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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
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)!

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[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]
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0 replies
jlacosta
May 1, 2025
0 replies
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Convergence of complex sequence
Rohit-2006   3
N an hour ago by solyaris
Suppose $z_1, z_2,\cdots,z_k$ are complex numbers with absolute value $1$. For $n=1,2,\cdots$ define $w_n=z_1^n+z_2^n+\cdots+z_k^n$. Given that the sequence $(w_n)_{n\geq1}$ converges. Show that,
$$z_1=z_2=\cdots=z_k=1$$.
3 replies
Rohit-2006
Saturday at 7:56 PM
solyaris
an hour ago
J
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System of Equations
P162008   0
2 hours ago
If $a,b$ and $c$ are complex numbers such that

$\frac{ab}{b + c} + \frac{bc}{c + a} + \frac{ca}{a + b} = -9$

$\frac{ab}{c + a} + \frac{bc}{a + b} + \frac{ca}{b + c} = 10$

Compute $\frac{a}{c + a} + \frac{b}{a + b} + \frac{c}{b + c}.$
0 replies
P162008
2 hours ago
0 replies
J
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System of Equations
P162008   0
2 hours ago
If $a,b$ and $c$ are complex numbers such that

$\sum_{cyc} ab = 23$

$\frac{a}{c + a} + \frac{b}{a + b} + \frac{c}{b + c} = -1$

$\frac{a^2b}{b + c} + \frac{b^2c}{c + a} + \frac{c^2a}{a + b} = 202$

Compute $\sum_{cyc} a^2.$
0 replies
P162008
2 hours ago
0 replies
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2022 MARBLE - Mock ARML I -8 \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=32
parmenides51   3
N 2 hours ago by P162008
Let $a,b,c$ complex numbers with $ab+ +bc+ca = 61$ such that
$$\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{a+b}= 5$$$$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=32.$$Find the value of $abc$.
3 replies
parmenides51
Jan 14, 2024
P162008
2 hours ago
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Calculus
youochange   4
N 2 hours ago by Mathzeus1024
Find the area enclosed by the curves $e^x,e^{-x},x^2+y^2=1$

4 replies
youochange
May 10, 2025
Mathzeus1024
2 hours ago
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ISI 2025
Zeroin   1
N 2 hours ago by alexheinis
Let $\mathbb{N}$ denote the set of natural numbers and let $(a_i,b_i),1 \leq i \leq 9$ denote $9$ ordered pairs in $\mathbb{N} \times \mathbb{N}$. Prove that there exist $3$ distinct elements in the set $2^{a_i}3^{b_i}$ for $1 \leq i \leq 9$ whose product is a perfect cube.
1 reply
Zeroin
Yesterday at 2:29 PM
alexheinis
2 hours ago
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Pell's Equation
Entrepreneur   1
N 3 hours ago by MihaiT
A Pells Equation is defined as follows $$x^2-1=ky^2.$$Where $x,y$ are positive integers and $k$ is a non-square positive integer. If $(x_n,y_n)$ denotes the n-th set of solution to the equation with $(x_0,y_0)=(1,0).$ Then, prove that $$x_{n+1}x_n-ky_{n+1}y_n=x_1,$$$$x_n\pm y_n\sqrt k=(x_1\pm y_1\sqrt k)^n.$$
1 reply
Entrepreneur
4 hours ago
MihaiT
3 hours ago
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Inequalities
sqing   15
N 4 hours ago by sqing
Let $a,b,c >2 $ and $ ab+bc+ca \leq 75.$ Show that
$$\frac{1}{a-2}+\frac{1}{b-2}+\frac{1}{c-2}\geq 1$$Let $a,b,c >2 $ and $ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq \frac{6}{7}.$ Show that
$$\frac{1}{a-2}+\frac{1}{b-2}+\frac{1}{c-2}\geq 2$$
15 replies
sqing
May 13, 2025
sqing
4 hours ago
J
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D1035 : Super TVI 2
Dattier   1
N 5 hours ago by alexheinis
Source: les dattes à Dattier
Let $f \in C([0,1])$. Is it true that $\exists a \in \left[0;\dfrac 13\right] \cup \left[\dfrac 23; 1 \right] , |f(a)| \leq 4 |f(1-a)|$ ?
1 reply
Dattier
Yesterday at 10:54 PM
alexheinis
5 hours ago
J
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D1034 : Super TVI
Dattier   2
N 5 hours ago by alexheinis
Source: les dattes à Dattier
Let $f \in C([0,1])$. Is it true that $\exists a \in \left[0;\dfrac 12\right], |f(a)| \leq 8 |f(1-a)|$ ?
2 replies
Dattier
Yesterday at 10:21 PM
alexheinis
5 hours ago
J
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Pertenacious Polynomial Problem
BadAtCompetitionMath21420   6
N Today at 3:51 AM by lbh_qys
Let the polynomial $P(x) = x^3-x^2+px-q$ have real roots and real coefficients with $q>0$. What is the maximum value of $p+q$?

This is a problem I made for my math competition, and I wanted to see if someone would double-check my work (No Mike allowed):

solution
Is this solution good?
6 replies
BadAtCompetitionMath21420
May 17, 2025
lbh_qys
Today at 3:51 AM
J
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Vieta's Formula.
BlackOctopus23   4
N Today at 3:11 AM by compoly2010
Can someone help me understand Vieta's Formula? I am currently learning it for my class. I learned that for a polynomial of degree $n$, all the roots added will give $-\frac{a_{n-1}}{a_n}$. I also learned that if every single root, multiplies every single root, it will give $\frac{a_{n-2}}{a_n}$. I also learned that if all the roots are multiplied, it will give $-\frac{a_0}{a_n}$. Is this right? And is there any purpose for these equations?
4 replies
BlackOctopus23
Yesterday at 11:10 PM
compoly2010
Today at 3:11 AM
J
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The sum of 335 distinct positive integers
Streit31415   1
N Today at 12:36 AM by Bocabulary142857
The sum of 335 distinct positive integers is equal to 100000
a) what is the minimum number of odd numbers among them ?
b) what is the maximum number of odd numbers among them ?
1 reply
Streit31415
Yesterday at 11:38 PM
Bocabulary142857
Today at 12:36 AM
J
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Diophantine Equation (cousin of Mordell)
urfinalopp   4
N Yesterday at 10:54 PM by FoeverResentful
Find pairs of integers $(x;y)$ such that:

$x^2=y^5+32$
4 replies
urfinalopp
Yesterday at 6:38 PM
FoeverResentful
Yesterday at 10:54 PM
J
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J
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Prove this recursion!
Entrepreneur   3
N Apr 20, 2025 by quasar_lord
Source: Amit Agarwal
Let $$I_n=\int z^n e^{\frac 1z}dz.$$Prove that $$\color{blue}{I_n=(n+1)!I_0+e^{\frac 1z}\sum_{n=1}^n n! z^{n+1}.}$$
3 replies
Entrepreneur
Jul 31, 2024
quasar_lord
Apr 20, 2025
J
Prove this recursion!
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Reveal topic tags
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Source: Amit Agarwal
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Entrepreneur
1177 posts
Entrepreneur
#1 h  Jul 31, 2024, 12:27 PM
Y by
Let $$I_n=\int z^n e^{\frac 1z}dz.$$Prove that $$\color{blue}{I_n=(n+1)!I_0+e^{\frac 1z}\sum_{n=1}^n n! z^{n+1}.}$$
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Evan484Ashoori
119 posts
Evan484Ashoori
#2 h  Aug 3, 2024, 3:23 PM
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sangsidhya
23 posts
sangsidhya
#3 h  Apr 20, 2025, 5:59 AM
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is he the author of arihant series?
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quasar_lord
313 posts
quasar_lord
#4 h  Apr 20, 2025, 11:06 AM
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Evan484Ashoori wrote:
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