Start the New Year strong with our problem-based courses! Enroll today!

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k a January Highlights and 2025 AoPS Online Class Information
jlacosta   0
Jan 1, 2025
Happy New Year!!! Did you know, 2025 is the first perfect square year that any AoPS student has experienced? The last perfect square year was 1936 and the next one will be 2116! Let’s make it a perfect year all around by tackling new challenges, connecting with more problem-solvers, and staying curious!

We have some fun new things happening at AoPS in 2025 with new courses, such as self-paced Introduction to Algebra B, more coding, more physics, and, well, more!

There are a number of upcoming events, so be sure to mark your calendars for the following:

[list][*]Accelerated AIME Problem Series classes start on January 6th and 7th. These AIME classes will run three times a week throughout the month of January. With this accelerated track, you can fit three months of contest tips and training into four weeks finishing right in time for the AIME I on February 6th.
[*]Join our Math Jam on January 7th to learn about our Spring course options. We'll work through a few sample problems, discuss how the courses work, and answer your questions.
[*]RSVP for our New Year, New Challenges webinar on January 9th. We’ll discuss how you can meet your goals, useful strategies for your problem solving journey, and what classes and resources are available.
Have questions? Our Academic Success team is only an email away, write to us at success@aops.com.[/list]
AoPS Spring classes are open for enrollment. Get a jump on 2025 and enroll in our math, contest prep, coding, and science classes today! Need help finding the right plan for your goals? Check out our recommendations page!

Don’t forget: Highlight your AoPS Education on LinkedIn!
Many of you are beginning to build your education and achievements history on LinkedIn. Now, you can showcase your courses from Art of Problem Solving (AoPS) directly on your LinkedIn profile! Don't miss this opportunity to stand out and connect with fellow problem-solvers in the professional world and be sure to follow us at: https://www.linkedin.com/school/art-of-problem-solving/mycompany/ Check out our job postings, too, if you are interested in either full-time, part-time, or internship opportunities!

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
Sunday, Jan 5 - Apr 20
Wednesday, Jan 15 - Apr 30
Monday, Feb 3 - May 19
Sunday, Mar 2 - Jun 22
Friday, Mar 28 - Jul 18
Sunday, Apr 13 - Aug 10

Prealgebra 1 Self-Paced

Prealgebra 2
Wednesday, Jan 8 - Apr 23
Sunday, Jan 19 - May 4 (1:00 - 2:15 pm ET/10:00 - 11:15 am PT)
Monday, Jan 27 - May 12
Tuesday, Jan 28 - May 13 (4:30 - 5:45 pm ET/1:30 - 2:45 pm PT)
Sunday, Feb 16 - Jun 8
Tuesday, Mar 25 - Jul 8
Sunday, Apr 13 - Aug 10

Prealgebra 2 Self-Paced

Introduction to Algebra A
Tuesday, Jan 7 - Apr 22
Wednesday, Jan 29 - May 14
Sunday, Feb 16 - Jun 8 (3:30 - 5:00 pm ET/12:30 - 2:00 pm PT)
Sunday, Mar 23 - Jul 20
Monday, Apr 7 - Jul 28

Introduction to Algebra A Self-Paced

Introduction to Counting & Probability
Wednesday, Jan 8 - Mar 26
Thursday, Jan 30 - Apr 17
Sunday, Feb 9 - Apr 27 (3:30 - 5:00 pm ET/12:30 - 2:00 pm PT)
Sunday, Mar 16 - Jun 8
Wednesday, Apr 16 - Jul 2

Introduction to Counting & Probability Self-Paced

Introduction to Number Theory
Tuesday, Jan 28 - Apr 15
Sunday, Feb 16 - May 4
Monday, Mar 17 - Jun 9
Thursday, Apr 17 - Jul 3

Introduction to Algebra B
Tuesday, Jan 28 - May 13
Thursday, Feb 13 - May 29
Sunday, Mar 2 - Jun 22
Monday, Mar 17 - Jul 7
Wednesday, Apr 16 - Jul 30

Introduction to Geometry
Wednesday, Jan 8 - Jun 18
Thursday, Jan 30 - Jul 10
Friday, Feb 14 - Aug 1
Tuesday, Mar 4 - Aug 12
Sunday, Mar 23 - Sep 21
Wednesday, Apr 23 - Oct 1

Intermediate: Grades 8-12

Intermediate Algebra
Friday, Jan 17 - Jun 27
Wednesday, Feb 12 - Jul 23
Sunday, Mar 16 - Sep 14
Tuesday, Mar 25 - Sep 2
Monday, Apr 21 - Oct 13

Intermediate Counting & Probability
Monday, Feb 10 - Jun 16
Sunday, Mar 23 - Aug 3

Intermediate Number Theory
Thursday, Feb 20 - May 8
Friday, Apr 11 - Jun 27

Precalculus
Wednesday, Jan 8 - Jun 4
Tuesday, Feb 25 - Jul 22
Sunday, Mar 16 - Aug 24
Wednesday, Apr 9 - Sep 3

Advanced: Grades 9-12

Olympiad Geometry
Wednesday, Mar 5 - May 21

Calculus
Friday, Feb 28 - Aug 22
Sunday, Mar 30 - Oct 5

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Tuesday, Feb 4 - Apr 22
Sunday, Mar 23 - Jun 15
Wednesday, Apr 16 - Jul 2

MATHCOUNTS/AMC 8 Advanced
Sunday, Feb 16 - May 4
Friday, Apr 11 - Jun 27

Special AMC 8 Problem Seminar A
Sat & Sun, Jan 11 - Jan 12 (4:00 - 7:00 pm ET/1:00 - 4:00 pm PT)

Special AMC 8 Problem Seminar B
Sat & Sun, Jan 18 - Jan 19 (4:00 - 7:00 pm ET/1:00 - 4:00 pm PT)

AMC 10 Problem Series
Sunday, Feb 9 - Apr 27
Tuesday, Mar 4 - May 20
Monday, Mar 31 - Jun 23

AMC 10 Final Fives
Sunday, Feb 9 - Mar 2 (3:30 - 5:00 pm ET/12:30 - 2:00 pm PT)

AMC 12 Problem Series
Sunday, Feb 23 - May 11

AMC 12 Final Fives
Sunday, Feb 9 - Mar 2 (3:30 - 5:00 pm ET/12:30 - 2:00 pm PT)

AIME Problem Series A
Tue, Thurs & Sun, Jan 7 - Feb (meets three times each week!)

AIME Problem Series B
Mon, Wed & Fri, Jan 6 - Jan 31 (meets three times each week!)

Special AIME Problem Seminar A
Sat & Sun, Jan 25 - Jan 26 (4:00 - 7:00 pm ET/1:00 - 4:00 pm PT)

Special AIME Problem Seminar B
Sat & Sun, Feb 1 - Feb 2 (4:00 - 7:00 pm ET/1:00 - 4:00 pm PT)

F=ma Problem Series
Wednesday, Feb 19 - May 7

Programming

Introduction to Programming with Python
Friday, Jan 17 - Apr 4
Sunday, Feb 16 - May 4
Monday, Mar 24 - Jun 16

Intermediate Programming with Python
Tuesday, Feb 25 - May 13

USACO Bronze Problem Series
Sunday, Jan 5 - Mar 23
Thursday, Feb 6 - Apr 24

Physics

Introduction to Physics
Friday, Feb 7 - Apr 25
Sunday, Mar 30 - Jun 22

Physics 1: Mechanics
Sunday, Feb 9 - Aug 3
Tuesday, Mar 25 - Sep 2

Relativity
Sat & Sun, Dec 14 - Dec 15 (4:00 - 7:00 pm ET/1:00 - 4:00pm PT)
0 replies
jlacosta
Jan 1, 2025
0 replies
GCD Functional Equation
pinetree1   57
N 2 minutes ago by D4N13LCarpenter
Source: USA TSTST 2019 Problem 7
Let $f: \mathbb Z\to \{1, 2, \dots, 10^{100}\}$ be a function satisfying
$$\gcd(f(x), f(y)) = \gcd(f(x), x-y)$$for all integers $x$ and $y$. Show that there exist positive integers $m$ and $n$ such that $f(x) = \gcd(m+x, n)$ for all integers $x$.

Ankan Bhattacharya
57 replies
pinetree1
Jun 25, 2019
D4N13LCarpenter
2 minutes ago
Minimum of an expresion with squares and square roots
DensSv   2
N 23 minutes ago by RagvaloD
Source: 2023 Shortlist for NMO in Romania, for 8th grade
For $a$ and $b$ real numbers, define
$$E(a,b)=\sqrt{\bigg(\frac{1}{2}-a\bigg)^{2}+\bigg(\frac{1}{2}-b\bigg)^{2}}+\sqrt{\bigg(\frac{1}{2}+a\bigg)^{2}+\bigg(\frac{1}{2}+b\bigg)^{2}}$$a) Prove that $E(a,b)\geq \sqrt{2}$ for every real number $a,b$ and determine the equality case.
b) Find $min\{E(a,a+2)|a\in \mathbb{R}\}$.
2 replies
DensSv
Yesterday at 9:45 PM
RagvaloD
23 minutes ago
Iran geometry
Dadgarnia   18
N 31 minutes ago by cursed_tangent1434
Source: Iranian TST 2020, second exam day 1, problem 3
Given a triangle $ABC$ with circumcircle $\Gamma$. Points $E$ and $F$ are the foot of angle bisectors of $B$ and $C$, $I$ is incenter and $K$ is the intersection of $AI$ and $EF$. Suppose that $T$ be the midpoint of arc $BAC$. Circle $\Gamma$ intersects the $A$-median and circumcircle of $AEF$ for the second time at $X$ and $S$. Let $S'$ be the reflection of $S$ across $AI$ and $J$ be the second intersection of circumcircle of $AS'K$ and $AX$. Prove that quadrilateral $TJIX$ is cyclic.

Proposed by Alireza Dadgarnia and Amir Parsa Hosseini
18 replies
Dadgarnia
Mar 11, 2020
cursed_tangent1434
31 minutes ago
Vector units
youochange   1
N an hour ago by youochange
\[
\left( \frac{\hat{\jmath}}{\hat{k}} \right) \times \hat{k} = ?
\]
1 reply
youochange
2 hours ago
youochange
an hour ago
Inequalities
sqing   7
N 3 hours ago by yellowmoose1024
Let $ a,b,c > 0 $ such that $ b^2 \geq ca $ and $ c^2 \geq ab. $ Prove that
$$  \frac{ 2a}{b+c}+\frac{c}{b}+\frac{ 4c}{c+a}  \geq 6\sqrt{2}-5$$$$   \frac{3a}{b+c}+\frac{c}{b}+\frac{9c}{c+a}   \geq 8 \sqrt{3} -7$$https://artofproblemsolving.com/community/c6h3473344p33629670
7 replies
sqing
Today at 3:46 AM
yellowmoose1024
3 hours ago
Random algebra
Sedro   8
N 4 hours ago by PaixiaoLover
Let $a$ and $b$ be real numbers satisfying $a^2+a-b^2-b = \tfrac{7}{4}$ and $2ab+a+b=\tfrac{11}{2}$. Find the sum of all possible values of $a^2+b^2$.
8 replies
Sedro
Yesterday at 12:38 AM
PaixiaoLover
4 hours ago
Inequality
hangb6pbc   2
N 5 hours ago by sqing
Let $a;b>0$, prove: $\dfrac{a^2-2ab+1}{a(2ab+1)}+\dfrac{1+3b^2}{3b} \geq \dfrac{4}{3}$
2 replies
hangb6pbc
Apr 13, 2022
sqing
5 hours ago
Classic 3 variable equation
Ali_Onat   3
N 5 hours ago by Ali_Onat
For real numbers x, y, z it is given that
$x^2 = yz + 3$
$y^2 = zx + 4$
$z^2 = xy + 5$
Find x + y + z
3 replies
Ali_Onat
Yesterday at 9:48 PM
Ali_Onat
5 hours ago
Inequalities
sqing   1
N 5 hours ago by sqing
Let $ a,b,c\in [0,1] .$ Prove that$$ (a +b)\sqrt{1-c}+ (b +c)\sqrt{1-a}  +(c +a)\sqrt{1-b} \leq \frac{64}{27}$$
1 reply
sqing
6 hours ago
sqing
5 hours ago
number theory
mathloverrrrr_3007-.-   2
N Today at 10:16 AM by mathloverrrrr_3007-.-
let $p$ is a prime number and $p>2$. For $k\in\{1,2,...,p-2\}$, $S_k=\displaystyle \sum_{1\le j\le p-1} j^k$. Know that exist $a\in\mathbb{Z}:a^k\ne 1(mod p)$ . Proof that $p\mid S_k$
2 replies
mathloverrrrr_3007-.-
Today at 3:16 AM
mathloverrrrr_3007-.-
Today at 10:16 AM
Inequality
hangb6pbc   7
N Today at 9:18 AM by sqing
Let $a;b;c>0$ such that $ab+bc+ca=1$
Find (and prove) the minimum value of: $P=(a+b+c)\left (\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}  \right )$
7 replies
hangb6pbc
Jun 23, 2020
sqing
Today at 9:18 AM
Given three non-negative real numbers a,b,c change so that a+b+c = 2 Prove that
hoangvu1009   1
N Today at 9:08 AM by sqing
Given three non-negative real numbers a,b,c change so that a+b+c = 2 Prove that
(a) ab(a+b)+bc(b+c)+ca(c+a) ⩽ 1/4.
(b) 12 ⩽a^2+b^2+c^2+2√(3abc)⩽1
1 reply
hoangvu1009
Today at 8:47 AM
sqing
Today at 9:08 AM
Given non-negative real numbers a,b,c changing to satisfy a^2 +3 (b^2
hoangvu1009   6
N Today at 7:19 AM by sqing
Given non-negative real numbers a,b,c changing to satisfy a^2 +3 (b^2 +bc+c^2) = 1 Find the maximum value and the minimum value of the expression P = a+b+c
6 replies
hoangvu1009
Yesterday at 2:20 PM
sqing
Today at 7:19 AM
Any fast way to solve this polynomial question?
SurvivingInEnglish   5
N Today at 7:16 AM by Scilyse
I was shocked when my friend told me his brother got this on his regular grade 10 non-calculator math test as the bonus question. According to him his brother's teacher gave out this question at the end of the test and only allowed 5 minutes. :skull:

Solve for exact values of $x$ if:
$$(x^2-3x+1)^2 - 3(x^2-3x+1) + 1 = x$$
5 replies
SurvivingInEnglish
Today at 4:51 AM
Scilyse
Today at 7:16 AM
AB_|_ MO in tetrahedron ABCD with <BAC +<BAD=<ABC+<ABD=90^&omicron;
parmenides51   0
Jul 8, 2021
Source: 2001 Ukraine NMO 11.6
The tetrahedron $ABCD$ is known to have $\angle BAC + \angle BAD = \angle ABC +\angle ABD=90^o$. Let $O$ be the center of the circumcircle of triangle $ABC$, $M$ be the midpoint of the edge $CD$. Prove that the lines $AB$ and $MO$ are perpendicular.
0 replies
parmenides51
Jul 8, 2021
0 replies
AB_|_ MO in tetrahedron ABCD with <BAC +<BAD=<ABC+<ABD=90^&omicron;
G H J
G H BBookmark kLocked kLocked NReply
Source: 2001 Ukraine NMO 11.6
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parmenides51
30514 posts
#1
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The tetrahedron $ABCD$ is known to have $\angle BAC + \angle BAD = \angle ABC +\angle ABD=90^o$. Let $O$ be the center of the circumcircle of triangle $ABC$, $M$ be the midpoint of the edge $CD$. Prove that the lines $AB$ and $MO$ are perpendicular.
This post has been edited 3 times. Last edited by parmenides51, Dec 22, 2022, 1:07 PM
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