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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
8 degree polynomial
Adywastaken   0
a minute ago
Source: NMTC Junior 2024/3
Let $a, b, c, d, e, f\in \mathbb{R}$ such that the polynomial $p(x)=x^8-4x^7+7x^6+ax^5+bx^4+cx^3+dx^2+ex+f$ has 8 linear factors of the form $x-x_i$ with $x_i>0$ for $i=1, 2, 3, 4, 5, 6, 7, 8$. Find all possible values of the constant $f$.
0 replies
Adywastaken
a minute ago
0 replies
System of equations
Adywastaken   0
6 minutes ago
Source: NMTC 2024/2
$(1+4^{2x-y})5^{1-2x+y}=1+2^{2x-y+1}$
$y^3+4x+1+\log(y^2+2x)=0$
0 replies
Adywastaken
6 minutes ago
0 replies
Inequality, inequality, inequality...
Assassino9931   8
N 8 minutes ago by sqing
Source: Al-Khwarizmi Junior International Olympiad 2025 P6
Let $a,b,c$ be real numbers such that \[ab^2+bc^2+ca^2=6\sqrt{3}+ac^2+cb^2+ba^2.\]Find the smallest possible value of $a^2 + b^2 + c^2$.

Binh Luan and Nhan Xet, Vietnam
8 replies
Assassino9931
6 hours ago
sqing
8 minutes ago
Easy counting
Adywastaken   0
11 minutes ago
Source: NMTC Junior 2024/1
Find the number of sets of $4$ positive integers, less than or equal to $25$, such that the difference between any $2 $ elements in the set is at least $3$.
0 replies
Adywastaken
11 minutes ago
0 replies
Preparing for Putnam level entrance examinations
Cats_on_a_computer   4
N 2 hours ago by Cats_on_a_computer
Non American high schooler in the equivalent of grade 12 here. Where I live, two the best undergraduates program in the country accepts students based on a common entrance exam. The first half of the exam is “screening”, with 4 options being presented per question, each of which one has to assign a True or False. This first half is about the difficulty of an average AIME, or JEE Adv paper, and it is a requirement for any candidate to achieve at least 24/40 on this half for the examiners to even consider grading the second part. The second part consists of long form questions, and I have, no joke, seen them literally rip off, verbatim, Putnam A6s. Some of the problems are generally standard textbook problems in certain undergrad courses but obviously that doesn’t translate it to being doable for high school students. I’ve effectively got to prepare for a slightly nerfed Putnam, if you will, and so I’ve been looking for resources (not just problems) for Putnam level questions. Does anyone have any suggestions?
4 replies
Cats_on_a_computer
Yesterday at 8:32 AM
Cats_on_a_computer
2 hours ago
Marginal Profit
NC4723   1
N 5 hours ago by Juno_34
Please help me solve this
1 reply
NC4723
Dec 11, 2015
Juno_34
5 hours ago
Romania NMO 2023 Grade 11 P1
DanDumitrescu   15
N Today at 5:46 AM by anudeep
Source: Romania National Olympiad 2023
Determine twice differentiable functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which verify relation

\[
    \left( f'(x) \right)^2 + f''(x) \leq 0, \forall x \in \mathbb{R}.
    \]
15 replies
DanDumitrescu
Apr 14, 2023
anudeep
Today at 5:46 AM
Subset Ordered Pairs of {1, 2, ..., 10}
ahaanomegas   11
N Today at 5:27 AM by cappucher
Source: Putnam 1990 A6
If $X$ is a finite set, let $X$ denote the number of elements in $X$. Call an ordered pair $(S,T)$ of subsets of $ \{ 1, 2, \cdots, n \} $ $ \emph {admissible} $ if $ s > |T| $ for each $ s \in S $, and $ t > |S| $ for each $ t \in T $. How many admissible ordered pairs of subsets $ \{ 1, 2, \cdots, 10 \} $ are there? Prove your answer.
11 replies
ahaanomegas
Jul 12, 2013
cappucher
Today at 5:27 AM
Putnam 2000 B4
ahaanomegas   6
N Today at 1:53 AM by mqoi_KOLA
Let $f(x)$ be a continuous function such that $f(2x^2-1)=2xf(x)$ for all $x$. Show that $f(x)=0$ for $-1\le x \le 1$.
6 replies
ahaanomegas
Sep 6, 2011
mqoi_KOLA
Today at 1:53 AM
Another integral limit
RobertRogo   1
N Yesterday at 8:37 PM by alexheinis
Source: "Traian Lalescu" student contest 2025, Section A, Problem 3
Let $f \colon [0, \infty) \to \mathbb{R}$ be a function differentiable at 0 with $f(0) = 0$. Find
$$\lim_{n \to \infty} \frac{1}{n} \int_{2^n}^{2^{n+1}} f\left(\frac{\ln x}{x}\right) dx$$
1 reply
RobertRogo
Yesterday at 2:28 PM
alexheinis
Yesterday at 8:37 PM
AB=BA if A-nilpotent
KevinDB17   3
N Yesterday at 7:51 PM by loup blanc
Let A,B 2 complex n*n matrices such that AB+I=A+B+BA
If A is nilpotent prove that AB=BA
3 replies
KevinDB17
Mar 30, 2025
loup blanc
Yesterday at 7:51 PM
Very nice equivalence in matrix equations
RobertRogo   3
N Yesterday at 5:45 PM by Etkan
Source: "Traian Lalescu" student contest 2025, Section A, Problem 4
Let $A, B \in \mathcal{M}_n(\mathbb{C})$ Show that the following statements are equivalent:

i) For every $C \in \mathcal{M}_n(\mathbb{C})$ there exist $X, Y \in \mathcal{M}_n(\mathbb{C})$ such that $AX + YB = C$
ii) For every $C \in \mathcal{M}_n(\mathbb{C})$ there exist $U, V \in \mathcal{M}_n(\mathbb{C})$ such that $A^2 U + V B^2 = C$

3 replies
RobertRogo
Yesterday at 2:34 PM
Etkan
Yesterday at 5:45 PM
Miklos Schweitzer 1971_5
ehsan2004   2
N Yesterday at 5:25 PM by pi_quadrat_sechstel
Let $ \lambda_1 \leq \lambda_2 \leq...$ be a positive sequence and let $ K$ be a constant such that \[  \sum_{k=1}^{n-1} \lambda^2_k < K \lambda^2_n \;(n=1,2,...).\] Prove that there exists a constant $ K'$ such that \[  \sum_{k=1}^{n-1} \lambda_k < K' \lambda_n \;(n=1,2,...).\]

L. Leindler
2 replies
ehsan2004
Oct 29, 2008
pi_quadrat_sechstel
Yesterday at 5:25 PM
Cute matrix equation
RobertRogo   1
N Yesterday at 4:46 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 $$2025A^{2025}=A^{2024}+A^{2023}+\ldots+A$$
1 reply
RobertRogo
Yesterday at 2:23 PM
loup blanc
Yesterday at 4:46 PM
A lot of numbers and statements
nAalniaOMliO   2
N Mar 29, 2025 by nAalniaOMIiO
Source: Belarusian National Olympiad 2025
101 numbers are written in a circle. Near the first number the statement "This number is bigger than the next one" is written, near the second "This number is bigger that the next two" and etc, near the 100th "This number is bigger than the next 100 numbers".
What is the maximum possible amount of the statements that can be true?
2 replies
nAalniaOMliO
Mar 28, 2025
nAalniaOMIiO
Mar 29, 2025
A lot of numbers and statements
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G H BBookmark kLocked kLocked NReply
Source: Belarusian National Olympiad 2025
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nAalniaOMliO
297 posts
#1
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101 numbers are written in a circle. Near the first number the statement "This number is bigger than the next one" is written, near the second "This number is bigger that the next two" and etc, near the 100th "This number is bigger than the next 100 numbers".
What is the maximum possible amount of the statements that can be true?
Z K Y
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RagvaloD
4913 posts
#3
Y by
Let biggest (or one of the biggest)number is $i$-th
If $j<i \leq 2j$ then statement for $j$-th number is wrong, so there are $[\frac{i}{2}]$ wrong statements for $j<i$
If $j>i$ and $i \leq 2j-101$ then statement for $j$-th number is wrong, so there are $101-[\frac{i+102}{2}]=50-[\frac{i}{2}]$ wrong statements for $j>i$
So at least $50$ statements are false, and so not more than $50$ is true.
Now just set $a_i=101-i$ to prove, that $50$ is possible
Z K Y
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nAalniaOMIiO
2 posts
#4
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Very beautiful solution.
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