time :)

Get excited, but carefully!!!
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cj13609517288
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cj13609517288
#47 h  Dec 9, 2021, 2:24 PM
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Let's generalize this! Express this in terms of $n$: $$\sum_{i=2}^{n}(-1)^i(i^2-i).$$
This post has been edited 1 time. Last edited by cj13609517288, Dec 9, 2021, 2:26 PM
Reason: Let's not try to, let's do it! :D
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cj13609517288
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cj13609517288
#48 h  Dec 9, 2021, 2:32 PM • 1 Y
Y by asdf334
Generalization Solution

Now we can find this for $n=10000$ with ease! :)
This post has been edited 9 times. Last edited by cj13609517288, Dec 9, 2021, 2:40 PM
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Mr.Sharkman
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Mr.Sharkman
#49 h  Sep 12, 2022, 6:13 PM
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This can easily be found to be 2-6+12=8.
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Math4Life7
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Math4Life7
#50 h  Sep 12, 2022, 10:59 PM • 1 Y
Y by NegativeZeroPlusOne
cj13609517288 wrote:
Let's generalize this! Express this in terms of $n$: $$\sum_{i=2}^{n}(-1)^i(i^2-i).$$

dont use $i$ in a summation it's confusing
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Jessicachan
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Jessicachan
#51 h  Sep 12, 2022, 11:03 PM
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TheEvilMathematician
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TheEvilMathematician
#52 h  Sep 13, 2022, 2:44 AM
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Using common sense, the ans is $8$
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mathmax12
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mathmax12
#53 h  Sep 3, 2023, 2:11 PM
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Let $3=x,$ then, $((x-1)^2-2)-(x^2-3)+((x+1)^2-4)=x^2-2x+1-x^2+x^2+2x+1-2+3-4=x^2+2-2+3-4=x^2-1=3^2-1=\boxed{8}.$
This post has been edited 1 time. Last edited by mathmax12, Sep 3, 2023, 2:11 PM
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joshualiu315
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joshualiu315
#54 h  Sep 21, 2023, 8:12 PM
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We have $(2^2-2) - (3^2-3) + (4^2-4)=2-6+12=\boxed{8}$.
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JH_K2IMO
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JH_K2IMO
#55 h  Oct 21, 2024, 1:54 PM
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2^2-2=2, 3^2-3=6, 4^2-4=12.
2-6+12=8
The answer is D.
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JetFire008
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JetFire008
#56 h  Mar 23, 2025, 12:15 PM
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$$2^2-2=4-2=2$$$$3^2-3=9-3=6$$$$4^2-4=16-4$$So we get $2-6+12=2+6=8$
Hence the answer is (D) 8.
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Pengu14
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Pengu14
#57 h  Mar 23, 2025, 12:37 PM
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JetFire008 wrote:
$$2^2-2=4-2=2$$$$3^2-3=9-3=6$$$$4^2-4=16-4$$So we get $2-6+12=2+6=8$
Hence the answer is (D) 8.

(HOW)
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