Plan ahead for the next school year. Schedule your class today!

Middle School Math Grades 5-8, Ages 10-13, MATHCOUNTS, AMC 8
Grades 5-8, Ages 10-13, MATHCOUNTS, AMC 8
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Middle School Math Grades 5-8, Ages 10-13, MATHCOUNTS, AMC 8
Grades 5-8, Ages 10-13, MATHCOUNTS, AMC 8
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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
and train with the best! Please note that early bird pricing ends August 19th!
Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

Our full course list for upcoming classes is below:
All classes start 7:30pm ET/4:30pm PT unless otherwise noted.

Introductory: Grades 5-10

Prealgebra 1 Self-Paced

Prealgebra 1
Wednesday, Jul 16 - Oct 29
Sunday, Aug 17 - Dec 14
Tuesday, Aug 26 - Dec 16
Friday, Sep 5 - Jan 16
Monday, Sep 8 - Jan 12
Tuesday, Sep 16 - Jan 20 (4:30 - 5:45 pm ET/1:30 - 2:45 pm PT)
Sunday, Sep 21 - Jan 25
Thursday, Sep 25 - Jan 29
Wednesday, Oct 22 - Feb 25
Tuesday, Nov 4 - Mar 10
Friday, Dec 12 - Apr 10

Prealgebra 2 Self-Paced

Prealgebra 2
Friday, Jul 25 - Nov 21
Sunday, Aug 17 - Dec 14
Tuesday, Sep 9 - Jan 13
Thursday, Sep 25 - Jan 29
Sunday, Oct 19 - Feb 22
Monday, Oct 27 - Mar 2
Wednesday, Nov 12 - Mar 18

Introduction to Algebra A Self-Paced

Introduction to Algebra A
Tuesday, Jul 15 - Oct 28
Sunday, Aug 17 - Dec 14
Wednesday, Aug 27 - Dec 17
Friday, Sep 5 - Jan 16
Thursday, Sep 11 - Jan 15
Sunday, Sep 28 - Feb 1
Monday, Oct 6 - Feb 9
Tuesday, Oct 21 - Feb 24
Sunday, Nov 9 - Mar 15
Friday, Dec 5 - Apr 3

Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
Wednesday, Jul 2 - Sep 17
Sunday, Jul 27 - Oct 19
Monday, Aug 11 - Nov 3
Wednesday, Sep 3 - Nov 19
Sunday, Sep 21 - Dec 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Friday, Oct 3 - Jan 16
Sunday, Oct 19 - Jan 25
Tuesday, Nov 4 - Feb 10
Sunday, Dec 7 - Mar 8

Introduction to Number Theory
Tuesday, Jul 15 - Sep 30
Wednesday, Aug 13 - Oct 29
Friday, Sep 12 - Dec 12
Sunday, Oct 26 - Feb 1
Monday, Dec 1 - Mar 2

Introduction to Algebra B Self-Paced

Introduction to Algebra B
Friday, Jul 18 - Nov 14
Thursday, Aug 7 - Nov 20
Monday, Aug 18 - Dec 15
Sunday, Sep 7 - Jan 11
Thursday, Sep 11 - Jan 15
Wednesday, Sep 24 - Jan 28
Sunday, Oct 26 - Mar 1
Tuesday, Nov 4 - Mar 10
Monday, Dec 1 - Mar 30

Introduction to Geometry
Monday, Jul 14 - Jan 19
Wednesday, Aug 13 - Feb 11
Tuesday, Aug 26 - Feb 24
Sunday, Sep 7 - Mar 8
Thursday, Sep 11 - Mar 12
Wednesday, Sep 24 - Mar 25
Sunday, Oct 26 - Apr 26
Monday, Nov 3 - May 4
Friday, Dec 5 - May 29

Paradoxes and Infinity
Mon, Tue, Wed, & Thurs, Jul 14 - Jul 16 (meets every day of the week!)
Sat & Sun, Sep 13 - Sep 14 (1:00 - 4:00 PM PT/4:00 - 7:00 PM ET)

Intermediate: Grades 8-12

Intermediate Algebra
Sunday, Jul 13 - Jan 18
Thursday, Jul 24 - Jan 22
Friday, Aug 8 - Feb 20
Tuesday, Aug 26 - Feb 24
Sunday, Sep 28 - Mar 29
Wednesday, Oct 8 - Mar 8
Sunday, Nov 16 - May 17
Thursday, Dec 11 - Jun 4

Intermediate Counting & Probability
Sunday, Sep 28 - Feb 15
Tuesday, Nov 4 - Mar 24

Intermediate Number Theory
Wednesday, Sep 24 - Dec 17

Precalculus
Wednesday, Aug 6 - Jan 21
Tuesday, Sep 9 - Feb 24
Sunday, Sep 21 - Mar 8
Monday, Oct 20 - Apr 6
Sunday, Dec 14 - May 31

Advanced: Grades 9-12

Calculus
Sunday, Sep 7 - Mar 15
Wednesday, Sep 24 - Apr 1
Friday, Nov 14 - May 22

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
Sunday, Aug 17 - Nov 9
Wednesday, Sep 3 - Nov 19
Tuesday, Sep 16 - Dec 9
Sunday, Sep 21 - Dec 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Monday, Oct 6 - Jan 12
Thursday, Oct 16 - Jan 22
Tues, Thurs & Sun, Dec 9 - Jan 18 (meets three times a week!)

MATHCOUNTS/AMC 8 Advanced
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
Sunday, Aug 17 - Nov 9
Tuesday, Aug 26 - Nov 11
Thursday, Sep 4 - Nov 20
Friday, Sep 12 - Dec 12
Monday, Sep 15 - Dec 8
Sunday, Oct 5 - Jan 11
Tues, Thurs & Sun, Dec 2 - Jan 11 (meets three times a week!)
Mon, Wed & Fri, Dec 8 - Jan 16 (meets three times a week!)

AMC 10 Problem Series
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
Sunday, Aug 10 - Nov 2
Thursday, Aug 14 - Oct 30
Tuesday, Aug 19 - Nov 4
Mon & Wed, Sep 15 - Oct 22 (meets twice a week!)
Mon, Wed & Fri, Oct 6 - Nov 3 (meets three times a week!)
Tue, Thurs & Sun, Oct 7 - Nov 2 (meets three times a week!)

AMC 10 Final Fives
Friday, Aug 15 - Sep 12
Sunday, Sep 7 - Sep 28
Tuesday, Sep 9 - Sep 30
Monday, Sep 22 - Oct 13
Sunday, Sep 28 - Oct 19 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Wednesday, Oct 8 - Oct 29
Thursday, Oct 9 - Oct 30

AMC 12 Problem Series
Wednesday, Aug 6 - Oct 22
Sunday, Aug 10 - Nov 2
Monday, Aug 18 - Nov 10
Mon & Wed, Sep 15 - Oct 22 (meets twice a week!)
Tues, Thurs & Sun, Oct 7 - Nov 2 (meets three times a week!)

AMC 12 Final Fives
Thursday, Sep 4 - Sep 25
Sunday, Sep 28 - Oct 19
Tuesday, Oct 7 - Oct 28

AIME Problem Series A
Thursday, Oct 23 - Jan 29

AIME Problem Series B
Tuesday, Sep 2 - Nov 18

F=ma Problem Series
Tuesday, Sep 16 - Dec 9
Friday, Oct 17 - Jan 30

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, Aug 14 - Oct 30
Sunday, Sep 7 - Nov 23
Tuesday, Dec 2 - Mar 3

Intermediate Programming with Python
Friday, Oct 3 - Jan 16

USACO Bronze Problem Series
Wednesday, Sep 3 - Dec 3
Thursday, Oct 30 - Feb 5
Tuesday, Dec 2 - Mar 3

Physics

Introduction to Physics
Tuesday, Sep 2 - Nov 18
Sunday, Oct 5 - Jan 11
Wednesday, Dec 10 - Mar 11

Physics 1: Mechanics
Sunday, Sep 21 - Mar 22
Sunday, Oct 26 - Apr 26
0 replies
jwelsh
Jul 1, 2025
0 replies
Random but useful theorems
booking   121
There have been all these random but useful theorems
Please post any theorems you know, random or not, but please say whether they are random or not.
I'll start give an example:
Random
I am just looking for some theorems to study.
121 replies
booking
Jul 16, 2025
AbhayAttarde01
21 minutes ago
Website to learn math
hawa   190
Hi, I'm kinda curious what website do yall use to learn math, like i dont find any website thats fun to learn math
190 replies
hawa
Apr 9, 2025
ZMB038
31 minutes ago
Bogus Proof Marathon
pifinity   7784
Hi!
I'd like to introduce the Bogus Proof Marathon.

In this marathon, simply post a bogus proof that is middle-school level and the next person will find the error. You don't have to post the real solution :P

Use classic Marathon format:
[hide=P#]a1b2c3[/hide]
[hide=S#]a1b2c3[/hide]


Example posts:

P(x)
-----
S(x)
P(x+1)
-----
Let's go!! Just don't make it too hard!
7784 replies
pifinity
Mar 12, 2018
happyfish0922
2 hours ago
Factorial fraction
Silverfalcon   47
$\frac{(3!)!}{3!} =$

$\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 6\qquad \text{(D)}\ 40\qquad \text{(E)}\ 120$
47 replies
Silverfalcon
Nov 21, 2005
geombi
3 hours ago
High School Math Grades 9-12, Ages 13-18
Grades 9-12, Ages 13-18
3
VNew Topic kLocked
High School Math Grades 9-12, Ages 13-18
Grades 9-12, Ages 13-18
3
VNew Topic kLocked
G
Topic
First Poster
Last Poster
k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
and train with the best! Please note that early bird pricing ends August 19th!
Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

Our full course list for upcoming classes is below:
All classes start 7:30pm ET/4:30pm PT unless otherwise noted.

Introductory: Grades 5-10

Prealgebra 1 Self-Paced

Prealgebra 1
Wednesday, Jul 16 - Oct 29
Sunday, Aug 17 - Dec 14
Tuesday, Aug 26 - Dec 16
Friday, Sep 5 - Jan 16
Monday, Sep 8 - Jan 12
Tuesday, Sep 16 - Jan 20 (4:30 - 5:45 pm ET/1:30 - 2:45 pm PT)
Sunday, Sep 21 - Jan 25
Thursday, Sep 25 - Jan 29
Wednesday, Oct 22 - Feb 25
Tuesday, Nov 4 - Mar 10
Friday, Dec 12 - Apr 10

Prealgebra 2 Self-Paced

Prealgebra 2
Friday, Jul 25 - Nov 21
Sunday, Aug 17 - Dec 14
Tuesday, Sep 9 - Jan 13
Thursday, Sep 25 - Jan 29
Sunday, Oct 19 - Feb 22
Monday, Oct 27 - Mar 2
Wednesday, Nov 12 - Mar 18

Introduction to Algebra A Self-Paced

Introduction to Algebra A
Tuesday, Jul 15 - Oct 28
Sunday, Aug 17 - Dec 14
Wednesday, Aug 27 - Dec 17
Friday, Sep 5 - Jan 16
Thursday, Sep 11 - Jan 15
Sunday, Sep 28 - Feb 1
Monday, Oct 6 - Feb 9
Tuesday, Oct 21 - Feb 24
Sunday, Nov 9 - Mar 15
Friday, Dec 5 - Apr 3

Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
Wednesday, Jul 2 - Sep 17
Sunday, Jul 27 - Oct 19
Monday, Aug 11 - Nov 3
Wednesday, Sep 3 - Nov 19
Sunday, Sep 21 - Dec 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Friday, Oct 3 - Jan 16
Sunday, Oct 19 - Jan 25
Tuesday, Nov 4 - Feb 10
Sunday, Dec 7 - Mar 8

Introduction to Number Theory
Tuesday, Jul 15 - Sep 30
Wednesday, Aug 13 - Oct 29
Friday, Sep 12 - Dec 12
Sunday, Oct 26 - Feb 1
Monday, Dec 1 - Mar 2

Introduction to Algebra B Self-Paced

Introduction to Algebra B
Friday, Jul 18 - Nov 14
Thursday, Aug 7 - Nov 20
Monday, Aug 18 - Dec 15
Sunday, Sep 7 - Jan 11
Thursday, Sep 11 - Jan 15
Wednesday, Sep 24 - Jan 28
Sunday, Oct 26 - Mar 1
Tuesday, Nov 4 - Mar 10
Monday, Dec 1 - Mar 30

Introduction to Geometry
Monday, Jul 14 - Jan 19
Wednesday, Aug 13 - Feb 11
Tuesday, Aug 26 - Feb 24
Sunday, Sep 7 - Mar 8
Thursday, Sep 11 - Mar 12
Wednesday, Sep 24 - Mar 25
Sunday, Oct 26 - Apr 26
Monday, Nov 3 - May 4
Friday, Dec 5 - May 29

Paradoxes and Infinity
Mon, Tue, Wed, & Thurs, Jul 14 - Jul 16 (meets every day of the week!)
Sat & Sun, Sep 13 - Sep 14 (1:00 - 4:00 PM PT/4:00 - 7:00 PM ET)

Intermediate: Grades 8-12

Intermediate Algebra
Sunday, Jul 13 - Jan 18
Thursday, Jul 24 - Jan 22
Friday, Aug 8 - Feb 20
Tuesday, Aug 26 - Feb 24
Sunday, Sep 28 - Mar 29
Wednesday, Oct 8 - Mar 8
Sunday, Nov 16 - May 17
Thursday, Dec 11 - Jun 4

Intermediate Counting & Probability
Sunday, Sep 28 - Feb 15
Tuesday, Nov 4 - Mar 24

Intermediate Number Theory
Wednesday, Sep 24 - Dec 17

Precalculus
Wednesday, Aug 6 - Jan 21
Tuesday, Sep 9 - Feb 24
Sunday, Sep 21 - Mar 8
Monday, Oct 20 - Apr 6
Sunday, Dec 14 - May 31

Advanced: Grades 9-12

Calculus
Sunday, Sep 7 - Mar 15
Wednesday, Sep 24 - Apr 1
Friday, Nov 14 - May 22

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
Sunday, Aug 17 - Nov 9
Wednesday, Sep 3 - Nov 19
Tuesday, Sep 16 - Dec 9
Sunday, Sep 21 - Dec 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Monday, Oct 6 - Jan 12
Thursday, Oct 16 - Jan 22
Tues, Thurs & Sun, Dec 9 - Jan 18 (meets three times a week!)

MATHCOUNTS/AMC 8 Advanced
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
Sunday, Aug 17 - Nov 9
Tuesday, Aug 26 - Nov 11
Thursday, Sep 4 - Nov 20
Friday, Sep 12 - Dec 12
Monday, Sep 15 - Dec 8
Sunday, Oct 5 - Jan 11
Tues, Thurs & Sun, Dec 2 - Jan 11 (meets three times a week!)
Mon, Wed & Fri, Dec 8 - Jan 16 (meets three times a week!)

AMC 10 Problem Series
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
Sunday, Aug 10 - Nov 2
Thursday, Aug 14 - Oct 30
Tuesday, Aug 19 - Nov 4
Mon & Wed, Sep 15 - Oct 22 (meets twice a week!)
Mon, Wed & Fri, Oct 6 - Nov 3 (meets three times a week!)
Tue, Thurs & Sun, Oct 7 - Nov 2 (meets three times a week!)

AMC 10 Final Fives
Friday, Aug 15 - Sep 12
Sunday, Sep 7 - Sep 28
Tuesday, Sep 9 - Sep 30
Monday, Sep 22 - Oct 13
Sunday, Sep 28 - Oct 19 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Wednesday, Oct 8 - Oct 29
Thursday, Oct 9 - Oct 30

AMC 12 Problem Series
Wednesday, Aug 6 - Oct 22
Sunday, Aug 10 - Nov 2
Monday, Aug 18 - Nov 10
Mon & Wed, Sep 15 - Oct 22 (meets twice a week!)
Tues, Thurs & Sun, Oct 7 - Nov 2 (meets three times a week!)

AMC 12 Final Fives
Thursday, Sep 4 - Sep 25
Sunday, Sep 28 - Oct 19
Tuesday, Oct 7 - Oct 28

AIME Problem Series A
Thursday, Oct 23 - Jan 29

AIME Problem Series B
Tuesday, Sep 2 - Nov 18

F=ma Problem Series
Tuesday, Sep 16 - Dec 9
Friday, Oct 17 - Jan 30

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, Aug 14 - Oct 30
Sunday, Sep 7 - Nov 23
Tuesday, Dec 2 - Mar 3

Intermediate Programming with Python
Friday, Oct 3 - Jan 16

USACO Bronze Problem Series
Wednesday, Sep 3 - Dec 3
Thursday, Oct 30 - Feb 5
Tuesday, Dec 2 - Mar 3

Physics

Introduction to Physics
Tuesday, Sep 2 - Nov 18
Sunday, Oct 5 - Jan 11
Wednesday, Dec 10 - Mar 11

Physics 1: Mechanics
Sunday, Sep 21 - Mar 22
Sunday, Oct 26 - Apr 26
0 replies
jwelsh
Jul 1, 2025
0 replies
Geometry Problem
ReticulatedPython   0
Consider a configuration of $n$ concentric circles at $O$ with radii $r_1, r_2, \cdots, r_n$ such that $r_1<r_2<\cdots<r_n.$ A single point $P_k$ is chosen on the circle with radius $r_k,$ for all integers $1\le k\le n.$ Prove that $\sum_{1 \le i<j \le n}^{n}(P_iP_j)^2 \le \sum_{k=1}^{n} nr_k^2.$

Bonus: Let $O$ be the origin of the complex plane. Let $z_k$ be the complex number corresponding to $P_k$, for all integers $1 \le k \le n.$ Prove that the maximum value occurs when $\sum_{k=1}^{n} z_k=0.$
0 replies
ReticulatedPython
an hour ago
0 replies
Cmath club selection test
justalonelyguy   0
Problem 1
a) Let \( a, b, c \in \mathbb{R} \) such that \( ab + bc + ca = 1 \). Compute the value of the expression:
\[
P = \frac{a}{1 + a^2} + \frac{b}{1 + b^2} + \frac{c}{1 + c^2} - \frac{2}{(a + b)(b + c)(c + a)}.
\]
b) Let \( x, y, z \in \mathbb{R} \) such that
\[
27(x + y + z)^3 = 240 + (3x + y - z)^3 + (3y + z - x)^3 + (3z + x - y)^3.
\]Prove that \( (x + 2y)(y + 2z)(z + 2x) = 10 \).

Problem 2

a) Let \( p \) be a prime number greater than 5. Prove that \( p^4 + 239 \) is divisible by 240.

b) Do there exist integers \( a, b \) such that
\[
(a + b\sqrt{2026})^2 = 2024 + 2025\sqrt{2026} \ ?
\]
\item[c)] Find all natural number pairs \( (x, y) \) such that
\[
2^x \cdot x^2 = 9y^2 + 6y + 88.
\]
Problem 3

a)Let \( a, b, c \geq 0 \) such that \( a + b + c = 3 \). Prove that
\[
\frac{a}{b^2 + 1} + \frac{b}{c^2 + 1} + \frac{c}{a^2 + 1} \geq \frac{3}{2}.
\]
b) Let \( a, b, c > 0 \). Prove that
\[
\frac{1 + bc + ca}{(1 + a + b)^2} + \frac{1 + ca + ab}{(1 + b + c)^2} + \frac{1 + ab + bc}{(1 + c + a)^2} \geq 1.
\]Problem 4

Let triangle \( ABC \) be acute with \( AB < CA < BC \). Let the altitudes \( BE \) and \( CF \) intersect at \( H \). The line through \( E \) perpendicular to \( EF \) intersects the extensions of \( BC \) and \( AB \) at points \( I \) and \( P \), respectively. The line through \( F \) perpendicular to \( EF \) intersects the extensions of \( BC \) and \( CA \) at points \( K \) and \( Q \), respectively. Let \( M \) be the midpoint of \( BC \).


a) Prove that triangles \( AEF \) and \( ABC \) are similar.

b) Prove that triangle \( MEF \) is isosceles and that \( BK = CI \).

c) Prove that \( \angle EHP = \angle FHQ \).


Problem 5

Mr. Quý chooses two distinct integers \( a \) and \( b \), then writes down the following six numbers:
\[
ab,\quad a(a + 4),\quad ab(a + 4),\quad b(a + 4),\quad b(b + 4),\quad (a + 4)(b + 4).
\]Among these six numbers, what is the maximum number of perfect squares that can appear? Explain why.

0 replies
justalonelyguy
2 hours ago
0 replies
Functional Equation
worthawholebean   7
The function $ f$ satisfies \[f(x) + f(2x + y) + 5xy = f(3x - y) + 2x^2 + 1\] for all real numbers $ x$, $ y$. Determine the value of $ f(10)$.
7 replies
worthawholebean
Mar 2, 2008
Sunshine_Paradise
2 hours ago
Inequality
Peter-G   3
Let \(a, b, c > 0\) and \(ab + bc + ca + 2abc = 1\)
Find the maximum value of
\[
\sqrt{ab} + \sqrt{bc} + \sqrt{ca}
\]
3 replies
Peter-G
4 hours ago
LayZee
3 hours ago
Contests & Programs AMC and other contests, summer programs, etc.
AMC and other contests, summer programs, etc.
3
VNew Topic kLocked
Contests & Programs AMC and other contests, summer programs, etc.
AMC and other contests, summer programs, etc.
3
VNew Topic kLocked
34,874 topics
w
18 users
G
Topic
First Poster
Last Poster
k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
and train with the best! Please note that early bird pricing ends August 19th!
Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

Our full course list for upcoming classes is below:
All classes start 7:30pm ET/4:30pm PT unless otherwise noted.

Introductory: Grades 5-10

Prealgebra 1 Self-Paced

Prealgebra 1
Wednesday, Jul 16 - Oct 29
Sunday, Aug 17 - Dec 14
Tuesday, Aug 26 - Dec 16
Friday, Sep 5 - Jan 16
Monday, Sep 8 - Jan 12
Tuesday, Sep 16 - Jan 20 (4:30 - 5:45 pm ET/1:30 - 2:45 pm PT)
Sunday, Sep 21 - Jan 25
Thursday, Sep 25 - Jan 29
Wednesday, Oct 22 - Feb 25
Tuesday, Nov 4 - Mar 10
Friday, Dec 12 - Apr 10

Prealgebra 2 Self-Paced

Prealgebra 2
Friday, Jul 25 - Nov 21
Sunday, Aug 17 - Dec 14
Tuesday, Sep 9 - Jan 13
Thursday, Sep 25 - Jan 29
Sunday, Oct 19 - Feb 22
Monday, Oct 27 - Mar 2
Wednesday, Nov 12 - Mar 18

Introduction to Algebra A Self-Paced

Introduction to Algebra A
Tuesday, Jul 15 - Oct 28
Sunday, Aug 17 - Dec 14
Wednesday, Aug 27 - Dec 17
Friday, Sep 5 - Jan 16
Thursday, Sep 11 - Jan 15
Sunday, Sep 28 - Feb 1
Monday, Oct 6 - Feb 9
Tuesday, Oct 21 - Feb 24
Sunday, Nov 9 - Mar 15
Friday, Dec 5 - Apr 3

Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
Wednesday, Jul 2 - Sep 17
Sunday, Jul 27 - Oct 19
Monday, Aug 11 - Nov 3
Wednesday, Sep 3 - Nov 19
Sunday, Sep 21 - Dec 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Friday, Oct 3 - Jan 16
Sunday, Oct 19 - Jan 25
Tuesday, Nov 4 - Feb 10
Sunday, Dec 7 - Mar 8

Introduction to Number Theory
Tuesday, Jul 15 - Sep 30
Wednesday, Aug 13 - Oct 29
Friday, Sep 12 - Dec 12
Sunday, Oct 26 - Feb 1
Monday, Dec 1 - Mar 2

Introduction to Algebra B Self-Paced

Introduction to Algebra B
Friday, Jul 18 - Nov 14
Thursday, Aug 7 - Nov 20
Monday, Aug 18 - Dec 15
Sunday, Sep 7 - Jan 11
Thursday, Sep 11 - Jan 15
Wednesday, Sep 24 - Jan 28
Sunday, Oct 26 - Mar 1
Tuesday, Nov 4 - Mar 10
Monday, Dec 1 - Mar 30

Introduction to Geometry
Monday, Jul 14 - Jan 19
Wednesday, Aug 13 - Feb 11
Tuesday, Aug 26 - Feb 24
Sunday, Sep 7 - Mar 8
Thursday, Sep 11 - Mar 12
Wednesday, Sep 24 - Mar 25
Sunday, Oct 26 - Apr 26
Monday, Nov 3 - May 4
Friday, Dec 5 - May 29

Paradoxes and Infinity
Mon, Tue, Wed, & Thurs, Jul 14 - Jul 16 (meets every day of the week!)
Sat & Sun, Sep 13 - Sep 14 (1:00 - 4:00 PM PT/4:00 - 7:00 PM ET)

Intermediate: Grades 8-12

Intermediate Algebra
Sunday, Jul 13 - Jan 18
Thursday, Jul 24 - Jan 22
Friday, Aug 8 - Feb 20
Tuesday, Aug 26 - Feb 24
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Sunday, Sep 28 - Feb 15
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Intermediate Number Theory
Wednesday, Sep 24 - Dec 17

Precalculus
Wednesday, Aug 6 - Jan 21
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Calculus
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Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
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AMC 10 Final Fives
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AMC 12 Final Fives
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AIME Problem Series A
Thursday, Oct 23 - Jan 29

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Physics

Introduction to Physics
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Physics 1: Mechanics
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0 replies
jwelsh
Jul 1, 2025
0 replies
AMC 10 Problem Series Recommendations
Amazingatmath.com   2
Hello,

I am thinking of taking the AMC 10 Problem Series from Aug 10-Nov 2 with the teacher Marco Figueroa Ibarra.

Does anyone have any feedback/advice on whether to take the course or not, related to the class, the teacher, or any other aspects?

For reference, I am going into 7th grade and got a 21 on 2025 AMC 8 and 79.5 on a mock 2024 AMC 10B. I am also doing AoPS vol 1, inter algebra, and intro to c&p.
2 replies
Amazingatmath.com
Jul 14, 2025
HiCalculus
a few seconds ago
The return of American geo
brianzjk   79
Source: USAJMO 2023/6
Isosceles triangle $ABC$, with $AB=AC$, is inscribed in circle $\omega$. Let $D$ be an arbitrary point inside $BC$ such that $BD\neq DC$. Ray $AD$ intersects $\omega$ again at $E$ (other than $A$). Point $F$ (other than $E$) is chosen on $\omega$ such that $\angle DFE = 90^\circ$. Line $FE$ intersects rays $AB$ and $AC$ at points $X$ and $Y$, respectively. Prove that $\angle XDE = \angle EDY$.

Proposed by Anton Trygub
79 replies
brianzjk
Mar 23, 2023
BS2012
6 minutes ago
otis application
Soupboy0   54
can you still get accepted if you dont get all the application questions right
54 replies
+1 w
Soupboy0
Jul 25, 2025
AbhayAttarde01
10 minutes ago
9 Most recommended resource
HiCalculus   6
For this poll, I'm going to let everyone choose 1 option to vote for so I can see which resource is the absolute best. Later, I'll make another poll with 2 options per user to find the best way to incorporate 2 resources to study for the test in around 3 months from now.
6 replies
HiCalculus
Yesterday at 1:32 AM
HiCalculus
12 minutes ago
High School Olympiads Regional, national, and international math olympiads
Regional, national, and international math olympiads
3
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High School Olympiads Regional, national, and international math olympiads
Regional, national, and international math olympiads
3
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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
and train with the best! Please note that early bird pricing ends August 19th!
Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

Our full course list for upcoming classes is below:
All classes start 7:30pm ET/4:30pm PT unless otherwise noted.

Introductory: Grades 5-10

Prealgebra 1 Self-Paced

Prealgebra 1
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Prealgebra 2 Self-Paced

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Introduction to Algebra A Self-Paced

Introduction to Algebra A
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Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
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Introduction to Algebra B Self-Paced

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Mon, Tue, Wed, & Thurs, Jul 14 - Jul 16 (meets every day of the week!)
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Intermediate: Grades 8-12

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Contest Preparation: Grades 6-12

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MATHCOUNTS/AMC 8 Advanced
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AMC 10 Problem Series
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AMC 10 Final Fives
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Wednesday, Oct 8 - Oct 29
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AMC 12 Problem Series
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Tues, Thurs & Sun, Oct 7 - Nov 2 (meets three times a week!)

AMC 12 Final Fives
Thursday, Sep 4 - Sep 25
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AIME Problem Series A
Thursday, Oct 23 - Jan 29

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Tuesday, Sep 2 - Nov 18

F=ma Problem Series
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Visit the pages linked for full schedule details for each of these programs!


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Programming

Introduction to Programming with Python
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Physics

Introduction to Physics
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Sunday, Sep 21 - Mar 22
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0 replies
jwelsh
Jul 1, 2025
0 replies
2024 SL C5
Twoisaprime   8
Source: 2024 IMO Shortlist C5
Let $N$ be a positive integer. Geoff and Ceri play a game in which they start by writing the numbers $1, 2, \dots, N$ on a board. They then take turns to make a move, starting with Geoff. Each move consists of choosing a pair of integers $(k, n)$, where $k \geq 0$ and $n$ is one of the integers on the board, and then erasing every integer $s$ on the board such that $2^k \mid n - s$. The game continues until the board is empty. The player who erases the last integer on the board loses.

Determine all values of $N$ for which Geoff can ensure that he wins, no matter how Ceri plays.
8 replies
Twoisaprime
Jul 16, 2025
Giabach298
3 minutes ago
Another Chinese projective cookie
MarkBcc168   33
Source: 2018 China Southeast MO Grade 10 P6
In the isosceles triangle $ABC$ with $AB=AC$, the center of $\odot O$ is the midpoint of the side $BC$, and $AB,AC$ are tangent to the circle at points $E,F$ respectively. Point $G$ is on $\odot O$ with $\angle AGE = 90^{\circ}$. A tangent line of $\odot O$ passes through $G$, and meets $AC$ at $K$. Prove that line $BK$ bisects $EF$.
33 replies
MarkBcc168
Jul 31, 2018
fearsum_fyz
5 minutes ago
Geometry is not dead, Geometry will live forever!
MathLuis   27
Source: GIMO 2022 Problem 3
On a triangle $\triangle ABC$ let $\omega$ be its circumcircle and let $H_A$ be a point on $\omega$ such that $AH_A \perp BC$. Let the projections from $B,C$ to $AC,AB$ be $E,F$ respectivily. Let $H_AE,H_AF$ meet $\omega$ again at $P,Q$ respecitivily and let $PQ$ hit $BC$ at $T$. Show that one of the tangents from $T$ to $\omega$ bisects the common chord of $(BFP),(CEQ)$.
Note: If 2 circles $\omega_1, \omega_2$ hit at $X,Y$, then the common chord of $\omega_1,\omega_2$ is segment $XY$

Proposed by Luis André Villán Gabriel, Bolivia

Expected Difficulty (in MOHS Scale): 35-40?

Proposer's AoPS username: MathLuis
27 replies
MathLuis
Sep 14, 2022
Emus
7 minutes ago
Functional equation
Eul12   4
Source: My creation
Any help for my problem
Let a be a positive integer. Find all increasing function f : IN---->IN such that f(f(n)) = (a^2)*n
for all positive integer n.
4 replies
Eul12
Yesterday at 5:27 PM
Eul12
11 minutes ago
College Math Topics in undergraduate and graduate studies
Topics in undergraduate and graduate studies
3
VNew Topic kLocked
College Math Topics in undergraduate and graduate studies
Topics in undergraduate and graduate studies
3
VNew Topic kLocked
G
Topic
First Poster
Last Poster
k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
and train with the best! Please note that early bird pricing ends August 19th!
Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

Our full course list for upcoming classes is below:
All classes start 7:30pm ET/4:30pm PT unless otherwise noted.

Introductory: Grades 5-10

Prealgebra 1 Self-Paced

Prealgebra 1
Wednesday, Jul 16 - Oct 29
Sunday, Aug 17 - Dec 14
Tuesday, Aug 26 - Dec 16
Friday, Sep 5 - Jan 16
Monday, Sep 8 - Jan 12
Tuesday, Sep 16 - Jan 20 (4:30 - 5:45 pm ET/1:30 - 2:45 pm PT)
Sunday, Sep 21 - Jan 25
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Prealgebra 2 Self-Paced

Prealgebra 2
Friday, Jul 25 - Nov 21
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Introduction to Algebra A Self-Paced

Introduction to Algebra A
Tuesday, Jul 15 - Oct 28
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Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
Wednesday, Jul 2 - Sep 17
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Introduction to Number Theory
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Introduction to Algebra B Self-Paced

Introduction to Algebra B
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Paradoxes and Infinity
Mon, Tue, Wed, & Thurs, Jul 14 - Jul 16 (meets every day of the week!)
Sat & Sun, Sep 13 - Sep 14 (1:00 - 4:00 PM PT/4:00 - 7:00 PM ET)

Intermediate: Grades 8-12

Intermediate Algebra
Sunday, Jul 13 - Jan 18
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Intermediate Counting & Probability
Sunday, Sep 28 - Feb 15
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Intermediate Number Theory
Wednesday, Sep 24 - Dec 17

Precalculus
Wednesday, Aug 6 - Jan 21
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Advanced: Grades 9-12

Calculus
Sunday, Sep 7 - Mar 15
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Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
Sunday, Aug 17 - Nov 9
Wednesday, Sep 3 - Nov 19
Tuesday, Sep 16 - Dec 9
Sunday, Sep 21 - Dec 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
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Tues, Thurs & Sun, Dec 9 - Jan 18 (meets three times a week!)

MATHCOUNTS/AMC 8 Advanced
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Mon, Wed & Fri, Dec 8 - Jan 16 (meets three times a week!)

AMC 10 Problem Series
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
Sunday, Aug 10 - Nov 2
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Mon & Wed, Sep 15 - Oct 22 (meets twice a week!)
Mon, Wed & Fri, Oct 6 - Nov 3 (meets three times a week!)
Tue, Thurs & Sun, Oct 7 - Nov 2 (meets three times a week!)

AMC 10 Final Fives
Friday, Aug 15 - Sep 12
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Monday, Sep 22 - Oct 13
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Wednesday, Oct 8 - Oct 29
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AMC 12 Problem Series
Wednesday, Aug 6 - Oct 22
Sunday, Aug 10 - Nov 2
Monday, Aug 18 - Nov 10
Mon & Wed, Sep 15 - Oct 22 (meets twice a week!)
Tues, Thurs & Sun, Oct 7 - Nov 2 (meets three times a week!)

AMC 12 Final Fives
Thursday, Sep 4 - Sep 25
Sunday, Sep 28 - Oct 19
Tuesday, Oct 7 - Oct 28

AIME Problem Series A
Thursday, Oct 23 - Jan 29

AIME Problem Series B
Tuesday, Sep 2 - Nov 18

F=ma Problem Series
Tuesday, Sep 16 - Dec 9
Friday, Oct 17 - Jan 30

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, Aug 14 - Oct 30
Sunday, Sep 7 - Nov 23
Tuesday, Dec 2 - Mar 3

Intermediate Programming with Python
Friday, Oct 3 - Jan 16

USACO Bronze Problem Series
Wednesday, Sep 3 - Dec 3
Thursday, Oct 30 - Feb 5
Tuesday, Dec 2 - Mar 3

Physics

Introduction to Physics
Tuesday, Sep 2 - Nov 18
Sunday, Oct 5 - Jan 11
Wednesday, Dec 10 - Mar 11

Physics 1: Mechanics
Sunday, Sep 21 - Mar 22
Sunday, Oct 26 - Apr 26
0 replies
jwelsh
Jul 1, 2025
0 replies
Please verify this proof that a sequence converges
seo-ha   0
Question: Prove the sequence $\bigg( 1 + \frac{1}{\sqrt{n} - \pi} \bigg)_{n\in \Bbb{N}}$ converges to $1$

Proof (please validate):

Given $\epsilon > 0$, choose $N = \max\Bigg(10,\ \bigg\lceil \bigg(\frac{1+\epsilon\pi}{\epsilon}\bigg)^{2}\ \bigg\rceil\Bigg)$

Then given $n > N$ we have:

$n > 10 \implies \sqrt{n} - \pi > 0$

So $\Big| a_n -1 \Big| = \bigg| \frac{1}{\sqrt{n} - \pi} \bigg| = \frac{1}{\sqrt{n}-\pi}$

$n > \bigg(\frac{1+\epsilon\pi}{\epsilon}\bigg)^2 \implies \sqrt{n} > \frac{1 + \epsilon\pi}{\epsilon} \implies \frac{1}{\sqrt{n}-\pi}<\epsilon$

Hence $\Big| a_n - 1 \Big | < \epsilon$

So $\lim_{n\to \infty} a_n = 1$

0 replies
seo-ha
an hour ago
0 replies
Prove that at most three of the \( x_i \)'s are nonzero, and at least one of the
Martin.s   0
Can anyone provide the source of this problem?


Suppose that \( x_1, x_2, \dots, x_n \), where \( n \geq 3 \), are nonnegative real numbers such that
\[
x_1 + x_2 + \cdots + x_n = 2 \quad \text{and} \quad x_1 x_2 + x_2 x_3 + \cdots + x_{n-1} x_n = 1.
\]Prove that at most three of the \( x_i \)'s are nonzero, and at least one of the \( x_i \)'s equals 1.
0 replies
Martin.s
2 hours ago
0 replies
Proof sequence converges
seo-ha   1
Hi, can someone check my proof, the question is to show that $\bigg(\frac{1}{2n+5\sin n}\bigg)_{n\in \Bbb{N}}$ converges to $0$.

My proof:

Given $\epsilon > 0$, choose $N = \bigg\lceil{\frac{1+5\epsilon}{2\epsilon}}\bigg\rceil$ s.t. given $n > N$, we have:

$n > \frac{1+5\epsilon}{2\epsilon} \iff \frac{1}{2n-5} < \epsilon$

Furthermore as: $ -1 \le \sin(n) \le 1$ we have $2n-5 \le 2n+5\sin n \le 2n+5$

In particular: $2n-5 \le 2n+5\sin n \iff \frac{1}{2n-5} \ge \frac{1}{2n+5\sin n}$

Thus: $\bigg | \frac{1}{2n+5\sin n} - 0 \bigg | = \frac{1}{2n+5\sin n} \le \frac{1}{2n-5} < \epsilon\ \ \ \square$
1 reply
seo-ha
4 hours ago
Hello_Kitty
3 hours ago
Expected value of product of min and max
Kempu33334   2
Source: Own
Let $n$ variables $X_1$, $X_2$, $\cdots$, $X_n$ be chosen uniformly at random in the range $[0,1]$. Show that the value of \[\mathbb{E}(\min(X)\cdot \max(X)) = \dfrac{1}{n+2}.\]I'm pretty sure this requires calculus (although would love a non-calculus based proof), hence why I posted in College Math.
2 replies
Kempu33334
Saturday at 9:08 PM
grupyorum
5 hours ago
Site Support Tech support and questions about AoPS classes and materials
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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
and train with the best! Please note that early bird pricing ends August 19th!
Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

Our full course list for upcoming classes is below:
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0 replies
jwelsh
Jul 1, 2025
0 replies
[RESOLVED] Cannot Load Homework Error
zhenghua   40
Summary of the problem: Some error is preventing me from finishing my homework. My brother is encountering the same problem.
Page URL: Your class homework
Steps to reproduce:
1. Open the classes tab.
2. Open homework
3. A window pops up.
Expected behavior: You should be able to access your homework.
Frequency: 100% after 18:00 PST
Operating system(s): Windows 10
Browser(s), including version: Firefox (and my brother tried google)
Additional information: Screenshot below (cannot upload screenshot for some reason...). As a side note, recently, I've been experiencing extreme lag on AoPS that doesn't seen to occur with other websites.
40 replies
zhenghua
Today at 3:14 AM
SoundersTID
2 hours ago
help how do i reset alcumus
dweaded   2
how do i reset my alcumus progress? does anyone know?
2 replies
dweaded
4 hours ago
Audreyma0321
4 hours ago
Error Connecting to Server
Zestra   1
Hey guys just wanted to share an issue I've always had (looks like more site support threads are getting looked at so perfect timing)

Whenever I am in my AoPS ebooks, just on the web page I am guaranteed to get a notification that there is an "Error connecting to server. Check internet connection. We'll try again in 10s." In fact it just happened right now (total 3 times overall while writing) as I am writing this on the community page. It doesn't really affect me much when I am in the ebook since nothing much should be changing on there, but if I am looking at my feed while on the page, it can be problematic bc I dont get the messages once the error begins, and I have to refresh if I want to find out if I got any messages or new feed to look at. Pretty sure it happens like every 2 minutes for me.

It can't be just because of my internet connection but I move around a lot in the room and like to different places. And can still replicate.

Does anybody else have this issue?
1 reply
Zestra
Today at 11:16 AM
jmr2010
Today at 11:27 AM
Emoji ....
princeofmath   7
Does anybody know all the codes for AOPS emojis?
7 replies
princeofmath
Today at 5:54 AM
Zestra
Today at 6:43 AM
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