Art of Problem Solving
AoPS Online AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy AoPS Academy
Small live classes for advanced math
and language arts learners in grades 1-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Stay ahead of learning milestones! Enroll in a class over the summer!
No tags match your search
M
G
Topic
First Poster
Last Poster
H
k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
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.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following events:
[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
[*]April 8th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS State Discussion
April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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.

Introductory: Grades 5-10

Prealgebra 1 Self-Paced

Prealgebra 1
Sunday, Apr 13 - Aug 10
Tuesday, May 13 - Aug 26
Thursday, May 29 - Sep 11
Sunday, Jun 15 - Oct 12
Monday, Jun 30 - Oct 20
Wednesday, Jul 16 - Oct 29

Prealgebra 2 Self-Paced

Prealgebra 2
Sunday, Apr 13 - Aug 10
Wednesday, May 7 - Aug 20
Monday, Jun 2 - Sep 22
Sunday, Jun 29 - Oct 26
Friday, Jul 25 - Nov 21

Introduction to Algebra A Self-Paced

Introduction to Algebra A
Monday, Apr 7 - Jul 28
Sunday, May 11 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Wednesday, May 14 - Aug 27
Friday, May 30 - Sep 26
Monday, Jun 2 - Sep 22
Sunday, Jun 15 - Oct 12
Thursday, Jun 26 - Oct 9
Tuesday, Jul 15 - Oct 28

Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
Wednesday, Apr 16 - Jul 2
Thursday, May 15 - Jul 31
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Wednesday, Jul 9 - Sep 24
Sunday, Jul 27 - Oct 19

Introduction to Number Theory
Thursday, Apr 17 - Jul 3
Friday, May 9 - Aug 1
Wednesday, May 21 - Aug 6
Monday, Jun 9 - Aug 25
Sunday, Jun 15 - Sep 14
Tuesday, Jul 15 - Sep 30

Introduction to Algebra B Self-Paced

Introduction to Algebra B
Wednesday, Apr 16 - Jul 30
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14

Introduction to Geometry
Wednesday, Apr 23 - Oct 1
Sunday, May 11 - Nov 9
Tuesday, May 20 - Oct 28
Monday, Jun 16 - Dec 8
Friday, Jun 20 - Jan 9
Sunday, Jun 29 - Jan 11
Monday, Jul 14 - Jan 19

Intermediate: Grades 8-12

Intermediate Algebra
Monday, Apr 21 - Oct 13
Sunday, Jun 1 - Nov 23
Tuesday, Jun 10 - Nov 18
Wednesday, Jun 25 - Dec 10
Sunday, Jul 13 - Jan 18
Thursday, Jul 24 - Jan 22

Intermediate Counting & Probability
Wednesday, May 21 - Sep 17
Sunday, Jun 22 - Nov 2

Intermediate Number Theory
Friday, Apr 11 - Jun 27
Sunday, Jun 1 - Aug 24
Wednesday, Jun 18 - Sep 3

Precalculus
Wednesday, Apr 9 - Sep 3
Friday, May 16 - Oct 24
Sunday, Jun 1 - Nov 9
Monday, Jun 30 - Dec 8

Advanced: Grades 9-12

Olympiad Geometry
Tuesday, Jun 10 - Aug 26

Calculus
Tuesday, May 27 - Nov 11
Wednesday, Jun 25 - Dec 17

Group Theory
Thursday, Jun 12 - Sep 11

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Wednesday, Apr 16 - Jul 2
Friday, May 23 - Aug 15
Monday, Jun 2 - Aug 18
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

MATHCOUNTS/AMC 8 Advanced
Friday, Apr 11 - Jun 27
Sunday, May 11 - Aug 10
Tuesday, May 27 - Aug 12
Wednesday, Jun 11 - Aug 27
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Problem Series
Friday, May 9 - Aug 1
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Tuesday, Jun 17 - Sep 2
Sunday, Jun 22 - Sep 21 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Monday, Jun 23 - Sep 15
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Final Fives
Sunday, May 11 - Jun 8
Tuesday, May 27 - Jun 17
Monday, Jun 30 - Jul 21

AMC 12 Problem Series
Tuesday, May 27 - Aug 12
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Wednesday, Aug 6 - Oct 22

AMC 12 Final Fives
Sunday, May 18 - Jun 15

F=ma Problem Series
Wednesday, Jun 11 - Aug 27

WOOT Programs
Visit the pages linked for full schedule details for each of these programs!

MathWOOT Level 1
MathWOOT Level 2
ChemWOOT
CodeWOOT
PhysicsWOOT

Programming

Introduction to Programming with Python
Thursday, May 22 - Aug 7
Sunday, Jun 15 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Tuesday, Jun 17 - Sep 2
Monday, Jun 30 - Sep 22

Intermediate Programming with Python
Sunday, Jun 1 - Aug 24
Monday, Jun 30 - Sep 22

USACO Bronze Problem Series
Tuesday, May 13 - Jul 29
Sunday, Jun 22 - Sep 1

Physics

Introduction to Physics
Wednesday, May 21 - Aug 6
Sunday, Jun 15 - Sep 14
Monday, Jun 23 - Sep 15

Physics 1: Mechanics
Thursday, May 22 - Oct 30
Monday, Jun 23 - Dec 15

Relativity
Sat & Sun, Apr 26 - Apr 27 (4:00 - 7:00 pm ET/1:00 - 4:00pm PT)
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)
0 replies
jlacosta
Apr 2, 2025
0 replies
J
H
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
J
H
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
J
H
3-digit palindrome and binary expansion \overline {xyx}
parmenides51   6
N an hour ago by imzzzzzz
Source: RMM Shortlist 2017 N2
Let $x, y$ and $k$ be three positive integers. Prove that there exist a positive integer $N$ and a set of $k + 1$ positive integers $\{b_0,b_1, b_2, ... ,b_k\}$, such that, for every $i = 0, 1, ... , k$ , the $b_i$-ary expansion of $N$ is a $3$-digit palindrome, and the $b_0$-ary expansion is exactly $\overline{\mbox{xyx}}$.

proposed by Bojan Basic, Serbia
6 replies
parmenides51
Jul 4, 2019
imzzzzzz
an hour ago
J
H
Bijection of the function x^3+x
chandru1   0
an hour ago
Let $n$ be a positive integer. Then for what values of $n$ is the function $f:\mathbb{Z}_{n}\to\mathbb{Z}_{n}$ defined by $f(x)=x^3+x$ bijective.
0 replies
chandru1
an hour ago
0 replies
J
H
Hard Functional Equation
yaybanana   3
N an hour ago by yaybanana
Source: own
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ , s.t :

$f(y^2+x)+f(x+yf(x))=f(y)f(y+x)+f(2x)$

for all $x,y \in \mathbb{R}$
3 replies
yaybanana
Yesterday at 3:35 PM
yaybanana
an hour ago
J
H
Orthocenter config once again
Assassino9931   6
N an hour ago by wassupevery1
Source: Bulgaria National Olympiad 2025, Day 2, Problem 4
Let \( ABC \) be an acute triangle with \( AB < AC \), midpoint $M$ of side $BC$, altitude \( AD \) (\( D \in BC \)), and orthocenter \( H \). A circle passes through points \( B \) and \( D \), is tangent to line \( AB \), and intersects the circumcircle of triangle \( ABC \) at a second point \( Q \). The circumcircle of triangle \( QDH \) intersects line \( BC \) at a second point \( P \). Prove that the lines \( MH \) and \( AP \) are perpendicular.
6 replies
Assassino9931
Tuesday at 1:53 PM
wassupevery1
an hour ago
J
H
Prove that x1=x2=....=x2025
Rohit-2006   5
N 2 hours ago by Rohit-2006
Source: A mock
The real numbers $x_1,x_2,\cdots,x_{2025}$ satisfy,
$$x_1+x_2=2\bar{x_1}, x_2+x_3=2\bar{x_2},\cdots, x_{2025}+x_1=2\bar{x_{2025}}$$Where {$\bar{x_1},\cdots,\bar{x_{2025}}$} is a permutation of $x_1,x_2,\cdots,x_{2025}$. Prove that $x_1=x_2=\cdots=x_{2025}$
5 replies
Rohit-2006
Yesterday at 5:22 AM
Rohit-2006
2 hours ago
J
H
Thanks u!
Ruji2018252   4
N 2 hours ago by teomihai
Let $a^2+b^2+c^2-2a-4b-4c=7(a,b,c\in\mathbb{R})$
Find minimum $T=2a+3b+6c$
4 replies
Ruji2018252
Yesterday at 5:52 PM
teomihai
2 hours ago
J
H
Integral solutions
KDS   4
N 2 hours ago by Maximilian113
Source: Romania TST 1993
Prove that the equation $ (x+y)^n=x^m+y^m$ has a unique solution in integers with $ x>y>0$ and $ m,n>1$.
4 replies
KDS
Jul 12, 2009
Maximilian113
2 hours ago
J
H
Romania NMO 2023 Grade 11 P1
DanDumitrescu   10
N 2 hours ago by Rohit-2006
Source: Romania National Olympiad 2023
Determine twice differentiable functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which verify relation

\[ \left( f'(x) \right)^2 + f''(x) \leq 0, \forall x \in \mathbb{R}. \]
10 replies
DanDumitrescu
Apr 14, 2023
Rohit-2006
2 hours ago
J
H
F=(F^3+F^3)/9-2F^3
Yiyj1   0
2 hours ago
Source: 101 Algebra Problems for the AMSP
Define the Fibonacci sequence $F_n$ as $$F_1=F_2=1, F_{n+1}+F_n=F_{n-1}$$fir $n \in \mathbb{N}$. Prove that $$F_{2n}=\dfrac{F_{2n+2}^3+F_{2n-2}^3}{9}-2F_{2n}^3$$for all $n \ge 2$.
0 replies
Yiyj1
2 hours ago
0 replies
J
H
4 lines concurrent
Zavyk09   2
N 2 hours ago by aidenkim119
Source: Homework
Let $ABC$ be triangle with circumcenter $(O)$ and orthocenter $H$. $BH, CH$ intersect $(O)$ again at $K, L$ respectively. Lines through $H$ parallel to $AB, AC$ intersects $AC, AB$ at $E, F$ respectively. Point $D$ such that $HKDL$ is a parallelogram. Prove that lines $KE, LF$ and $AD$ are concurrent at a point on $OH$.
2 replies
Zavyk09
Yesterday at 11:51 AM
aidenkim119
2 hours ago
J
H
Inequality => square
Rushil   13
N 2 hours ago by mqoi_KOLA
Source: INMO 1998 Problem 4
Suppose $ABCD$ is a cyclic quadrilateral inscribed in a circle of radius one unit. If $AB \cdot BC \cdot CD \cdot DA \geq 4$, prove that $ABCD$ is a square.
13 replies
Rushil
Oct 7, 2005
mqoi_KOLA
2 hours ago
J
H
where a, b, c are positive real numbers
eyesofgod1930   2
N 2 hours ago by sqing
where $a, b, c$ are positive real numbers.Prove that
$\frac{4}{\sqrt{a^{2}+b^{2}+c^{2}+4}}-\frac{9}{\sqrt{(a+b)\sqrt{(a+2c)(b+2c)}}}\leq \frac{5}{8}$
2 replies
eyesofgod1930
Jun 8, 2020
sqing
2 hours ago
J
H
Inequality, integrals, and permutations.
KAME06   0
5 hours ago
Source: Ecuador National Olympiad OMEC level U 2024 P6 Day 2
Let $n \in \mathbb{Z^+}$ such that $n \le e^{4047}$. Let $(f_1(x), f_2(x), ... ,f_n(x))$ and $(g_1(x), g_2(x), ... ,g_n(x))$ permutations of $(x, x^2, ... , x^n)$. Is possible that:
$$\int_0^1[xf_1(x)+x^2f_2(x)+...+x^nf_n(x)]dx > 2024\int_0^1[xg_1(x)+x^2g_2(x)+...+x^ng_n(x)]dx \text{ }?$$
0 replies
KAME06
5 hours ago
0 replies
J
H
maximal determinant
EthanWYX2009   2
N 6 hours ago by loup blanc
Source: 2023 Aug taca-9
Let matrix
\[A=\begin{bmatrix} 1&1&1&1&1\\1&-1&1&-1&1\\?&?&?&?&?\\?&?&?&?&?\\?&?&?&?&?\end{bmatrix}\in\mathbb R^{5\times 5}\]satisfy $\text{tr} (AA^T)=28.$ Determine the maximum value of $\det A.$
2 replies
EthanWYX2009
Yesterday at 9:55 AM
loup blanc
6 hours ago
J
H
J
H
symmetric linear combination of matrices
MetaphysicalWukong   2
N Apr 5, 2025 by rchokler
What conditions on $\alpha,\ \beta,\ \gamma$ will make M symmetric? Show all working.
2 replies
MetaphysicalWukong
Apr 2, 2025
rchokler
Apr 5, 2025
J
symmetric linear combination of matrices
G H J
Reveal topic tags
linear algebra
L
G H BBookmark kLocked kLocked NReply
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
MetaphysicalWukong
260 posts
MetaphysicalWukong
#1 h  Apr 2, 2025, 6:01 AM
Y by
What conditions on $\alpha,\ \beta,\ \gamma$ will make M symmetric? Show all working.
Attachments:
This post has been edited 1 time. Last edited by MetaphysicalWukong, Apr 2, 2025, 6:01 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
MetaphysicalWukong
260 posts
MetaphysicalWukong
#2 h  Apr 2, 2025, 7:11 AM
Y by
bumping Caizhong Qu
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
rchokler
2954 posts
rchokler
#3 h  Apr 5, 2025, 2:19 AM
Y by
Note that $(AB)^\top=B^\top A^\top$ in general, so $(KK^\top)^\top=(K^\top)^\top K^\top=KK^\top$.

$M=\alpha K+\beta K^\top+\gamma KK^\top\implies M^\top=\alpha K^\top+\beta K+\gamma KK^\top\implies M-M^\top=(\alpha-\beta)(K-K^\top)$.

This difference is skew-symmetric, and will thus only be the zero-matrix if either $\alpha=\beta$ or $K$ is symmetric.

Since we want $M$ symmetric for all square matrices $K$, we conclude $\alpha=\beta$.

The two choices you checked off are thus the only ones that work.
Z K Y
N Quick Reply
G
H
=
a