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

Free webinar 4/3 about Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor.  
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
Yesterday at 3:18 PM
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
Yesterday at 3:18 PM
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
n x n square and strawberries
pohoatza   18
N a minute ago by shanelin-sigma
Source: IMO Shortlist 2006, Combinatorics 4, AIMO 2007, TST 4, P2
A cake has the form of an $ n$ x $ n$ square composed of $ n^{2}$ unit squares. Strawberries lie on some of the unit squares so that each row or column contains exactly one strawberry; call this arrangement $\mathcal{A}$.

Let $\mathcal{B}$ be another such arrangement. Suppose that every grid rectangle with one vertex at the top left corner of the cake contains no fewer strawberries of arrangement $\mathcal{B}$ than of arrangement $\mathcal{A}$. Prove that arrangement $\mathcal{B}$ can be obtained from $ \mathcal{A}$ by performing a number of switches, defined as follows:

A switch consists in selecting a grid rectangle with only two strawberries, situated at its top right corner and bottom left corner, and moving these two strawberries to the other two corners of that rectangle.
18 replies
pohoatza
Jun 28, 2007
shanelin-sigma
a minute ago
J
H
Very interesting inequalities
sqing   1
N 2 minutes ago by SunnyEvan
Source: Own
Let $ x ,y \geq 0 $ and $ x^2 -x+ \frac{1}{2}y\leq 1.$ Prove that
$$x^2 + ky \leq \frac{k(5k-2)}{2k-1}$$Where $ k\in N^+.$
$$x^2 + y \leq 3$$$$x^2 + 2y \leq \frac{16}{3}$$
1 reply
+1 w
sqing
an hour ago
SunnyEvan
2 minutes ago
J
H
The Tetrahedral Space Partition
jannatiar   7
N 2 minutes ago by sami1618
Source: 2025 AlborzMO Day 2 P3
Is it possible to partition three-dimensional space into tetrahedra (not necessarily regular) such that there exists a plane that intersects the edges of each tetrahedron at exactly 4 or 0 points?

Proposed by Arvin Taheri
7 replies
jannatiar
Mar 9, 2025
sami1618
2 minutes ago
J
H
Nasty Floor Sum with Omega Function
Kezer   8
N 10 minutes ago by cubres
Source: Bulgaria 1989, Evan Chen's Summation Handout
Let $\Omega(n)$ denote the number of prime factors of $n$, counted with multiplicity. Evaluate \[ \sum_{n=1}^{1989} (-1)^{\Omega(n)}\left\lfloor \frac{1989}{n} \right \rfloor. \]
8 replies
Kezer
Jul 15, 2017
cubres
10 minutes ago
J
H
NT functional equation on positive integers
RootofUnityfilter   0
12 minutes ago
Source: my teacher
Let $\mathbb{Z^+}$ denote the set of positive integers.
Find all function $f: \mathbb{Z^+} \rightarrow \mathbb{Z^+}$ such that for all $m, n \in \mathbb{Z^+}$,
\[f(mn)^2 = f(m^2)f(f(n))f(mf(n))\]if and only if $\gcd(m, n) = 1$
0 replies
RootofUnityfilter
12 minutes ago
0 replies
J
H
Inspired by old results
sqing   0
21 minutes ago
Source: Own
Let $ a,b,c \ge \frac{1}{21} $ and $ a+b+c=1. $. Prove that
$$(a^2-ab+b^2)(b^2-bc+c^2)(c^2-ca+a^2)\geq\frac{1}{729} $$Let $ a,b,c \ge \frac{1}{10} $ and $ a+b+c=2. $. Prove that
$$(a^2-ab+b^2)(b^2-bc+c^2)(c^2-ca+a^2)\geq\frac{64}{729} $$Let $ a,b,c \ge \frac{1}{11} $ and $ a+b+c=2. $. Prove that
$$(a^2-ab+b^2)(b^2-bc+c^2)(c^2-ca+a^2)\geq \frac{145161}{1771561} $$
0 replies
1 viewing
sqing
21 minutes ago
0 replies
J
H
There is a tri-blued unit square
Ankoganit   18
N 34 minutes ago by alexanderhamilton124
Source: India TST 2016 Day 3 Problem 3
Let $n$ be an odd natural number. We consider an $n\times n$ grid which is made up of $n^2$ unit squares and $2n(n+1)$ edges. We colour each of these edges either $\color{red} \textit{red}$ or $\color{blue}\textit{blue}$. If there are at most $n^2$ $\color{red} \textit{red}$ edges, then show that there exists a unit square at least three of whose edges are $\color{blue}\textit{blue}$.
18 replies
Ankoganit
Jul 22, 2016
alexanderhamilton124
34 minutes ago
J
H
D is incenter
Layaliya   2
N 36 minutes ago by Lil_flip38
Source: From my friend in Indonesia
Given an acute triangle \( ABC \) where \( AB > AC \). Point \( O \) is the circumcenter of triangle \( ABC \), and \( P \) is the projection of point \( A \) onto line \( BC \). The midpoints of \( BC \), \( CA \), and \( AB \) are \( D \), \( E \), and \( F \), respectively. The line \( AO \) intersects \( DE \) and \( DF \) at points \( Q \) and \( R \), respectively. Prove that \( D \) is the incenter of triangle \( PQR \).
2 replies
Layaliya
3 hours ago
Lil_flip38
36 minutes ago
J
H
Mmo 9-10 graders P5
Bet667   4
N 40 minutes ago by Bet667
Let $a,b,c,d$ be real numbers less than 2.Then prove that $\frac{a^3}{b^2+4}+\frac{b^3}{c^2+4}+\frac{c^3}{d^2+4}+\frac{d^3}{a^2+4}\le4$
4 replies
Bet667
Today at 6:54 AM
Bet667
40 minutes ago
J
H
Medians and stuff regarding it
Konigsberg   2
N an hour ago by lightsynth123
Source: SMO(O) 2014 #2
Let $ABC$ be an acute-angled triangle and let $D$, $E$, and $F$ be the midpoints of $BC$, $CA$, and $AB$ respectively. Construct a circle, centered at the orthocenter of triangle $ABC$, such that triangle $ABC$ lies in the interior of the circle. Extend $EF$ to intersect the circle at $P$, $FD$ to intersect the circle at $Q$ and $DE$ to intersect the circle at $R$. Show that $AP=BQ=CR$.
2 replies
Konigsberg
Jan 22, 2015
lightsynth123
an hour ago
J
H
interesting nt
medhimdi   0
an hour ago
Source: OFM2021 Senior P1
Let $a_1, a_2, a_3, \dots$ and $b_1, b_2, b3, \dots$ be two sequences of integers such that $a{n+2}=a_{n+1}+an$ and $b{n+2}=b_{n+1}+b_n$ for all $n\geq1$. Suppose that $a_n$ divides $b_n$ for an infinity of integers $n\geq1$. Prove that there exist an integer $c$ such that $b_n=ca_n$ for all $n\geq1$
0 replies
medhimdi
an hour ago
0 replies
J
H
Inequality from China
sqing   0
an hour ago
Source: lemondian(https://kuing.cjhb.site/thread-13667-1-1.html)
Let $x\in (0,\frac{\pi}{2}) . $ Prove that $$tanx\ge x^k$$Where $ k=1,2,3,4.$
0 replies
sqing
an hour ago
0 replies
J
H
A problem
KhuongTrang   18
N an hour ago by KhuongTrang
Source: own
Problem. Let $a,b,c$ be positive real variables such that $a+b+c=3.$ Prove that $$\color{black}{\sqrt{2a^{2}-a+3}+\sqrt{2b^{2}-b+3}+\sqrt{2c^{2}-c+3}\le 3\sqrt{\frac{ab+bc+ca+9}{ab+bc+ca}}.}$$
18 replies
KhuongTrang
Dec 2, 2024
KhuongTrang
an hour ago
J
H
Circle !
ComplexPhi   6
N 2 hours ago by mathbetter
Let $AB$ and $CD$ be chords in a circle of center $O$ with $A , B , C , D$ distinct , and with the lines $AB$ and $CD$ meeting at a right angle at point $E$. Let also $M$ and $N$ be the midpoints of $AC$ and $BD$ respectively . If $MN \bot OE$ , prove that $AD \parallel BC$.
6 replies
ComplexPhi
Feb 4, 2015
mathbetter
2 hours ago
J
H
J
H
fraction sum
miiirz30   4
N Apr 1, 2025 by miiirz30
Source: 2025 Euler Olympiad, Round 1
Evaluate the following sum:
$$ \frac{1}{1} + \frac{1}{1 + 2} + \frac{1}{1 + 2 + 3} + \frac{1}{1 + 2 + 3 + 4} + \ldots + \frac{1}{1 + 2 + 3 + 4 + \dots + 2025} $$
Proposed by Prudencio Guerrero Fernández
4 replies
miiirz30
Mar 31, 2025
miiirz30
Apr 1, 2025
J
fraction sum
G H J
Reveal topic tags
Euler Olympiad
algebra
L
G H BBookmark kLocked kLocked NReply
Source: 2025 Euler Olympiad, Round 1
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
miiirz30
21 posts
miiirz30
#1 h  Mar 31, 2025, 5:52 PM
Y by
Evaluate the following sum:
$$ \frac{1}{1} + \frac{1}{1 + 2} + \frac{1}{1 + 2 + 3} + \frac{1}{1 + 2 + 3 + 4} + \ldots + \frac{1}{1 + 2 + 3 + 4 + \dots + 2025} $$
Proposed by Prudencio Guerrero Fernández
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
ehuseyinyigit
799 posts
ehuseyinyigit
#2 h  Mar 31, 2025, 6:43 PM
Y by
$$\sum_{k=1}^{2025}{\dfrac{1}{\sum_{j=1}^{k}{j}}}=2\left(\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{2026}\right)=2-\dfrac{1}{1013}=\dfrac{2025}{1013}$$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
NicoN9
94 posts
NicoN9
#3 h  Apr 1, 2025, 12:43 AM
Y by
Telescoping. Since $\frac{1}{1+\dots +n}= \frac{2}{n(n+1)}=2\left(\frac{1}{n}-\frac{1}{n+1}\right)$, this is equal to\[ 2\sum_{k=1}^{2025}\left(\frac{1}{k}-\frac{1}{k+1}\right)=2\left(1-\frac{1}{2026}\right) =\frac{2025}{1013}. \]
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
KAME06
128 posts
KAME06
#5 h  Apr 1, 2025, 1:12 AM
Y by
miiirz30 wrote:
Evaluate the following sum:
$$ \frac{1}{1} + \frac{1}{1 + 2} + \frac{1}{1 + 2 + 3} + \frac{1}{1 + 2 + 3 + 4} + \ldots + \frac{1}{1 + 2 + 3 + 4 + \dots + 2025} $$
Proposed by Prudencio Guerrero Fernández

Where does that competition come from?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
miiirz30
21 posts
miiirz30
#6 h  Apr 1, 2025, 6:19 AM
Y by
KAME06 wrote:
Where does that competition come from?

Georgia.
Z K Y
N Quick Reply
G
H
=
a