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
Very Cute Functional Equation :)
YLG_123   2
N 13 minutes ago by bin_sherlo
Source: Olimphíada 2021 - Problem 6
Let $\mathbb{Z}_{>0}$ be the set of positive integers. Find all functions $f : \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that, for all $m, n \in \mathbb{Z}_{>0 }$:
$$f(mf(n)) + f(n) | mn + f(f(n)).$$
2 replies
YLG_123
Jul 9, 2023
bin_sherlo
13 minutes ago
J
H
The order of colors
Entei   0
28 minutes ago
There are $3n$ balls, with $n$ red, $n$ green, and $n$ blue balls, randomly arranged in a row. Two observers, one at the front and one at the back, each record the order of the first appearance of each color. What is the probability that both observers record the same order of colors?

For example, the sequence RGGBRB would be read as RGB for the front observer and BRG for the back observer.
0 replies
Entei
28 minutes ago
0 replies
J
H
Romanian National Olympiad 1997 - Grade 10 - Problem 4
Filipjack   1
N 34 minutes ago by MS_asdfgzxcvb
Source: Romanian National Olympiad 1997 - Grade 10 - Problem 4
Let $a_0,$ $a_1,$ $\ldots,$ $a_n$ be complex numbers such that [center]$|a_nz^n+a_{n-1}z^{n-1}+\ldots+a_1z+a_0| \le 1,$ for any $z \in \mathbb{C}$ with $|z|=1.$[/center]

Prove that $|a_k| \le 1$ and $|a_0+a_1+\ldots+a_n-(n+1)a_k| \le n,$ for any $k=\overline{0,n}.$
1 reply
Filipjack
an hour ago
MS_asdfgzxcvb
34 minutes ago
J
H
Geometry
youochange   3
N 36 minutes ago by Double07
m:}
Let $\triangle ABC$ be a triangle inscribed in a circle, where the tangents to the circle at points $B$ and $C$ intersect at the point $P$. Let $M$ be a point on the arc $AC$ (not containing $B$) such that $M \neq A$ and $M \neq C$. Let the lines $BC$ and $AM$ intersect at point $K$. Let $P'$ be the reflection of $P$ with respect to the line $AM$. The lines $AP'$ and $PM$ intersect at point $Q$, and $PM$ intersects the circumcircle of $\triangle ABC$ again at point $N$.

Prove that the point $Q$ lies on the circumcircle of $\triangle ANK$.
3 replies
youochange
Today at 11:27 AM
Double07
36 minutes ago
J
H
Putnam 2017 A6
Kent Merryfield   9
N 3 hours ago by imzzzzzz
The $30$ edges of a regular icosahedron are distinguished by labeling them $1,2,\dots,30.$ How many different ways are there to paint each edge red, white, or blue such that each of the 20 triangular faces of the icosahedron has two edges of the same color and a third edge of a different color?
9 replies
Kent Merryfield
Dec 3, 2017
imzzzzzz
3 hours ago
J
H
Matrices and combinatorics
KAME06   1
N 4 hours ago by Rainbow1971
Source: Ecuador National Olympiad OMEC level U 2024 P1 Day 1
Let $n \in \mathbb{Z}$. A matrix is n-national if its size is $2 \times 2$ and their entries belong to the set $\{2, 2^2, 2^3, ..., 2^n\}$. For example:
$$\begin{bmatrix} 2 & 8 \\ 16 & 4 \end{bmatrix}, \begin{bmatrix} 4 & 4 \\ 8 & 8 \end{bmatrix}, \begin{bmatrix} 8 & 2 \\ 16 & 8 \end{bmatrix}$$For all $n \in \mathbb{Z}$, find the number of invertible n-national matrices.
1 reply
KAME06
Yesterday at 7:59 PM
Rainbow1971
4 hours ago
J
H
Putnam 2000 A6
ahaanomegas   15
N 4 hours ago by Levieee
Let $f(x)$ be a polynomial with integer coefficients. Define a sequence $a_0, a_1, \cdots $ of integers such that $a_0=0$ and $a_{n+1}=f(a_n)$ for all $n \ge 0$. Prove that if there exists a positive integer $m$ for which $a_m=0$ then either $a_1=0$ or $a_2=0$.
15 replies
ahaanomegas
Sep 6, 2011
Levieee
4 hours ago
J
H
Polynomial meets geometry
chirita.andrei   1
N 5 hours ago by AndreiVila
Source: Own. Proposed for Romanian National Olympiad 2025.
(a) Let $A,B,C$ be collinear points (in order) and $D$ a point in plane. Consider the disc $\mathcal{D}$ of center $D$ and radius $kBD$, for some $k\in(0,1)$. Prove that $\mathcal{D}\cap [AC]$ is either the empty set or a segment of length at most $2kAC$.
(b) Let $n$ be a positive integer and $P(X)\in\mathbb{C}[X]$ be a polynomial of degree $n$. Prove that \[\sup_{x\in[0,1]}|P(x)|\le(2n+1)^{n+1}\int\limits_{0}^{1}|P(x)|\mathrm{d}x.\]
1 reply
chirita.andrei
Apr 2, 2025
AndreiVila
5 hours ago
J
H
real analysis
ay19bme   1
N 5 hours ago by alexheinis
.................
1 reply
ay19bme
Today at 10:04 AM
alexheinis
5 hours ago
J
H
Romanian National Olympiad 1997 - Grade 11 - Problem 3
Filipjack   1
N 5 hours ago by MS_asdfgzxcvb
Source: Romanian National Olympiad 1997 - Grade 11 - Problem 3
Let $\mathcal{F}$ be the set of the differentiable functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x) \ge f(x+ \sin x)$ for any $x \in \mathbb{R}.$

a) Prove that there exist nonconstant functions in $\mathcal{F}.$

b) Prove that if $f \in \mathcal{F},$ then the set of solutions of the equation $f'(x)=0$ is infinite.
1 reply
Filipjack
Today at 10:11 AM
MS_asdfgzxcvb
5 hours ago
J
H
Maximizing absolute value of directional derivative of a scalar function
adityaguharoy   1
N 6 hours ago by Mathzeus1024
Source: own but possibly well known
Consider the function $f : \mathbb{R}^3 \to \mathbb{R}$ given by $f(x,y,z) = x + ye^z.$ Show that $\nabla f$ exists everywhere and find the direction along which the absolute value of the directional derivative is maximized at the point $(0,1,0).$


Hint
1 reply
adityaguharoy
Jul 27, 2023
Mathzeus1024
6 hours ago
J
H
Romanian National Olympiad 1997 - Grade 11 - Problem 1
Filipjack   0
Today at 10:48 AM
Source: Romanian National Olympiad 1997 - Grade 11 - Problem 1
Let $m \ge 2$ and $n \ge 1$ be integers and $A=(a_{ij})$ a square matrix of order $n$ with integer entries. Prove that for any permutation $\sigma \in S_n$ there is a function $\varepsilon : \{1,2,\ldots,n\} \to \{0,1\}$ such that replacing the entries $a_{\sigma(1)1},$ $a_{\sigma(2)2}, $ $\ldots,$ $a_{\sigma(n)n}$ of $A$ respectively by $$a_{\sigma(1)1}+\varepsilon(1), ~a_{\sigma(2)2}+\varepsilon(2), ~\ldots, ~a_{\sigma(n)n}+\varepsilon(n),$$the determinant of the matrix $A_{\varepsilon}$ thus obtained is not divisible by $m.$
0 replies
Filipjack
Today at 10:48 AM
0 replies
J
H
Differentiable function with a constant ratio
KAME06   1
N Today at 10:26 AM by Mathzeus1024
Source: Ecuador National Olympiad OMEC level U 2024 P2 Day 1
Let $\alpha >0$ a real number. Given a differentiable function $f: \mathbb{R^+} \rightarrow \mathbb{R^+}$, let $\gamma$ the curve $y=f(x)$ on the XY-plane. For all point $P$ on $\gamma$, the tangent to $\gamma$ on $P$ intersect the x-axis and the y-axis on $A$ and $B$, respectively, such $P \in AB$ and $\frac{BP}{PA}=\alpha$.
If $(20,24)$ belongs to $\gamma$, find all possible functions $f(x)$.
1 reply
KAME06
Yesterday at 8:13 PM
Mathzeus1024
Today at 10:26 AM
J
H
Null Traces of 2 Matrices
Saucepan_man02   2
N Today at 10:16 AM by loup blanc
Let $A,B\in \mathcal{M}_2(\mathbb{C})$ two non-zero matrices such that $AB+BA=O_2$ and $\det(A+B)=0$. Prove $A$ and $B$ have null traces.
2 replies
Saucepan_man02
Yesterday at 8:01 AM
loup blanc
Today at 10:16 AM
J
H
J
H
Thanks u!
Ruji2018252   1
N Apr 2, 2025 by alexheinis
Let $a_1,...,a_{2024}\in\mathbb{Z}$ and $1\leqslant a_1\leqslant a_2\leqslant ...\leqslant a_{2024}\leqslant 99$ and
\[P=a_1^2+a_2^2+...+a_{2024}^2-(a_1a_3+a_2a_4+...+a_{2022}a_{2024})\]Find maximum $P$
1 reply
Ruji2018252
Apr 1, 2025
alexheinis
Apr 2, 2025
J
Thanks u!
G H J
Reveal topic tags
inequalities
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.
Ruji2018252
381 posts
Ruji2018252
#1 h  Apr 1, 2025, 11:51 AM
Y by
Let $a_1,...,a_{2024}\in\mathbb{Z}$ and $1\leqslant a_1\leqslant a_2\leqslant ...\leqslant a_{2024}\leqslant 99$ and
\[P=a_1^2+a_2^2+...+a_{2024}^2-(a_1a_3+a_2a_4+...+a_{2022}a_{2024})\]Find maximum $P$
This post has been edited 1 time. Last edited by Ruji2018252, Apr 1, 2025, 11:52 AM
Reason: Hihi
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
alexheinis
10521 posts
alexheinis
#2 h  Apr 2, 2025, 5:34 PM • 1 Y
Y by Ruji2018252
With $n=2024$ we have $P=\sum_1^n a_k^2-\sum_1^{n-2}a_k a_{k+2}=a_n^2+a_{n-1}^2-\sum_1^{n-2} a_k (a_{k+2}-a_k)\le 2\cdot 99^2$ and equality when all $a_k=99$.
Z K Y
N Quick Reply
G
H
=
a