Join our FREE webinar on May 1st to learn about managing anxiety.

G
Topic
First Poster
Last Poster
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
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
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
4 variables with quadrilateral sides 2
mihaig   4
N 5 minutes ago by arqady
Source: Own
Let $a,b,c,d\geq0$ satisfying
$$\frac1{a+1}+\frac1{b+1}+\frac1{c+1}+\frac1{d+1}=2.$$Prove
$$\left(a+b+c+d-2\right)^2+8\geq3\left(abc+abd+acd+bcd\right).$$
4 replies
mihaig
Tuesday at 8:47 PM
arqady
5 minutes ago
BMO 2015 #1: Inequality on a,b,c.
MathKnight16   25
N 20 minutes ago by Rayvhs
Source: BMO 2015 problem 1
If ${a, b}$ and $c$ are positive real numbers, prove that

\begin{align*}
 a ^ 3b ^ 6 + b ^ 3c ^ 6 + c ^ 3a ^ 6 + 3a ^ 3b ^ 3c ^ 3 &\ge{ abc \left (a ^ 3b ^ 3 + b ^ 3c ^ 3 + c ^ 3a ^ 3 \right) + a ^ 2b ^ 2c ^ 2 \left (a ^ 3 + b ^ 3 + c ^ 3 \right)}.
\end{align*}

(Montenegro).
25 replies
MathKnight16
May 5, 2015
Rayvhs
20 minutes ago
4 lines concurrent
Zavyk09   5
N 22 minutes ago by tomsuhapbia
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 $OH$.
5 replies
Zavyk09
Apr 9, 2025
tomsuhapbia
22 minutes ago
SL 2015 G1: Prove that IJ=AH
Problem_Penetrator   136
N an hour ago by Mathgloggers
Source: IMO 2015 Shortlist, G1
Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.
136 replies
Problem_Penetrator
Jul 7, 2016
Mathgloggers
an hour ago
An inequality in geometry
MattArg   0
an hour ago
Let $ABC$ be a triangle such that $BC\ge AB$ and $BC\ge AC$
Let $M$ be any point in the plane. Prove that $BM+CM\ge AM$
0 replies
MattArg
an hour ago
0 replies
Question 4
Valentin Vornicu   128
N an hour ago by DensSv
Source: IMO Shortlist 2007, G1
In triangle $ ABC$ the bisector of angle $ BCA$ intersects the circumcircle again at $ R$, the perpendicular bisector of $ BC$ at $ P$, and the perpendicular bisector of $ AC$ at $ Q$. The midpoint of $ BC$ is $ K$ and the midpoint of $ AC$ is $ L$. Prove that the triangles $ RPK$ and $ RQL$ have the same area.

Author: Marek Pechal, Czech Republic
128 replies
Valentin Vornicu
Jul 26, 2007
DensSv
an hour ago
from Steiner-line to tangent
Zavyk09   1
N an hour ago by tomsuhapbia
A triangle $ABC$ is inscribed in circle $\omega$ and a point $P$ on $\omega$. $A'$ be the reflection of $A$ through $BC$. Let $Q$ be a point on the Steiner-line of $P$ wrt $\triangle ABC $. $AQ$ intersects the circumcircle of $\triangle BQC $ again at $Q'$. $PQ$ intersects $\omega$ again at $R$. Prove that $AR$ tangents to the circumcircle of $\triangle AA'Q' $.
1 reply
Zavyk09
Dec 11, 2024
tomsuhapbia
an hour ago
Necessary and sufficient condition
Fang-jh   7
N an hour ago by MathLuis
Source: Chinese TST 2009 3rd quiz P2
In convex quadrilateral $ ABCD$, $ CB,DA$ are external angle bisectors of $ \angle DCA,\angle CDB$, respectively. Points $ E,F$ lie on the rays $ AC,BD$ respectively such that $ CEFD$ is cyclic quadrilateral. Point $ P$ lie in the plane of quadrilateral $ ABCD$ such that $ DA,CB$ are external angle bisectors of $ \angle PDE,\angle PCF$ respectively. $ AD$ intersects $ BC$ at $ Q.$ Prove that $ P$ lies on $ AB$ if and only if $ Q$ lies on segment $ EF$.
7 replies
Fang-jh
Mar 22, 2009
MathLuis
an hour ago
expansion of polynomial
luutrongphuc   1
N an hour ago by luutrongphuc
Consider the sequence $(T_n)$ defined by:
\[
T_0 = 0,\quad T_1 = T_2 = 1,
\]and
\[
T_{n+3} = T_{n+2} + T_{n+1} + T_n \quad \text{for all } n \geq 0.
\]Prove that, for every positive integer $n$, we have:
\[
T_{3n} = a_0 T_0 + a_1 T_1 + a_2 T_2 + \cdots + a_{2n} T_{2n},
\]where, for each $i \in \{0, 1, 2, \ldots, 2n\}$, $a_i$ is the coefficient of $x^i$ in the expansion of the polynomial $(x^2 + x + 1)^n$.
1 reply
luutrongphuc
2 hours ago
luutrongphuc
an hour ago
calculation !
nttu   22
N an hour ago by lpieleanu
Source: IMO ShortList 1998, geometry problem 7
Let $ABC$ be a triangle such that $\angle ACB=2\angle ABC$. Let $D$ be the point on the side $BC$ such that $CD=2BD$. The segment $AD$ is extended to $E$ so that $AD=DE$. Prove that \[ \angle ECB+180^{\circ }=2\angle EBC. \]
22 replies
nttu
Oct 15, 2004
lpieleanu
an hour ago
Do not try to bash on beautiful geometry
ItzsleepyXD   6
N an hour ago by DottedCaculator
Source: Own , Mock Thailand Mathematic Olympiad P9
Let $ABC$be triangle with point $D,E$ and $F$ on $BC,AB,CA$
such that $BE=CF$ and $E,F$ are on the same side of $BC$
Let $M$ be midpoint of segment $BC$ and $N$ be midpoint of segment $EF$
Let $G$ be intersection of $BF$ with $CE$ and $\dfrac{BD}{DC}=\dfrac{AC}{AB}$
Prove that $MN\parallel DG$
6 replies
ItzsleepyXD
Yesterday at 9:30 AM
DottedCaculator
an hour ago
If ab+1 is divisible by A then so is a+b
ravengsd   2
N an hour ago by ravengsd
Source: Romania EGMO TST 2025 Day 2, Problem 4
Find the greatest positive integer $A$ such that, for all positive integers $a$ and $b$, if $A$ divides $ab+1$, then $A$ divides $a+b$.
2 replies
ravengsd
4 hours ago
ravengsd
an hour ago
Here is another solution which looks a bit like the one by p_square. Let r1, r2,
Techno0-8   0
an hour ago
Here is another solution which looks a bit like the one by p_square.
Let r1, r2, r3, be radii close to the vertices A, B and C, respectively. And Let ra be the radius of the circle which is tangent inwards to AB, AC and the line parallel to BC ( other than BC) and tangent to the incircle. Define rb and rc similarly. We claim that r1>=ra, r2>=rb and r3>=rc. Because clearly if radii are lower than these then it is not gonna be enough to touch the incircle.
Let h1, h2, h3 be altitudes and S the area of the ABC. Now
(r1+r2+r3)/r >= (ra+rb+rc)/r =
(h1-2r) /h1+
(h2-2r) /h2+
(h3-2r) /h3 =
3 - 2 • r • (1/h1 +1/h2 +1/h3) =
3 - 2 • 2S/(AB + BC+AC) • (AB+BC+CA) /2S=
3 - 2 = 1 and we are done.
0 replies
Techno0-8
an hour ago
0 replies
more incircles and tangents
rmtf1111   84
N 2 hours ago by cj13609517288
Source: EGMO 2019 P4
Let $ABC$ be a triangle with incentre $I$. The circle through $B$ tangent to $AI$ at $I$ meets side $AB$ again at $P$. The circle through $C$ tangent to $AI$ at $I$ meets side $AC$ again at $Q$. Prove that $PQ$ is tangent to the incircle of $ABC.$
84 replies
rmtf1111
Apr 10, 2019
cj13609517288
2 hours ago
NT Function with divisibility
oVlad   3
N Apr 10, 2025 by sangsidhya
Source: Romanian District Olympiad 2023 9.4
Determine all strictly increasing functions $f:\mathbb{N}_0\to\mathbb{N}_0$ which satisfy \[f(x)\cdot f(y)\mid (1+2x)\cdot f(y)+(1+2y)\cdot f(x)\]for all non-negative integers $x{}$ and $y{}$.
3 replies
oVlad
Mar 11, 2023
sangsidhya
Apr 10, 2025
NT Function with divisibility
G H J
G H BBookmark kLocked kLocked NReply
Source: Romanian District Olympiad 2023 9.4
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
oVlad
1742 posts
#1 • 1 Y
Y by tiendung2006
Determine all strictly increasing functions $f:\mathbb{N}_0\to\mathbb{N}_0$ which satisfy \[f(x)\cdot f(y)\mid (1+2x)\cdot f(y)+(1+2y)\cdot f(x)\]for all non-negative integers $x{}$ and $y{}$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
straight
413 posts
#2
Y by
$f(x) = 2x+1$ or $f(x) = 4x+2$.

You can prove for primes $2a+1$ that $f(a) = 2a+1$ or $f(a) = 4a+2$. Then use the strict increasing fact to show that in both cases we need every number between to primes to satisfy $f(b) = 2b+1$ or $f(b) = 4b+2$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
sanyalarnab
930 posts
#3 • 1 Y
Y by Mra123
Joy Bangla! The following proof is what @straight has pointed out.
The answer is $f(x)=2x+1$ and $f(x)=4x+2$ which can be easily verified.
Let $P(x,y)$ denote the given assertion.
Also consider $2<p_1<p_2< \cdots$ to be the infinite sequence of primes.
$P(0,0): f(0) \in \{1,2\}$.
Case 1: $f(0)=1$
$P(x,0):f(x)|2x+1$
Thus $f \left(\frac{p_k-1}{2}\right)=p_k$ for all $k\in \mathbb{N}$.
Also note that $P(x,0) \implies f(x)\text{ is odd}$ which in turn implies that $f(x+1) - f(x) \geq 2$ for all non negative integers $x$.Next we have the following chain(for any $k\in \mathbb{N}$):
$$p_k=\underbrace{f \left(\frac{p_k-1}{2}\right)<f \left(\frac{p_k+1}{2}\right)}_{\text{jump of length more than 1}}<\cdots < f \left(\frac{p_{k+1}-1}{2}\right)=p_{k+1}$$Thus all odd numbers between $p_k$ and $p_{k+1}$ are mapped with corresponding values of $f$. Also $\mathbb{N} = \bigcup_{k\in \mathbb{N}} \left[\frac{p_k-1}{2},\frac{p_{k+1}-1}{2}\right]$ which shows that $f(x)=2x+1$ is true for all natural numbers $x$.
Case 2: $f(0)=2$
Similar argument follows just like Case 1. $P(x,0): f(x)|4x+2$ and $2||f(x)$. Thus we get that $f(x+1) - f(x) \geq 4$ for all non negative integers $x$ and $f \left(\frac{p_k-1}{2}\right)=2p_k$ for all $k\in \mathbb{N}$.Next we have the following chain(for any $k\in \mathbb{N}$):
$$2p_k=\underbrace{f \left(\frac{p_k-1}{2}\right)<f \left(\frac{p_k+1}{2}\right)}_{\text{jump of length more than 3}}<\cdots < f \left(\frac{p_{k+1}-1}{2}\right)=2p_{k+1}$$Thus all numbers $\equiv 2 \pmod 4$ between $2p_k$ and $2p_{k+1}$ are mapped with corresponding values of $f$. Also $\mathbb{N} = \bigcup_{k\in \mathbb{N}} \left[\frac{p_k-1}{2},\frac{p_{k+1}-1}{2}\right]$ which shows that $f(x)=4x+2$ is true for all natural numbers $x$. :D
This post has been edited 1 time. Last edited by sanyalarnab, Mar 19, 2024, 7:48 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
sangsidhya
19 posts
#4
Y by
arnab da orz
Z K Y
N Quick Reply
G
H
=
a