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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.

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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
J
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Got what it takes to disprove Euler?
4everwise   18
N 2 hours ago by P162008
One of Euler's conjectures was disproved in then 1960s by three American mathematicians when they showed there was a positive integer $ n$ such that \[133^5 + 110^5 + 84^5 + 27^5 = n^5.\]Find the value of $ n$.
18 replies
4everwise
Feb 20, 2006
P162008
2 hours ago
J
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Inequalities
sqing   3
N 2 hours ago by sqing
Let $ a,b,c>0. $ Prove that$$a^2+b^2+c^2+abc-k(a+b+c)\geq 3k+2-2(k+1)\sqrt{k+1}$$Where $7\geq k \in N^+.$
$$a^2+b^2+c^2+abc-3(a+b+c)\geq-5$$
3 replies
sqing
Yesterday at 2:23 PM
sqing
2 hours ago
J
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Inequalities
sqing   14
N 3 hours ago by sqing
Let $ a,b>0 $ . Prove that
$$ \frac{a}{a^2+a +2b+1}+ \frac{b}{b^2+2a +b+1} \leq \frac{2}{5} $$$$ \frac{a}{a^2+2a +b+1}+ \frac{b}{b^2+a +2b+1} \leq \frac{2}{5} $$
14 replies
sqing
May 13, 2025
sqing
3 hours ago
J
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Prove the statement
Butterfly   11
N 4 hours ago by solyaris
Given an infinite sequence $\{x_n\} \subseteq [0,1]$, there exists some constant $C$, for any $r>0$, among the sequence $x_n$ and $x_m$ could be chosen to satisfy $|n-m|\ge r $ and $|x_n-x_m|<\frac{C}{|n-m|}$.
11 replies
Butterfly
May 7, 2025
solyaris
4 hours ago
J
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Maximum number of empty squares
Ecrin_eren   1
N 4 hours ago by Ecrin_eren


There are 16 kangaroos on a giant 4×4 chessboard, with exactly one kangaroo on each square. In each round, every kangaroo jumps to a neighboring square (up, down, left, or right — but not diagonally). All kangaroos stay on the board. More than one kangaroo can occupy the same square. What is the maximum number of empty squares that can exist after 100 rounds?



1 reply
Ecrin_eren
Yesterday at 6:35 PM
Ecrin_eren
4 hours ago
J
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External Direct Sum
We2592   0
Today at 2:45 AM
Q) 1. Let $V$ be external direct sum of vector spaces $U$ and $W$ over a field $\mathbb{F}$.let $\hat{U}={\{(u,0):u\in U\}}$ and $\hat{W}={\{(0,w):w\in W\}}$
show that
i) $\hat{U}$ and $\hat{W}$ is subspaces.
ii)$V=\hat{U}\oplus\hat{W}$

Q)2. Suppose $V=U+W$. Let $\hat{V}$ be the external direct sum of $U$ and $W$. show that $V$ is isomorphic to $\hat{V}$ under the correspondence $v=u+w\leftrightarrow(u,w)$

I face some trouble to solve this problems help me for understanding.
thank you.

0 replies
We2592
Today at 2:45 AM
0 replies
J
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Definite integration
girishpimoli   2
N Yesterday at 11:59 PM by Amkan2022
If $\displaystyle g(t)=\int^{t^{2}}_{2t}\cot^{-1}\bigg|\frac{1+x}{(1+t)^2-x}\bigg|dx.$ Then $\displaystyle \frac{g(5)}{g(3)}$ is
2 replies
girishpimoli
Apr 6, 2025
Amkan2022
Yesterday at 11:59 PM
J
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Putnam 1968 A6
sqrtX   11
N Yesterday at 11:47 PM by ohiorizzler1434
Source: Putnam 1968
Find all polynomials whose coefficients are all $\pm1$ and whose roots are all real.
11 replies
sqrtX
Feb 19, 2022
ohiorizzler1434
Yesterday at 11:47 PM
J
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Affine variety
YamoSky   1
N Yesterday at 9:01 PM by amplreneo
Let $A=\left\{z\in\mathbb{C}|Im(z)\geq0\right\}$. Is it possible to equip $A$ with a finitely generated k-algebra with one generator such that make $A$ be an affine variety?
1 reply
YamoSky
Jan 9, 2020
amplreneo
Yesterday at 9:01 PM
J
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Reducing the exponents for good
RobertRogo   0
Yesterday at 6:38 PM
Source: The national Algebra contest (Romania), 2025, Problem 3/Abstract Algebra (a bit generalized)
Let $A$ be a ring with unity such that for every $x \in A$ there exist $t_x, n_x \in \mathbb{N}^*$ such that $x^{t_x+n_x}=x^{n_x}$. Prove that
a) If $t_x \cdot 1 \in U(A), \forall x \in A$ then $x^{t_x+1}=x, \forall x \in A$
b) If there is an $x \in A$ such that $t_x \cdot 1 \notin U(A)$ then the result from a) may no longer hold.

Authors: Laurențiu Panaitopol, Dorel Miheț, Mihai Opincariu, me, Filip Munteanu
0 replies
RobertRogo
Yesterday at 6:38 PM
0 replies
J
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Differential equations , Matrix theory
c00lb0y   3
N Yesterday at 12:26 PM by loup blanc
Source: RUDN MATH OLYMP 2024 problem 4
Any idea?? Diff equational system combined with Matrix theory.
Consider the equation dX/dt=X^2, where X(t) is an n×n matrix satisfying the condition detX=0. It is known that there are no solutions of this equation defined on a bounded interval, but there exist non-continuable solutions defined on unbounded intervals of the form (t ,+∞) and (−∞,t). Find n.
3 replies
c00lb0y
Apr 17, 2025
loup blanc
Yesterday at 12:26 PM
J
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The matrix in some degree is a scalar
FFA21   4
N Yesterday at 12:06 PM by FFA21
Source: MSU algebra olympiad 2025 P2
$A\in M_{3\times 3}$ invertible, for an infinite number of $k$:
$tr(A^k)=0$
Is it true that $\exists n$ such that $A^n$ is a scalar
4 replies
FFA21
Yesterday at 12:11 AM
FFA21
Yesterday at 12:06 PM
J
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Weird integral
Martin.s   0
Yesterday at 9:33 AM
\[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{1 - e^{-2} \cos\left(2\left(u + \tan u\right)\right)} {1 - 2e^{-2} \cos\left(2\left(u + \tan u\right)\right) + e^{-4}} \, \mathrm{d}u \]
0 replies
Martin.s
Yesterday at 9:33 AM
0 replies
J
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hard number theory problem
danilorj   4
N Yesterday at 9:01 AM by c00lb0y
Let \( a \) and \( b \) be positive integers. Prove that
\[ a^2 + \left\lceil \frac{4a^2}{b} \right\rceil \]is not a perfect square.
4 replies
danilorj
May 18, 2025
c00lb0y
Yesterday at 9:01 AM
J
a