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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
1 viewing
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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IMO Shortlist 2009 - Problem C5
April   38
N an hour ago by MathematicalArceus
Five identical empty buckets of $2$-liter capacity stand at the vertices of a regular pentagon. Cinderella and her wicked Stepmother go through a sequence of rounds: At the beginning of every round, the Stepmother takes one liter of water from the nearby river and distributes it arbitrarily over the five buckets. Then Cinderella chooses a pair of neighbouring buckets, empties them to the river and puts them back. Then the next round begins. The Stepmother goal's is to make one of these buckets overflow. Cinderella's goal is to prevent this. Can the wicked Stepmother enforce a bucket overflow?

Proposed by Gerhard Woeginger, Netherlands
38 replies
April
Jul 5, 2010
MathematicalArceus
an hour ago
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Inspired by Austria 2025
sqing   4
N an hour ago by Tkn
Source: Own
Let $ a,b\geq 0 ,a,b\neq 1$ and $ a^2+b^2=1. $ Prove that$$ (a + b ) \left( \frac{a}{(b -1)^2} + \frac{b}{(a - 1)^2} \right) \geq 12+8\sqrt 2$$
4 replies
sqing
Today at 2:01 AM
Tkn
an hour ago
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Erasing the difference of two numbers
BR1F1SZ   1
N 2 hours ago by BR1F1SZ
Source: Austria National MO Part 1 Problem 3
Consider the following game for a positive integer $n$. Initially, the numbers $1, 2, \ldots, n$ are written on a board. In each move, two numbers are selected such that their difference is also present on the board. This difference is then erased from the board. (For example, if the numbers $3,6,11$ and $17$ are on the board, then $3$ can be erased as $6 - 3=3$, or $6$ as $17 - 11=6$, or $11$ as $17 - 6=11$.)

For which values of $n$ is it possible to end with only one number remaining on the board?

(Michael Reitmeir)
1 reply
BR1F1SZ
Yesterday at 9:48 PM
BR1F1SZ
2 hours ago
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3-var inequality
sqing   1
N 2 hours ago by Natrium
Source: Own
Let $ a,b,c\geq 0 ,a+b+c =1. $ Prove that
$$\frac{ab}{2c+1} +\frac{bc}{2a+1} +\frac{ca}{2b+1}+\frac{27}{20} abc\leq \frac{1}{4} $$
1 reply
sqing
May 3, 2025
Natrium
2 hours ago
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mathemetics
Pangbowen   0
2 hours ago
Let a,b,c≥0 and a+b+c=7. Prove that : a/b+b/c+c/a+abc≥ab+bc+ca-2
0 replies
Pangbowen
2 hours ago
0 replies
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Property of a function
Ritangshu   1
N 3 hours ago by Natrium
Let \( f(x, y) = xy \), where \( x \geq 0 \) and \( y \geq 0 \).
Prove that the function \( f \) satisfies the following property:

\[ f\left( \lambda x + (1 - \lambda)x',\; \lambda y + (1 - \lambda)y' \right) > \min\{f(x, y),\; f(x', y')\} \]
for all \( (x, y) \ne (x', y') \) and for all \( \lambda \in (0, 1) \).

1 reply
Ritangshu
May 3, 2025
Natrium
3 hours ago
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max value
Bet667   2
N 3 hours ago by Natrium
Let $a,b$ be a real numbers such that $a^2+ab+b^2\ge a^3+b^3.$Then find maximum value of $a+b$
2 replies
Bet667
4 hours ago
Natrium
3 hours ago
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Geometry
gggzul   2
N 3 hours ago by gggzul
In trapezoid $ABCD$ segments $AB$ and $CD$ are parallel. Angle bisectors of $\angle A$ and $\angle C$ meet at $P$. Angle bisectors of $\angle B$ and $\angle D$ meet at $Q$. Prove that $ABPQ$ is cyclic
2 replies
gggzul
4 hours ago
gggzul
3 hours ago
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thank you !
Piwbo   2
N 3 hours ago by Piwbo
Given positive integers $a,b$ such that $a$ is even , $b$ is odd and $ab(a+b)^{2023}$ is divisible by $a^{2024}+b^{2024}$ .Prove that there exists a prime number $p$ such that $a^{2024}+b^{2024}$ is divisible by $p^{2025}$
2 replies
+1 w
Piwbo
4 hours ago
Piwbo
3 hours ago
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Inequality involving square root cube root and 8th root
bamboozled   2
N 3 hours ago by bamboozled
If $a,b,c,d,e,f,g,h,k\in R^+$ and $a+b+c=d+e+f=g+h+k=8$, then find the maximum value of $\sqrt{ad^3 g^4} +\sqrt[3]{be^3 h^4} + \sqrt[8]{cf^3 k^4}$
2 replies
bamboozled
Today at 4:46 AM
bamboozled
3 hours ago
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old problem from an open contest
Darealzolt   1
N 4 hours ago by alexheinis
Given that $a, b \in \mathbb{R}$ satisfy
\[ a + \frac{1}{a + 2015} = b - 4030 + \frac{1}{b - 2015} \]and $|a - b| > 5000$. Determine the value of
\[ \frac{ab}{2015} - a + b. \]
1 reply
Darealzolt
Yesterday at 1:41 AM
alexheinis
4 hours ago
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A Collection of Good Problems from my end
SomeonecoolLovesMaths   12
N 4 hours ago by SomeonecoolLovesMaths
This is a collection of good problems and my respective attempts to solve them. I would like to encourage everyone to post their solutions to these problems, if any. This will not only help others verify theirs but also perhaps bring forward a different approach to the problem. I will constantly try to update the pool of questions.

The difficulty level of these questions vary from AMC 10 to AIME. (Although the main pool of questions were prepared as a mock test for IOQM over the years)

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5
12 replies
SomeonecoolLovesMaths
May 4, 2025
SomeonecoolLovesMaths
4 hours ago
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Geometry Proof
strongstephen   2
N 5 hours ago by greenturtle3141
Proof that choosing four distinct points at random has an equal probability of getting a convex quadrilateral vs a concave one.
not cohesive proof alert!

NOTE: By choosing four distinct points, that means no three points lie on the same line on the Gaussian Plane.

Start by picking three of the four points. Next, graph the regions where the fourth point would make the quadrilateral convex or concave. In diagram 1 below, you can see the regions where the fourth point would be convex or concave. Of course, there is the centre region (the shaded triangle), but in an infinite plane, the probability the fourth point ends up in the finite region approaches 0.

Next, I want to prove to you the area of convex/concave, or rather, the probability a point ends up in each area, is the same. Referring to the second diagram, you can flip each concave region over the line perpendicular to the angle bisector of which the region is defined. (Just look at it and you'll get what it means.) Now, each concave region has an almost perfect 1:1 probability correspondence to another convex region. The only difference is the finite region (the triangle, shaded). Again, however, the actual significance (probability) of this approaches 0.

If I call each of the convex region's probability P(a), P(c), and P(e) and the concave ones P(b), P(d), P(f), assuming areas a and b are on opposite sides (same with c and d, e and f) you can get:
P(a) = P(b)
P(c) = P(d)
P(e) = P(f)

and P(a) + P(c) + P(e) = P(convex)
and P(b) + P(d) + P(f) = P(concave)

therefore:
P(convex) = P(concave)
2 replies
strongstephen
Today at 4:54 AM
greenturtle3141
5 hours ago
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BABBAGE'S THEOREM EXTENSION
Mathgloggers   4
N 5 hours ago by Mathgloggers
A few days ago I came across. this interesting result is someone interested in proving this.

$\boxed{\sum_{k=1}^{p-1} \frac{1}{k} \equiv \sum_{k=p+1}^{2p-1} \frac{1}{k} \equiv \sum_{k=2p+1}^{3p-1}\frac{1}{k} \equiv.....\sum_{k=p(p-1)+1}^{p^2-1}\frac{1}{k} \equiv 0(mod p^2)}$
4 replies
Mathgloggers
Apr 29, 2025
Mathgloggers
5 hours ago
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Base 9 Repunits = Triangular Numbers
MCrawford   4
N May 9, 2006 by Iron Fist
Prove that every positive integer that can be written using only 1 as each digit in base 9 is a triangular number:
\begin{eqnarray*} 1_9 &=& 1 \\ 11_9 &=& 1 + 2 + 3 + 4 \\ 111_9 &=& 1 + 2 + \cdots + 13 \\ 1111_9 &=& 1 + 2 + \cdots + 40 \\ & \vdots & \end{eqnarray*}
4 replies
MCrawford
May 8, 2006
Iron Fist
May 9, 2006
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Base 9 Repunits = Triangular Numbers
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MCrawford
6325 posts
MCrawford
#1 h  May 8, 2006, 9:24 PM • 1 Y
Y by Adventure10
Prove that every positive integer that can be written using only 1 as each digit in base 9 is a triangular number:
\begin{eqnarray*} 1_9 &=& 1 \\ 11_9 &=& 1 + 2 + 3 + 4 \\ 111_9 &=& 1 + 2 + \cdots + 13 \\ 1111_9 &=& 1 + 2 + \cdots + 40 \\ & \vdots & \end{eqnarray*}
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chess64
4794 posts
chess64
#2 h  May 8, 2006, 10:00 PM • 3 Y
Y by Adventure10, Adventure10, Mango247
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towersfreak2006
3012 posts
towersfreak2006
#3 h  May 8, 2006, 10:06 PM • 2 Y
Y by Adventure10, Mango247
MCrawford wrote:
Prove that every positive integer that can be written using only 1 as each digit in base 9 is a triangular number:
\begin{eqnarray*} 1_9 &=& 1 \\ 11_9 &=& 1 + 2 + 3 + 4 \\ 111_9 &=& 1 + 2 + \cdots + 13 \\ 1111_9 &=& 1 + 2 + \cdots + 40 \\ & \vdots & \end{eqnarray*}
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maniacman384
282 posts
maniacman384
#4 h  May 9, 2006, 12:16 AM • 2 Y
Y by Adventure10, Mango247
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Iron Fist
98 posts
Iron Fist
#5 h  May 9, 2006, 12:20 AM • 2 Y
Y by Adventure10, Mango247
maniacman384 wrote:
I just noticed something, any number that starts with a 1 and has the rest of its digits as zeros is a triangular number.

Last time I checked, 1000 was a triangle number, not in base 10 nor base 9...
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