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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
Yesterday at 11:16 PM
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
Yesterday at 11:16 PM
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
Hard inequality
ys33   5
N 6 minutes ago by sqing
Let $a, b, c, d>0$. Prove that
$\sqrt[3]{ab}+ \sqrt[3]{cd} < \sqrt[3]{(a+b+c)(b+c+d)}$.
5 replies
+1 w
ys33
4 hours ago
sqing
6 minutes ago
China Northern MO 2009 p4 CNMO
parkjungmin   1
N 13 minutes ago by exoticc
Source: China Northern MO 2009 p4 CNMO
China Northern MO 2009 p4 CNMO

The problem is too difficult.
Is there anyone who can help me?
1 reply
parkjungmin
Apr 30, 2025
exoticc
13 minutes ago
Hojoo Lee problem 73
Leon   25
N 17 minutes ago by sqing
Source: Belarus 1998
Let $a$, $b$, $c$ be real positive numbers. Show that \[\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq \frac{a+b}{b+c}+\frac{b+c}{a+b}+1\]
25 replies
+1 w
Leon
Aug 21, 2006
sqing
17 minutes ago
Almost Squarefree Integers
oVlad   3
N 24 minutes ago by Primeniyazidayi
Source: Romania Junior TST 2025 Day 1 P1
A positive integer $n\geqslant 3$ is almost squarefree if there exists a prime number $p\equiv 1\bmod 3$ such that $p^2\mid n$ and $n/p$ is squarefree. Prove that for any almost squarefree positive integer $n$ the ratio $2\sigma(n)/d(n)$ is an integer.
3 replies
oVlad
Apr 12, 2025
Primeniyazidayi
24 minutes ago
Math camp combi
ErTeeEs06   3
N 24 minutes ago by genius_007
Source: BxMO 2025 P2
Let $N\geq 2$ be a natural number. At a mathematical olympiad training camp the same $N$ courses are organised every day. Each student takes exactly one of the $N$ courses each day. At the end of the camp, every student has takes each course exactly once, and any two students took the same course on at least one day, but took different courses on at least one other day. What is, in terms of $N$, the largest possible number of students at the camp?
3 replies
ErTeeEs06
Apr 26, 2025
genius_007
24 minutes ago
Benelux fe
ErTeeEs06   10
N 34 minutes ago by genius_007
Source: BxMO 2025 P1
Does there exist a function $f:\mathbb{R}\to \mathbb{R}$ such that $$f(x^2+f(y))=f(x)^2-y$$for all $x, y\in \mathbb{R}$?
10 replies
1 viewing
ErTeeEs06
Apr 26, 2025
genius_007
34 minutes ago
IMO Shortlist Problems
ABCD1728   0
an hour ago
Source: IMO official website
Where can I get the official solution for ISL before 2005? The official website only has solutions after 2006. Thanks :)
0 replies
ABCD1728
an hour ago
0 replies
Geometric inequality in quadrilateral
BBNoDollar   0
an hour ago
Source: Romanian Mathematical Gazette 2025
Let ABCD be a convex quadrilateral with angles BAD and BCD obtuse, and let the points E, F ∈ BD, such that AE ⊥ BD and CF ⊥ BD.
Prove that 1/(AE*CF) ≥ 1/(AB*BC) + 1/(AD*CD) .
0 replies
BBNoDollar
an hour ago
0 replies
A coincidence about triangles with common incenter
flower417477   2
N an hour ago by flower417477
$\triangle ABC,\triangle ADE$ have the same incenter $I$.Prove that $BCDE$ is concyclic iff $BC,DE,AI$ is concurrent
2 replies
flower417477
Wednesday at 2:08 PM
flower417477
an hour ago
Function equation
LeDuonggg   5
N 2 hours ago by luutrongphuc
Find all functions $f: \mathbb{R^+} \rightarrow \mathbb{R^+}$ , such that for all $x,y>0$:
\[ f(x+f(y))=\dfrac{f(x)}{1+f(xy)}\]
5 replies
LeDuonggg
Yesterday at 2:59 PM
luutrongphuc
2 hours ago
Consecutive sum of integers sum up to 2020
NicoN9   2
N 2 hours ago by NicoN9
Source: Japan Junior MO Preliminary 2020 P2
Let $a$ and $b$ be positive integers. Suppose that the sum of integers between $a$ and $b$, including $a$ and $b$, are equal to $2020$.
All among those pairs $(a, b)$, find the pair such that $a$ achieves the minimum.
2 replies
NicoN9
Today at 6:09 AM
NicoN9
2 hours ago
Range of a^3+b^3-3c
Kunihiko_Chikaya   1
N 2 hours ago by Mathzeus1024
Let $a,\ b,\ c$ be real numbers such that $b<\frac{1}{c}<a$ and

$$\begin{cases}a+b+c=1 \ \\ a^2+b^2+c^2=23	

\end{cases}$$
Find the range of $a^3+b^3-3c.$


Proposed by Kunihiko Chikaya/September 23, 2020
1 reply
Kunihiko_Chikaya
Sep 23, 2020
Mathzeus1024
2 hours ago
equations
kjhgyuio   1
N 2 hours ago by mashumaro
........
1 reply
kjhgyuio
2 hours ago
mashumaro
2 hours ago
powers of 2
ErTeeEs06   3
N 2 hours ago by GeorgeMetrical123
Source: BxMO 2025 P4
Let $a_0, a_1, \ldots, a_{10}$ be integers such that, for each $i \in \{0,1,\ldots,2047\}$, there exists a subset $S \subseteq \{0,1,\ldots,10\}$ with
\[
\sum_{j \in S} a_j \equiv i \pmod{2048}.
\]Show that for each $i \in \{0,1,\ldots,10\}$, there is exactly one $j \in \{0,1,\ldots,10\}$ such that $a_j$ is divisible by $2^i$ but not by $2^{i+1}$.

Note: $\sum_{j \in S} a_j$ is the summation notation, for instance, $\sum_{j \in \{2,5\}} a_j = a_2 + a_5$, while for the empty set $\varnothing$, one defines $\sum_{j \in \varnothing} a_j = 0$.
3 replies
ErTeeEs06
Apr 26, 2025
GeorgeMetrical123
2 hours ago
NT with repeating decimal digits
oVlad   1
N Apr 21, 2025 by kokcio
Source: Romania EGMO TST 2019 Day 1 P2
Determine the digits $0\leqslant c\leqslant 9$ such that for any positive integer $k{}$ there exists a positive integer $n$ such that the last $k{}$ digits of $n^9$ are equal to $c{}.$
1 reply
oVlad
Apr 21, 2025
kokcio
Apr 21, 2025
NT with repeating decimal digits
G H J
G H BBookmark kLocked kLocked NReply
Source: Romania EGMO TST 2019 Day 1 P2
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oVlad
1742 posts
#1
Y by
Determine the digits $0\leqslant c\leqslant 9$ such that for any positive integer $k{}$ there exists a positive integer $n$ such that the last $k{}$ digits of $n^9$ are equal to $c{}.$
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kokcio
69 posts
#2
Y by
We cannot have $c=2,4,6,8$, because last four digits would have to be divisible by $16$, but in each of this cases numbers $2222, 4444, 6666, 8888$ are not. Similarly, we can see that $c=5$ leads to contradiction.
Now, assume that $c$ is coprime to $10$. We know that $9$ is coprime with $\phi(5^n)$ and $\phi(2^n)$ for all $n$, so function $f(x)=x^9$ is bijective on integers relatively prime to $10^n$ (we count modulo $10^n$). To see that this function is bijective, we can also see that $a^9\equiv b^9\mod 5^n$ iff $5^n$ divides $(a-b)(a^2+ab+b^2)(a^6+a^3b^3+b^6)$, but $5$ cannot divide neither $a^2+ab+b^2$, nor $a^6+a^3b^3+b^6$ if $ab$ is not divisible by $5$, so we would have to have $a\equiv b\mod5^n$. The same argument works modulo $2^n$. Therefore, we can have $c=1,3,7,9$. Obviously, we can also have $c=0$, so our answer is $c\in\{0,1,3,7,9\}$.
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