Stay ahead of learning milestones! Enroll in a class over the summer!

G
Topic
First Poster
Last Poster
k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
Thursday at 11:16 PM
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. 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 upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/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
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
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
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
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
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
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14

Introduction to Geometry
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

Paradoxes and Infinity
Mon, Tue, Wed, & Thurs, Jul 14 - Jul 16 (meets every day of the week!)

Intermediate: Grades 8-12

Intermediate Algebra
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
Sunday, Jun 1 - Aug 24
Wednesday, Jun 18 - Sep 3

Precalculus
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
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
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

AIME Problem Series A
Thursday, May 22 - Jul 31

AIME Problem Series B
Sunday, Jun 22 - Sep 21

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
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)
0 replies
jlacosta
Thursday at 11:16 PM
0 replies
Summation
Saucepan_man02   0
20 minutes ago
If $P = \sum_{r=1}^{50} \sum_{k=1}^{r} (-1)^{r-1} \frac{\binom{50}{r}}{k}$, then find the value of $P$.

Ans
0 replies
Saucepan_man02
20 minutes ago
0 replies
Equation of Matrices which have same rank
PureRun89   3
N an hour ago by pi_quadrat_sechstel
Source: Gazeta Mathematica

Let $A,B \in \mathbb{C}_{n \times n}$ and $rank(A)=rank(B)$.
Given that there exists positive integer $k$ such that
$$A^{k+1} B^k=A.$$Prove that
$$B^{k+1} A^k=B.$$
(Note: The submition of the problem is end so I post this)
3 replies
PureRun89
May 18, 2023
pi_quadrat_sechstel
an hour ago
D1023 : MVT 2.0
Dattier   1
N 2 hours ago by Dattier
Source: les dattes à Dattier
Let $f \in C(\mathbb R)$ derivable on $\mathbb R$ with $$\forall x \in \mathbb R,\forall h \geq 0, f(x)-3f(x+h)+3f(x+2h)-f(x+3h) \geq 0$$
Is it true that $$\forall (a,b) \in\mathbb R^2, |f(a)-f(b)|\leq \max\left(\left|f'\left(\dfrac{a+b} 2\right)\right|,\dfrac {|f'(a)+f'(b)|}{2}\right)\times |a-b|$$
1 reply
Dattier
Apr 29, 2025
Dattier
2 hours ago
Equivalent condition of the uniformly continuous fo a function
Alphaamss   0
2 hours ago
Source: Personal
Let $f_{a,b}(x)=x^a\cos(x^b),x\in(0,\infty)$. Get all the $(a,b)\in\mathbb R^2$ such that $f_{a,b}$ is uniformly continuous on $(0,\infty)$.
0 replies
Alphaamss
2 hours ago
0 replies
No more topics!
Two times derivable real function
Valentin Vornicu   13
N Apr 24, 2025 by solyaris
Source: RMO 2008, 11th Grade, Problem 3
Let $ f: \mathbb R \to \mathbb R$ be a function, two times derivable on $ \mathbb R$ for which there exist $ c\in\mathbb R$ such that
\[ \frac { f(b)-f(a) }{b-a} \neq f'(c) ,\] for all $ a\neq b \in \mathbb R$.

Prove that $ f''(c)=0$.
13 replies
Valentin Vornicu
Apr 30, 2008
solyaris
Apr 24, 2025
Two times derivable real function
G H J
Source: RMO 2008, 11th Grade, Problem 3
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Valentin Vornicu
7301 posts
#1 • 2 Y
Y by Adventure10, Mango247
Let $ f: \mathbb R \to \mathbb R$ be a function, two times derivable on $ \mathbb R$ for which there exist $ c\in\mathbb R$ such that
\[ \frac { f(b)-f(a) }{b-a} \neq f'(c) ,\] for all $ a\neq b \in \mathbb R$.

Prove that $ f''(c)=0$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
harazi
5526 posts
#2 • 2 Y
Y by Adventure10, Mango247
The image of the function $ g(a,b)=\frac{f(a)-f(b)}{a-b}$ defined for $ a\ne b$ being an interval (connected subset of the real line) and $ f'(c)$ not being in this image, it follows that we may assume that $ f'(c)<g(a,b)$ for all $ a\ne b$. But then $ f'(c)\leq f'(x)$ for all $ x$ and so $ c$ is a minimum point of $ f'$. Of course, this can be written in 11-th grade vocabulary. :D
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Svejk
663 posts
#3 • 1 Y
Y by Adventure10
I tried to solve it harazi's way but I didn't get maximum since I didn't manage to prove that $ g(a,b)-f'(c)$ has the same sign for all $ a,b$.Can you be give me more detailes please :lol: ?The official solution is based on the fact that the function $ g(x)=f(x)-f'(c)\cdot x$ is injective ,hence monotonic.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
harazi
5526 posts
#4 • 1 Y
Y by Adventure10
The set of pairs $ (a,b)$ such that $ a\ne b$ is a connected subset of the plane and the function $ g$ is continuous on this domain, thus its image is a connected subset of the line, thus an interval. I agree however that this is not a solution of an 11-th grade student, but that's how life is. :D I will not be amazed if in a few years I see complex analysis, Lebesgue integration and other such stuff at RMO. It's quite à pity, it gives huges advantages to some people. :(
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
enescu
741 posts
#5 • 2 Y
Y by Adventure10, Mango247
harazi wrote:
But then $ f'(c)\leq f'(x)$ for all $ x$ and so $ c$ is a minimum point of $ f'$.
Why for all $ x$?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
harazi
5526 posts
#6 • 2 Y
Y by Adventure10, Mango247
Well, $ f'(c)\leq g(a,x)$ for all $ a\ne x$ and now make $ a$ close to $ x$ and use the definition of derivative.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
harazi
5526 posts
#7 • 1 Y
Y by Adventure10
Well, I said however a very stupid thing: the function $ g$ should be defined on the set of pairs $ (a,b)$ such that $ a<b$ to ensure that its domain is connected. Of course, all the rest works with this modification, I don't know how I could write such a stupid thing. :D
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
enescu
741 posts
#8 • 4 Y
Y by adityaguharoy, Adventure10, Mango247, RobertRogo
Well, when I created this problem, I was thinking to the obvious geometric meaning: if $ f''(c) \ne 0$, then $ f$ is strictly concave up or down on some neighbourhood of the point $ c$, thus one can draw a close enough parallel to the line tangent at $ c$ to the function's graph that intersects the graph in two points $ (a,f(a))$ and $ (b,f(b))$. The slope of that tangent would be $ \frac{f(b)-f(a)}{b-a}$, equal to the slope of the tangent at $ c$, that is,$ f'(c)$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
subham1729
1479 posts
#9 • 1 Y
Y by Adventure10
One of above solutions uses that $C=\mathbb{R}^2-\{(a,a) \mid a \in \mathbb{R}\}$ is connected, but why $C$ is connected ? $C$ has clearly two connected components. However with this spirit we can also solve the problem, extend $g(a,b)=\frac{f(a)-f(b)}{a-b}$ to whole plane defining $g(a,a)=f'(a)$ and now $g$ is continuous on whole plane and do similar thing.
This post has been edited 1 time. Last edited by subham1729, Jun 14, 2016, 4:53 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Raunii
28 posts
#10
Y by
Svejk wrote:
I tried to solve it harazi's way but I didn't get maximum since I didn't manage to prove that $ g(a,b)-f'(c)$ has the same sign for all $ a,b$.Can you be give me more detailes please :lol: ?The official solution is based on the fact that the function $ g(x)=f(x)-f'(c)\cdot x$ is injective ,hence monotonic.

where did you find the official solution?
This post has been edited 1 time. Last edited by Raunii, Mar 15, 2020, 6:05 PM
Reason: .
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Rohit-2006
237 posts
#11
Y by
Too easy for grade 11....
Attachments:
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
solyaris
632 posts
#12
Y by
@above: This is not a valid argument. From the MVT you only get for every $(a,b)$ there exists an $m$ with the desired property. So you get $f'(m) \neq f'(c)$ only for values $m$ in some set $M$, which has to be dense in the reals, but $M$ need not be an interval, so the IVP you use later on in your proof does not give a contradiction. (See the solutions above for proofs that avoid this problem.)
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Rohit-2006
237 posts
#13
Y by
Solyaris....can you please elaborate what you are trying to say....I can't get it what you are trying to say....I am just interested that $f'$ is continuous on $\mathbb{R}$ and that is true because $f$ is twice differentiable.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
solyaris
632 posts
#15
Y by
@above. To elaborate: Let $M = \{x \in R : f'(x) \neq f'(c)\}$. In the first paragraph you show that for all $a < b$ $M$ has to contain some $m \in (a,b)$ (which means that $M$ is a dense subset of the real numbers). This part of your proof is fine. But in order to make the proof of the green claim work you need to show that $M$ contains all real numbers. This is missing in your proof (if I interpret you proof correctly).
Z K Y
N Quick Reply
G
H
=
a