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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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[*]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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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
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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
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official solution of IGO
ABCD1728   7
N 8 minutes ago by ABCD1728
Source: IGO official website
Where can I get the official solution of IGO for 2023 and 2024, there are some inhttps://imogeometry.blogspot.com/p/iranian-geometry-olympiad.html, but where can I find them on the official website, thanks :)
7 replies
1 viewing
ABCD1728
May 4, 2025
ABCD1728
8 minutes ago
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Combo geo with circles
a_507_bc   10
N 11 minutes ago by EthanWYX2009
Source: 239 MO 2024 S8
There are $2n$ points on the plane. No three of them lie on the same straight line and no four lie on the same circle. Prove that it is possible to split these points into $n$ pairs and cover each pair of points with a circle containing no other points.
10 replies
a_507_bc
May 22, 2024
EthanWYX2009
11 minutes ago
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Vietnam TST #5
IMOStarter   2
N 17 minutes ago by cursed_tangent1434
Source: Vietnam TST 2022 P5
A fractional number $x$ is called pretty if it has finite expression in base$-b$ numeral system, $b$ is a positive integer in $[2;2022]$. Prove that there exists finite positive integers $n\geq 4$ that with every $m$ in $(\frac{2n}{3}; n)$ then there is at least one pretty number between $\frac{m}{n-m}$ and $\frac{n-m}{m}$
2 replies
IMOStarter
Apr 27, 2022
cursed_tangent1434
17 minutes ago
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Squares consisting of digits 0, 4, 9
VicKmath7   4
N 32 minutes ago by NicoN9
Source: Bulgaria MO Regional round 2024, 9.3
A positive integer $n$ is called a $\textit{supersquare}$ if there exists a positive integer $m$, such that $10 \nmid m$ and the decimal representation of $n=m^2$ consists only of digits among $\{0, 4, 9\}$. Are there infinitely many $\textit{supersquares}$?
4 replies
VicKmath7
Feb 13, 2024
NicoN9
32 minutes ago
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Combinatorial Sum
P162008   0
42 minutes ago
Source: ARML
Compute the greatest integer $k$ such that $2^k$ divides

$\sum_{0 \leq i < j \leq 2024} \left[\binom{2024}{i}\binom{2034}{j} - \binom{2024}{j}\binom{2034}{i}\right]^2$
0 replies
P162008
42 minutes ago
0 replies
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(3x+y)(3y+z)(3z+x) \ge 64xyz if x,y,z>0
parmenides51   4
N an hour ago by AylyGayypow009
Source: Greece JBMO TST 2015 p1
If $x,y,z>0$, prove that $(3x+y)(3y+z)(3z+x) \ge 64xyz$. When we have equality;
4 replies
parmenides51
Apr 29, 2019
AylyGayypow009
an hour ago
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Need proof for greedy algorithm for array merging
avighnac   1
N an hour ago by avighnac
Source: Baltic Olympiad in Informatics 2025: Day 2, Problem 2
I'm working on the following problem:

[size=150]Problem[/size]
You have an array of $n$ numbers $a_1, \dots, a_n$. You repeatedly merge two adjacent numbers $x$ and $y$ into a single number $\max(x,y)+1$, until only one number remains. Find the minimum final value that can be obtained.

Note: $a_i \ge 0$, and $a_i \in \mathbb{Z}^+_0$. So each $a_i$ is a non-negative integer.

[size=150]Greedy algorithm[/size]
I need help proving (or disproving) the following greedy algorithm: at each step merge $a_i$ with $a_{i+1}$ such that $\max(a_i, a_{i+1})+1$ is minimized across all choices of $i \in [1, n)$. In case of ties, choose the _smallest_ $i$.

I understand how to rephrase any merge sequence as a complete binary tree of depth $d_i$ at leaf $i$, and show that the final root value equals

$$\max_{1\le i \le n} a_i+d_i$$

Note that this also means the answer has to be $\le M+\log_2(n)$, where $M$ is the maximum value in the array.

However, I'm struggling to make the exchange argument fully rigourous. In particular, after swapping the first merge of an assumed-optimal strategy with the greedy-first merge, the resulting multiset of intermediate values changes. How do I argue that "continuing the same tree shape" on this new multiset still yields a no-worse maximum $a_i+d_i$, since it changes?

I’ve posted this on Math Stack Exchange but haven't received any feedback yet. It seems that the focus there is more on formal proofs and textbook-style problems. I think AoPS might be a better place for more creative and exploratory questions like this. If you have any ideas, please let me know!
1 reply
avighnac
an hour ago
avighnac
an hour ago
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Really fun geometry problem
Sadigly   6
N an hour ago by farhad.fritl
Source: Azerbaijan Senior MO 2025 P6
In an acute triangle $ABC$ with $AB<AC$, the foot of altitudes from $A,B,C$ to the sides $BC,CA,AB$ are $D,E,F$, respectively. $H$ is the orthocenter. $M$ is the midpoint of segment $BC$. Lines $MH$ and $EF$ intersect at $K$. Let the tangents drawn to circumcircle $(ABC)$ from $B$ and $C$ intersect at $T$. Prove that $T;D;K$ are colinear
6 replies
Sadigly
Yesterday at 4:29 PM
farhad.fritl
an hour ago
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the epitome of olympiad nt
youlost_thegame_1434   31
N an hour ago by MR.1
Source: 2023 IMO Shortlist N3
For positive integers $n$ and $k \geq 2$, define $E_k(n)$ as the greatest exponent $r$ such that $k^r$ divides $n!$. Prove that there are infinitely many $n$ such that $E_{10}(n) > E_9(n)$ and infinitely many $m$ such that $E_{10}(m) < E_9(m)$.
31 replies
youlost_thegame_1434
Jul 17, 2024
MR.1
an hour ago
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help!!!!!!!!!!!!
Cobedangiu   6
N an hour ago by MathsII-enjoy
help
6 replies
Cobedangiu
Mar 23, 2025
MathsII-enjoy
an hour ago
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Number Theory
VicKmath7   4
N an hour ago by AylyGayypow009
Source: Archimedes Junior 2014
Let $p$ prime and $m$ a positive integer. Determine all pairs $( p,m)$ satisfying the equation: $ p(p+m)+p=(m+1)^3$
4 replies
VicKmath7
Mar 17, 2020
AylyGayypow009
an hour ago
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JBMO Shortlist 2023 A4
Orestis_Lignos   6
N 2 hours ago by MR.1
Source: JBMO Shortlist 2023, A4
Let $a,b,c,d$ be positive real numbers with $abcd=1$. Prove that

$$\sqrt{\frac{a}{b+c+d^2+a^3}}+\sqrt{\frac{b}{c+d+a^2+b^3}}+\sqrt{\frac{c}{d+a+b^2+c^3}}+\sqrt{\frac{d}{a+b+c^2+d^3}} \leq 2$$
6 replies
Orestis_Lignos
Jun 28, 2024
MR.1
2 hours ago
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60 posts!(and a question )
kjhgyuio   1
N 2 hours ago by Pal702004
Finally 60 posts :D
1 reply
kjhgyuio
3 hours ago
Pal702004
2 hours ago
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|a^2-b^2-2abc|<2c implies abc EVEN!
tom-nowy   1
N 2 hours ago by Tkn
Source: Own
Prove that if integers $a, b$ and $c$ satisfy $\left| a^2-b^2-2abc \right| <2c $, then $abc$ is an even number.
1 reply
tom-nowy
May 3, 2025
Tkn
2 hours ago
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nice system of equations
outback   4
N Apr 23, 2025 by Raj_singh1432
Solve in positive numbers the system

$ x_1+\frac{1}{x_2}=4, x_2+\frac{1}{x_3}=1, x_3+\frac{1}{x_4}=4, ..., x_{99}+\frac{1}{x_{100}}=4, x_{100}+\frac{1}{x_1}=1$
4 replies
outback
Oct 8, 2008
Raj_singh1432
Apr 23, 2025
J
nice system of equations
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Reveal topic tags
inequalities
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outback
293 posts
outback
#1 h  Oct 8, 2008, 2:41 PM • 2 Y
Y by Adventure10, Mango247
Solve in positive numbers the system

$ x_1+\frac{1}{x_2}=4, x_2+\frac{1}{x_3}=1, x_3+\frac{1}{x_4}=4, ..., x_{99}+\frac{1}{x_{100}}=4, x_{100}+\frac{1}{x_1}=1$
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Bovy11
151 posts
Bovy11
#2 h  Oct 8, 2008, 7:14 PM • 3 Y
Y by Adventure10, Mango247, ehuseyinyigit
outback wrote:
Solve in positive numbers the system

$ x_1 + \frac {1}{x_2} = 4, x_2 + \frac {1}{x_3} = 1, x_3 + \frac {1}{x_4} = 4, ..., x_{99} + \frac {1}{x_{100}} = 4, x_{100} + \frac {1}{x_1} = 1$

solution
This post has been edited 2 times. Last edited by Bovy11, Oct 9, 2008, 1:33 PM
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t0rajir0u
12167 posts
t0rajir0u
#3 h  Oct 8, 2008, 7:37 PM • 2 Y
Y by Adventure10, Mango247
Was $ x_{100} + \frac{1}{x_1} = 1$ a typo?
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outback
293 posts
outback
#4 h  Oct 9, 2008, 12:32 AM • 2 Y
Y by Adventure10, Mango247
t0rajir0u wrote:
Was $ x_{100} + \frac {1}{x_1} = 1$ a typo?

Why do you think it is?
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Raj_singh1432
3 posts
Raj_singh1432
#5 h  Apr 23, 2025, 4:57 PM
Y by
t0rajir0u wrote:
Was $ x_{100} + \frac{1}{x_1} = 1$ a typo?

No
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