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

Join our free webinar April 22 to learn about competitive programming!
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
domino question
kjhgyuio   0
8 minutes ago
........
0 replies
kjhgyuio
8 minutes ago
0 replies
J
H
demonic monic polynomial problem
iStud   0
20 minutes ago
Source: Monthly Contest KTOM April P4 Essay
(a) Let $P(x)$ be a monic polynomial so that there exists another real coefficients $Q(x)$ that satisfy
\[P(x^2-2)=P(x)Q(x)\]Determine all complex roots that are possible from $P(x)$
(b) For arbitrary polynomial $P(x)$ that satisfies (a), determine whether $P(x)$ should have real coefficients or not.
0 replies
iStud
20 minutes ago
0 replies
J
H
fun set problem
iStud   0
23 minutes ago
Source: Monthly Contest KTOM April P2 Essay
Given a set $S$ with exactly 9 elements that is subset of $\{1,2,\dots,72\}$. Prove that there exist two subsets $A$ and $B$ that satisfy the following:
- $A$ and $B$ are non-empty subsets from $S$,
- the sum of all elements in each of $A$ and $B$ are equal, and
- $A\cap B$ is an empty subset.
0 replies
iStud
23 minutes ago
0 replies
J
H
two tangent circles
KPBY0507   3
N 39 minutes ago by Sanjana42
Source: FKMO 2021 Problem 5
The incenter and $A$-excenter of $\triangle{ABC}$ is $I$ and $O$. The foot from $A,I$ to $BC$ is $D$ and $E$. The intersection of $AD$ and $EO$ is $X$. The circumcenter of $\triangle{BXC}$ is $P$.
Show that the circumcircle of $\triangle{BPC}$ is tangent to the $A$-excircle if $X$ is on the incircle of $\triangle{ABC}$.
3 replies
+1 w
KPBY0507
May 8, 2021
Sanjana42
39 minutes ago
J
H
basically INAMO 2010/6
iStud   0
39 minutes ago
Source: Monthly Contest KTOM April P1 Essay
Call $n$ kawaii if it satisfies $d(n)+\varphi(n)+1=n$ ($d(n)$ is the number of positive factors of $n$, while $\varphi(n)$ is the number of integers not more than $n$ that are relatively prime with $n$). Find all $n$ that is kawaii.
0 replies
iStud
39 minutes ago
0 replies
J
H
trolling geometry problem
iStud   0
43 minutes ago
Source: Monthly Contest KTOM April P3 Essay
Given a cyclic quadrilateral $ABCD$ with $BC<AD$ and $CD<AB$. Lines $BC$ and $AD$ intersect at $X$, and lines $CD$ and $AB$ intersect at $Y$. Let $E,F,G,H$ be the midpoints of sides $AB,BC,CD,DA$, respectively. Let $S$ and $T$ be points on segment $EG$ and $FH$, respectively, so that $XS$ is the angle bisector of $\angle{DXA}$ and $YT$ is the angle bisector of $\angle{DYA}$. Prove that $TS$ is parallel to $BD$ if and only if $AC$ divides $ABCD$ into two triangles with equal area.
0 replies
iStud
43 minutes ago
0 replies
J
H
My hardest algebra ever created (only one solve in the contest)
mshtand1   6
N an hour ago by mshtand1
Source: Ukraine IMO TST P9
Find all functions \( f: (0, +\infty) \to (0, +\infty) \) for which, for all \( x, y > 0 \), the following identity holds:
\[ f(x) f(yf(x)) + y f(xy) = \frac{f\left(\frac{x}{y}\right)}{y} + \frac{f\left(\frac{y}{x}\right)}{x} \]
Proposed by Mykhailo Shtandenko
6 replies
mshtand1
Apr 19, 2025
mshtand1
an hour ago
J
H
Killer NT that nobody solved (also my hardest NT ever created)
mshtand1   4
N 2 hours ago by mshtand1
Source: Ukraine IMO 2025 TST P8
A positive integer number \( a \) is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence \( \{b_k\}_{k=1}^{\infty} \), where
\[ b_k = a^{k^k} \cdot 2^{2^k - k} + 1. \]
Proposed by Arsenii Nikolaev and Mykhailo Shtandenko
4 replies
mshtand1
Apr 19, 2025
mshtand1
2 hours ago
J
H
Advanced topics in Inequalities
va2010   22
N 2 hours ago by Primeniyazidayi
So a while ago, I compiled some tricks on inequalities. You are welcome to post solutions below!
22 replies
va2010
Mar 7, 2015
Primeniyazidayi
2 hours ago
J
H
Funny easy transcendental geo
qwerty123456asdfgzxcvb   0
3 hours ago
Let $\mathcal{S}$ be a logarithmic spiral centered at the origin (ie curve satisfying for any point $X$ on it, line $OX$ makes a fixed angle with the tangent to $\mathcal{S}$ at $X$). Let $\mathcal{H}$ be a rectangular hyperbola centered at the origin, scaled such that it is tangent to the logarithmic spiral at some point.

Prove that for a point $P$ on the spiral, the polar of $P$ wrt. $\mathcal{H}$ is tangent to the spiral.
0 replies
qwerty123456asdfgzxcvb
3 hours ago
0 replies
J
H
Nice problem about a trapezoid
manlio   1
N 3 hours ago by kiyoras_2001
Have you a nice solution for this problem?
Thank you very much
1 reply
manlio
Apr 19, 2025
kiyoras_2001
3 hours ago
J
H
product of the first n terms
FireBreathers   5
N 3 hours ago by ihategeo_1969
Does there exist an infinite sequence of positive integers $a_i$ such that every positive integer appears exactly once and the product of the first $n$ terms is a perfect $n - th$ power ?
5 replies
FireBreathers
Dec 16, 2024
ihategeo_1969
3 hours ago
J
H
Inequality with three conditions
oVlad   1
N 3 hours ago by Haris1
Source: Romania EGMO TST 2019 Day 1 P3
Let $a,b,c$ be non-negative real numbers such that \[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]Prove that $a^2+b^2+c^2\leqslant 2abc+1.$
1 reply
oVlad
Today at 1:48 PM
Haris1
3 hours ago
J
H
Paint and Optimize: A Grid Strategy Problem
mojyla222   2
N 3 hours ago by sami1618
Source: Iran 2025 second round p2
Ali and Shayan are playing a turn-based game on an infinite grid. Initially, all cells are white. Ali starts the game, and in the first turn, he colors one unit square black. In the following turns, each player must color a white square that shares at least one side with a black square. The game continues for exactly 2808 turns, after which each player has made 1404 moves. Let $A$ be the set of black cells at the end of the game. Ali and Shayan respectively aim to minimize and maximise the perimeter of the shape $A$ by playing optimally. (The perimeter of shape $A$ is defined as the total length of the boundary segments between a black and a white cell.)

What are the possible values of the perimeter of $A$, assuming both players play optimally?
2 replies
mojyla222
Yesterday at 4:25 AM
sami1618
3 hours ago
J
H
J
H
fractions question
kjhgyuio   2
N Apr 7, 2025 by Filipjack

........
2 replies
kjhgyuio
Apr 7, 2025
Filipjack
Apr 7, 2025
J
fractions question
G H J
Reveal topic tags
Fractions
solved by unknown method
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.
kjhgyuio
50 posts
kjhgyuio
#1 h  Apr 7, 2025, 9:02 AM
Y by
........
Attachments:
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
kjhgyuio
50 posts
kjhgyuio
#2 h  Apr 7, 2025, 9:05 AM
Y by
Click to reveal hidden text





........
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Filipjack
872 posts
Filipjack
#3 h  Apr 7, 2025, 9:13 AM
Y by
$\frac{7}{2009}<\frac{1}{2003}+\ldots+\frac{1}{2009} < \frac{7}{2003},$ so $286<\frac{2003}{7}<\frac{1}{\frac{1}{2003}+\ldots+\frac{1}{2009}}< \frac{2009}{7}=287,$ so the integer part is $286.$
This post has been edited 1 time. Last edited by Filipjack, Apr 7, 2025, 9:13 AM
Z K Y
N Quick Reply
G
H
=
a