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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.
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[*]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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Inspired by m4thbl3nd3r
sqing   1
N 14 minutes ago by sqing
Source: Own
Let $ a, b,c>0,b+c>a$. Prove that$$\sqrt{\frac{a}{b+c-a}}-\frac{2a^2-b^2-c^2}{(a+b)(a+c)}\geq 1$$$$\frac{a}{b+c-a}-\frac{2a^2-b^2-c^2}{(a+b)(a+c)} \geq \frac{4\sqrt 2}{3}-1$$
1 reply
sqing
29 minutes ago
sqing
14 minutes ago
J
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2-var inequality
sqing   6
N 39 minutes ago by sqing
Source: Own
Let $ a,b> 0 $ and $2a+2b+ab=5. $ Prove that
$$ \frac{a^2}{b^2}+\frac{1}{a^2}-a^2\geq 1$$$$ \frac{a^3}{b^3}+\frac{1}{a^3}-a^3\geq 1$$
6 replies
1 viewing
sqing
Yesterday at 1:19 PM
sqing
39 minutes ago
J
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Inequality
Amin12   8
N an hour ago by A.H.H
Source:  Iran 3rd round-2017-Algebra final exam-P3
Let $a,b$ and $c$ be positive real numbers. Prove that
$$\sum_{cyc} \frac {a^3b}{(3a+2b)^3} \ge \sum_{cyc} \frac {a^2bc}{(2a+2b+c)^3} $$
8 replies
Amin12
Sep 2, 2017
A.H.H
an hour ago
J
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Not so beautiful
m4thbl3nd3r   0
an hour ago
Let $a, b,c>0$ such that $b+c>a$. Prove that $$2 \sqrt[4]{\frac{a}{b+c-a}}\ge 2 +\frac{2a^2-b^2-c^2}{(a+b)(a+c)}.$$
0 replies
m4thbl3nd3r
an hour ago
0 replies
J
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Problem 2, Grade 12th RMO Shortlist - Year 2002
sticknycu   5
N Yesterday at 4:07 PM by P_Fazioli
Let $A \in M_2(C), A \neq O_2, A \neq I_2, n \in \mathbb{N}^*$ and $S_n = \{ X \in M_2(C) | X^n = A \}$.
Show:
a) $S_n$ with multiplication of matrixes operation is making an isomorphic-group structure with $U_n$.
b) $A^2 = A$.

Marian Andronache
5 replies
sticknycu
Jan 3, 2020
P_Fazioli
Yesterday at 4:07 PM
J
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3rd AKhIMO for University Students, P1
UzbekMathematician   1
N Yesterday at 3:42 PM by KAME06
Source: AKhIMO 2025, P1
There are two circles in the $xy-$plane centered on the $y-$axis which are tangent to both the parabola $y=x^2$ and the line $y=2025$. Determine the lengths of the circles' diameters.
1 reply
UzbekMathematician
Yesterday at 1:44 PM
KAME06
Yesterday at 3:42 PM
J
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3rd AKhIMO for university students, p4
UzbekMathematician   1
N Yesterday at 2:29 PM by grupyorum
Source: AKhIMO 2025, P4
Define a sequence $a_1, a_2, a_3, ... $ by $a_1=2, a_2=5$ and $a_{n+2}=f(a_n, a_{n+1})$ for all $n \ge 1$, where $$f(x,y)=5(x+y)+2\sqrt{6x^2+15xy+6y^2}.$$Show that $a_n$ is an integer for all $n\ge 1$.
1 reply
UzbekMathematician
Yesterday at 2:03 PM
grupyorum
Yesterday at 2:29 PM
J
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Double Integral
namesis   3
N Yesterday at 2:27 PM by Mathzeus1024
The area of integration, $D$, is defined in plane polar coordinates $(r, \phi)$ by the inequality
$r-2 \leq r \leq r_1$, where $r_1 = 1 + \cos(\phi)$ and $r_2 = 3/2$.

Evaluate:

$\iint_D \frac{x+y+xy}{x^2 + y^2} dx dy $

I tried evaluating the integral in polar, with $r$ from $\frac{3}{2}$ to $1+ \cos \phi$ and $\phi$ from $- \frac{\pi}{3}$ to $ \frac{\pi}{3}$ but in vain.
3 replies
namesis
Dec 16, 2015
Mathzeus1024
Yesterday at 2:27 PM
J
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Japanese Olympiad
parkjungmin   8
N Yesterday at 2:22 PM by parkjungmin
It's about the Japanese Olympiad

I can't solve it no matter how much I think about it.

If there are people who are good at math

Please help me.
8 replies
parkjungmin
May 10, 2025
parkjungmin
Yesterday at 2:22 PM
J
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3rd AKhIMO for university students, P5
UzbekMathematician   0
Yesterday at 2:10 PM
Source: AKhIMO 2025, P5
Show that for every positive integer $n$ there exist nonnegative integers $p, q$ and integers $a_1, a_2, ... , a_p, b_1, b_2, ... , b_q \ge 2$ such that $$ n=\frac{(a_1^3-1)(a_2^3-1)...(a_p^3-1)}{(b_1^3-1)(b_2^3-1)...(b_q^3-1)} $$
0 replies
UzbekMathematician
Yesterday at 2:10 PM
0 replies
J
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Double integral on polar coordinates
Kurchi   1
N Yesterday at 1:54 PM by Mathzeus1024
How to find the volume inside z=r above r=3+cosθ, with the help of double integral?
1 reply
Kurchi
Jul 2, 2020
Mathzeus1024
Yesterday at 1:54 PM
J
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Recurrence trouble
SomeonecoolLovesMaths   0
Yesterday at 11:27 AM
Let $0 < x_0 < y_0$ be real numbers. Define $x_{n+1} = \frac{x_n + y_n}{2}$ and $y_{n+1} = \sqrt{x_{n+1}y_n}$.
Prove that $\lim_{n \to \infty} x_n = \lim_{n \to \infty} y_n$ and hence find the limit.
0 replies
SomeonecoolLovesMaths
Yesterday at 11:27 AM
0 replies
J
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nice integral
Martin.s   1
N Yesterday at 10:10 AM by Svyatoslav
$$\int_0^\infty \left(\frac{1}{\log t}+\frac{1}{1-t}\right)^3 \frac{dt}{1+t^2}$$
1 reply
Martin.s
Tuesday at 11:24 AM
Svyatoslav
Yesterday at 10:10 AM
J
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Sequence and Series
P162008   1
N Yesterday at 9:24 AM by alexheinis
Source: Coaching Test
Let $a_n = 3n + \sqrt{n^2 - 1}$ and $b_n = 2\left(\sqrt{n^2 - n} + \sqrt{n^2 + n}\right)$
If $\sum_{i=1}^{49} \sqrt{a_i - b_i} = A + B\sqrt{2}$ for some integers $A$ and $B.$ Find the value of $A^2 + B^2.$
1 reply
P162008
Tuesday at 6:22 AM
alexheinis
Yesterday at 9:24 AM
J
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J
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Number theory
XAN4   1
N Apr 22, 2025 by NTstrucker
Source: own
Prove that there exists infinitely many positive integers $x,y,z$ such that $x,y,z\ne1$ and $x^x\cdot y^y=z^z$.
1 reply
XAN4
Apr 20, 2025
NTstrucker
Apr 22, 2025
J
Number theory
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Reveal topic tags
number theory
number theory unsolved
Diophantine equation
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XAN4
61 posts
XAN4
#1 h  Apr 20, 2025, 9:44 AM
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Prove that there exists infinitely many positive integers $x,y,z$ such that $x,y,z\ne1$ and $x^x\cdot y^y=z^z$.
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NTstrucker
164 posts
NTstrucker
#2 h  Apr 22, 2025, 10:28 AM
Y by
https://www.mathnet.ru/links/a298bf5eafa2440220a2e98417c92670/tm2337.pdf
This post has been edited 1 time. Last edited by NTstrucker, Apr 22, 2025, 10:33 AM
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