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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)!

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]
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0 replies
jlacosta
May 1, 2025
0 replies
Bang's Lemma
EthanWYX2009   1
N 10 minutes ago by EthanWYX2009
Source: Bang's Lemma
Let $v_1,$ $v_2,$ $\ldots,$ $v_t$ be nonzero vectors in $d$-dimensional space. $m_1,$ $m_2,$ $\ldots ,$ $m_t$ are real numbers. Show that there exists $\varepsilon_1,$ $\varepsilon_2,$ $\ldots ,$ $\varepsilon_t\in\{\pm 1\},$ such that\[\left|\left\langle\sum_{i=1}^t\varepsilon_iv_i,\frac{v_k}{|v_k|}\right\rangle-m_k\right|\ge |v_k|\]holds for all $k=1,$ ${}{}{}2,$ $\ldots ,$ $t.$
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EthanWYX2009
2 hours ago
EthanWYX2009
10 minutes ago
Thailand MO 2025 P3
Kaimiaku   3
N 12 minutes ago by AblonJ
Let $a,b,c,x,y,z$ be positive real numbers such that $ay+bz+cx \le az+bx+cy$. Prove that $$ \frac{xy}{ax+bx+cy}+\frac{yz}{by+cy+az}+\frac{zx}{cz+az+bx} \le \frac{x+y+z}{a+b+c}$$
3 replies
Kaimiaku
Today at 6:48 AM
AblonJ
12 minutes ago
Grouping angles in a pentagon with bisectors
Assassino9931   1
N 23 minutes ago by nabodorbuco2
Source: Al-Khwarizmi International Junior Olympiad 2025 P2
Let $ABCD$ be a convex quadrilateral with \[\angle ADC = 90^\circ, \ \ \angle BCD = \angle ABC > 90^\circ, \mbox{ and } AB = 2CD.\]The line through \(C\), parallel to \(AD\), intersects the external angle bisector of \(\angle ABC\) at point \(T\). Prove that the angles $\angle ATB$, $\angle TBC$, $\angle BCD$, $\angle CDA$, $\angle DAT$ can be divided into two groups, so that the angles in each group have a sum of $270^{\circ}$.

Miroslav Marinov, Bulgaria
1 reply
Assassino9931
May 9, 2025
nabodorbuco2
23 minutes ago
Anything real in this system must be integer
Assassino9931   5
N 30 minutes ago by ballyyev.sapar
Source: Al-Khwarizmi International Junior Olympiad 2025 P1
Determine the largest integer $c$ for which the following statement holds: there exists at least one triple $(x,y,z)$ of integers such that
\begin{align*} x^2 + 4(y + z) = y^2 + 4(z + x) = z^2 + 4(x + y) = c \end{align*}and all triples $(x,y,z)$ of real numbers, satisfying the equations, are such that $x,y,z$ are integers.

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5 replies
Assassino9931
May 9, 2025
ballyyev.sapar
30 minutes ago
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Silver08   56
N 2 hours ago by Aiden-1089
Regular Round

Quarterfinals

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56 replies
Silver08
May 9, 2025
Aiden-1089
2 hours ago
Polynomial with integer coefficients
smartvong   1
N 3 hours ago by alexheinis
Source: UM Mathematical Olympiad 2024
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smartvong
3 hours ago
alexheinis
3 hours ago
Existence of scalars
smartvong   0
3 hours ago
Source: UM Mathematical Olympiad 2024
Let $U$ be a finite subset of $\mathbb{R}$ such that $U = -U$. Let $f,g : \mathbb{R} \to \mathbb{R}$ be functions satisfying
$$g(x) - g(y ) = (x - y)f(x + y)$$for all $x,y \in \mathbb{R} \backslash U$.
Show that there exist scalars $\alpha, \beta, \gamma \in \mathbb{R}$ such that
$$f(x) = \alpha x + \beta$$for all $x \in \mathbb{R}$,
$$g(x) = \alpha x^2 + \beta x + \gamma$$for all $x \in \mathbb{R} \backslash U$.
0 replies
smartvong
3 hours ago
0 replies
Invertible matrices in F_2
smartvong   1
N 4 hours ago by alexheinis
Source: UM Mathematical Olympiad 2024
Let $n \ge 2$ be an integer and let $\mathcal{S}_n$ be the set of all $n \times n$ invertible matrices in which their entries are $0$ or $1$. Let $m_A$ be the number of $1$'s in the matrix $A$. Determine the minimum and maximum values of $m_A$ in terms of $n$, as $A$ varies over $S_n$.
1 reply
smartvong
Today at 12:41 AM
alexheinis
4 hours ago
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SomeonecoolLovesMaths   13
N 4 hours ago by iced_tea
Source: ISI UGB 2025 P3
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SomeonecoolLovesMaths
May 11, 2025
iced_tea
4 hours ago
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Stephen123980   3
N Yesterday at 9:01 PM by BadAtMath23
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3 replies
Stephen123980
May 9, 2025
BadAtMath23
Yesterday at 9:01 PM
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youochange   2
N Yesterday at 7:46 PM by tom-nowy
$\int_{\alpha}^{\theta} \frac{d\theta}{\sqrt{cos\theta-cos\alpha}}$
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youochange
Yesterday at 2:26 PM
tom-nowy
Yesterday at 7:46 PM
ISI UGB 2025 P1
SomeonecoolLovesMaths   6
N Yesterday at 5:10 PM by SomeonecoolLovesMaths
Source: ISI UGB 2025 P1
Suppose $f \colon \mathbb{R} \longrightarrow \mathbb{R}$ is differentiable and $| f'(x)| < \frac{1}{2}$ for all $x \in \mathbb{R}$. Show that for some $x_0 \in \mathbb{R}$, $f \left( x_0 \right) = x_0$.
6 replies
SomeonecoolLovesMaths
May 11, 2025
SomeonecoolLovesMaths
Yesterday at 5:10 PM
Cute matrix equation
RobertRogo   3
N Yesterday at 2:23 PM by loup blanc
Source: "Traian Lalescu" student contest 2025, Section A, Problem 2
Find all matrices $A \in \mathcal{M}_n(\mathbb{Z})$ such that $$2025A^{2025}=A^{2024}+A^{2023}+\ldots+A$$Edit: Proposed by Marian Vasile (congrats!).
3 replies
RobertRogo
May 9, 2025
loup blanc
Yesterday at 2:23 PM
Integration Bee Kaizo
Calcul8er   63
N Yesterday at 1:50 PM by MS_asdfgzxcvb
Hey integration fans. I decided to collate some of my favourite and most evil integrals I've written into one big integration bee problem set. I've been entering integration bees since 2017 and I've been really getting hands on with the writing side of things over the last couple of years. I hope you'll enjoy!
63 replies
Calcul8er
Mar 2, 2025
MS_asdfgzxcvb
Yesterday at 1:50 PM
An interesting inequality
karis   1
N Oct 16, 2010 by kuing
Let $ x, y: 0<y\leq x\leq 1$.Prove that:
\[\frac{x^3y^2+y^3+x^2}{x^2+y^2+1}\geq xy\]
1 reply
karis
Oct 16, 2010
kuing
Oct 16, 2010
An interesting inequality
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karis
265 posts
#1 • 2 Y
Y by Adventure10, Mango247
Let $ x, y: 0<y\leq x\leq 1$.Prove that:
\[\frac{x^3y^2+y^3+x^2}{x^2+y^2+1}\geq xy\]
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kuing
1008 posts
#2 • 2 Y
Y by Adventure10, Mango247
karis wrote:
Let $ x, y: 0<y\leq x\leq 1$.Prove that:
\[\frac{x^3y^2+y^3+x^2}{x^2+y^2+1}\geq xy\]

because $0<y\leq x\leq 1$, we can let $x=\frac{a}{a+b},y=\frac{a}{a+b+c}$, where $a>0$ and $b,c\ge 0$.

then $x^3y^2+y^3+x^2-xy(x^2+y^2+1)$ $=\frac{a^4 \left(b^2+b c+c^2\right)+a^3 \left(b^3+3 b^2 c+4 b c^2+c^3\right)+a^2b c (b+c)^2}{(a+b)^3 (a+b+c)^3}\ge0$
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