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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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[*]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
k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
Graph Theory
ABCD1728   0
an hour ago
Can anyone provide the PDF version of "Graphs: an introduction" by Radu Bumbacea (XYZ press), thanks!
0 replies
ABCD1728
an hour ago
0 replies
Number theory for people who love theory
Assassino9931   3
N 2 hours ago by NamelyOrange
Source: Bulgaria RMM TST 2019
Prove that there is no positive integer $n$ such that $2^n + 1$ divides $5^n-1$.
3 replies
Assassino9931
Jul 31, 2024
NamelyOrange
2 hours ago
Interesting inequalities
sqing   0
2 hours ago
Source: Own
Let $ a,b> 0 ,   a+b+a^2+b^2=2.$ Prove that
$$ab+ \frac{k}{a+b+ab} \geq \frac{3-k+(k-1)\sqrt{5}}{2}$$Where $ k\geq 2. $
$$ab+ \frac{2}{a+b+ab} \geq \frac{1+\sqrt{5}}{2}$$$$ab+ \frac{3}{a+b+ab} \geq  \sqrt{5} $$
0 replies
sqing
2 hours ago
0 replies
Convergence of complex sequence
Rohit-2006   9
N 3 hours ago by Saucitom
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,
$$z_1=z_2=\cdots=z_k=1$$.
9 replies
1 viewing
Rohit-2006
May 17, 2025
Saucitom
3 hours ago
Stability of Additive Cauchy Equation
doanquangdang   1
N 3 hours ago by jasperE3
Show that if $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfies
$$
|f(x+y)-f(x)-f(y)-x y| \leq \varepsilon\left(|x|^p+|y|^p\right)
$$for some $\varepsilon>0,$ $p \in[0,1)$ and for all $x, y \in \mathbb{R}$, then there exists a unique solution $a: \mathbb{R} \rightarrow \mathbb{R}$ of the functional equation $a(x+y)=$ $a(x)+a(y)$ for all $x, y \in \mathbb{R}$ such that
$$
\left|f(x)-a(x)-\frac{1}{2} x^2\right| \leq \frac{2}{2-2^p} \varepsilon|x|^p
$$for all $x \in \mathbb{R}$.
1 reply
doanquangdang
Aug 16, 2024
jasperE3
3 hours ago
Polynomials with common roots and coefficients
VicKmath7   10
N 4 hours ago by math-olympiad-clown
Source: Balkan MO SL 2020 A3
Let $P(x), Q(x)$ be distinct polynomials of degree $2020$ with non-zero coefficients. Suppose that they have $r$ common real roots counting multiplicity and $s$ common coefficients. Determine the maximum possible value of $r + s$.

Demetres Christofides, Cyprus
10 replies
VicKmath7
Sep 9, 2021
math-olympiad-clown
4 hours ago
Serbian selection contest for the IMO 2025 - P3
OgnjenTesic   1
N 4 hours ago by korncrazy
Source: Serbian selection contest for the IMO 2025
Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that:
- $f$ is strictly increasing,
- there exists $M \in \mathbb{N}$ such that $f(x+1) - f(x) < M$ for all $x \in \mathbb{N}$,
- for every $x \in \mathbb{Z}$, there exists $y \in \mathbb{Z}$ such that
\[
            f(y) = \frac{f(x) + f(x + 2024)}{2}.
        \]Proposed by Pavle Martinović
1 reply
OgnjenTesic
Yesterday at 4:06 PM
korncrazy
4 hours ago
Interesting inequalities
sqing   0
4 hours ago
Source: Own
Let $ a,b,c \geq  0 , a+b+c+abc = 4.$ Prove that
$$ a+ab^2+\frac{15}{4}ab^2c^3  \leq \frac{2(100+13\sqrt{13})}{27}$$$$ 2a+ab^2+ 4ab^2c^3\leq \frac{4(68+5\sqrt{10})}{27}$$$$ 3a+ab^2+ \frac{9}{2}ab^2c^3\leq \frac{2(172+7\sqrt{7})}{27}$$$$a+ab^2+ 3.75982ab^2c^3 \leq \frac{2(100+13\sqrt{13})}{27}$$$$ 2a+ab^2+ 4.21981ab^2c^3\leq \frac{4(68+5\sqrt{10})}{27}$$$$ 3a+ab^2+4.73626ab^2c^3\leq \frac{2(172+7\sqrt{7})}{27}$$
0 replies
sqing
4 hours ago
0 replies
Inspired by old results
sqing   0
4 hours ago
Source: Own
Let $ a,b,c \geq  0 , a+b+c+abc = 4.$ Prove that
$$ a+ab+2ab^2c^3  \leq \frac{25}{4}$$$$ 2a+ab+\frac{29}{10}ab^2c^3 \leq 9$$$$3a+ab+4ab^2c^3 \leq \frac{49}{4}$$$$ 2a+ab+2.9746371ab^2c^3 \leq 9$$$$3a+ab+4.062494ab^2c^3 \leq \frac{49}{4}$$
0 replies
sqing
4 hours ago
0 replies
Interesting inequalities
sqing   3
N 5 hours ago by sqing
Source: Own
Let $ a,b,c,d\geq  0 , a+b+c+d \leq 4.$ Prove that
$$a(kbc+bd+cd)  \leq \frac{64k}{27}$$$$a (b+c) (kb c+  b d+  c d) \leq \frac{27k}{4}$$Where $ k\geq 2. $
3 replies
sqing
Yesterday at 12:44 PM
sqing
5 hours ago
Functional equation with powers
tapir1729   14
N 5 hours ago by Mathandski
Source: TSTST 2024, problem 6
Determine whether there exists a function $f: \mathbb{Z}_{> 0} \rightarrow \mathbb{Z}_{> 0}$ such that for all positive integers $m$ and $n$,
\[f(m+nf(m))=f(n)^m+2024! \cdot m.\]Jaedon Whyte
14 replies
tapir1729
Jun 24, 2024
Mathandski
5 hours ago
Invertible Matrices
Mateescu Constantin   8
N Yesterday at 8:44 PM by loup blanc
Source: Romanian District Olympiad 2018 - Grade XI - Problem 1
Show that if $n\ge 2$ is an integer, then there exist invertible matrices $A_1, A_2, \ldots, A_n \in \mathcal{M}_2(\mathbb{R})$ with non-zero entries such that:

\[A_1^{-1} + A_2^{-1} + \ldots + A_n^{-1} = (A_1 + A_2 + \ldots + A_n)^{-1}.\]
Edit.
8 replies
Mateescu Constantin
Mar 10, 2018
loup blanc
Yesterday at 8:44 PM
2024 Miklós-Schweitzer problem 3
Martin.s   2
N Yesterday at 7:01 PM by NODIRKHON_UZ
Do there exist continuous functions $f, g: \mathbb{R} \to \mathbb{R}$, both nowhere differentiable, such that $f \circ g$ is differentiable?
2 replies
Martin.s
Dec 5, 2024
NODIRKHON_UZ
Yesterday at 7:01 PM
2024 Mikl&oacute;s Schweitzer problem 2
Martin.s   1
N Yesterday at 6:43 PM by NODIRKHON_UZ
Does there exist a nowhere dense, nonempty compact set $C \subset [0,1]$ such that
\[
\liminf_{h \to 0^+} \frac{\lambda(C \cap (x, x+h))}{h} > 0 \quad \text{or} \quad \liminf_{h \to 0^+} \frac{\lambda(C \cap (x-h, x))}{h} > 0
\]holds for every point $x \in C$, where $\lambda(A)$ denotes the Lebesgue measure of $A$?
1 reply
Martin.s
Dec 5, 2024
NODIRKHON_UZ
Yesterday at 6:43 PM
A Typical Determinant Problem
Saucepan_man02   4
N Apr 26, 2025 by loup blanc
Source: Romania Contest, 2010
Let $A, B \in M_n(\mathbb R)$ with $B^2 = O_n$. Show that: $\det(AB+BA+I_n) \ge 0$.
4 replies
Saucepan_man02
Apr 17, 2025
loup blanc
Apr 26, 2025
A Typical Determinant Problem
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G H BBookmark kLocked kLocked NReply
Source: Romania Contest, 2010
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Saucepan_man02
1356 posts
#1
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Let $A, B \in M_n(\mathbb R)$ with $B^2 = O_n$. Show that: $\det(AB+BA+I_n) \ge 0$.
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loup blanc
3600 posts
#2 • 1 Y
Y by Tip_pay
$B$ is similar, over $\mathbb{R}$, to $diag(0_{n-2r},J_1,\cdots,J_r)$, where $J_i=J=\begin{pmatrix}0&1\\0&0\end{pmatrix}$.
Then we may assume that $B$ is in Jordan form.
$\bullet$ Here, we consider only the case $n=2r$.
Then $C=AB+BA$ is a $r\times r$ block-matrix where the entries are $2\times 2$ matrices in the form $C_{i,j}=p_{i,j}I_2+q_{i,j}J$.
Let $t\in\mathbb{R}$; since the $C_{i,j}$ pairwise commute, $\det(AB+BA+tI_n)=\det(u_0I_2+u_1J+u_2J^2+\cdots)$, where
$\sum_i u_iJ^i$ is obtained by developing the determinant by considering the $C_{i,j}$ as elements.
Finally $\det(AB+BA+tI)=\det(u_0I_2)={u_0}^2$, where $u_0$ is a polynomial of degree $r$ in $t$. $\square$
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loup blanc
3600 posts
#3
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$\bullet$ The general case -the sequel of post #2-
We may assume (see above) that $B=diag(0_{n-2r},K_{2r})$ and $A=\begin{pmatrix}P_{n-2r}&Q\\R&S_{2r}\end{pmatrix}$.
Then $AB+BA+I=\begin{pmatrix}I&QK\\KR&SK+KS+I\end{pmatrix}$ and
$\det(AB+BA+I)=\det(SK+KS+I-KTK)$, where $T=RQ\in M_{2r}$.
Note that $SK+KS$ is the matrix $C$ in post #2 and that $KTK$ is a $r\times r$ block-matrix where the entries are
$2\times 2$ matrices in the form $(KTK)_{i,j}=q_{i,j}J$ (then nilpotent).
As in post #2, $\det(SK+KS+I-KTK)=\det(v_0I_2+v_1J+v_2J^2+\cdots)=\det(v_0 I_2)$ but here $v_0$ does not depend on
the matrix $KTK$, that is, $\det(AB+BA+I)=\det(SK+KS+I)=\det(v_0 I_2)={v_0}^2$. $\square$
PS. If $t>0$, then $\det(AB+BA+tI)=t^n\det(A'B+BA'+I)$, where $A'=\dfrac{1}{t}A$.
Then $\det(AB+BA+tI)\geq 0$ and, by continuity in $0^+$, $\det(AB+BA)\geq 0$.
This post has been edited 1 time. Last edited by loup blanc, Apr 17, 2025, 8:59 PM
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removablesingularity
569 posts
#4 • 3 Y
Y by loup blanc, GreenKeeper, RobertRogo
Attempt for shorter answer : Using fact $B^2 = 0$ we have $AB+BA + I = AB^2A + AB + BA + I = AB.BA + AB + BA + I = \left(AB+I\right).\left(BA+I\right)$.
We have $\det\left(AB+BA+I\right) =  \det\left(AB+I\right) \det\left(BA+I\right).$ Meanwhile, recall that $\det \left(I+AB\right)=\det \left(I+BA\right)$.Proceed.
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loup blanc
3600 posts
#5
Y by
@ removablesingul ; well done.
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