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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
Thursday 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)!

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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
Thursday at 11:16 PM
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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Solution needed ASAP
UglyScientist   0
a minute ago
$ABC$ is acute triangle. $H$ is orthocenter, $M$ is midpoint, $L$ is the midpoint of smaller arc $BC$. Point $K$ is on $AH$ such that, $MK$ is perpendicular to $AL$. Prove that: $HMLK$ is paralelogram

0 replies
UglyScientist
a minute ago
0 replies
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Product is a perfect square( very easy)
Nuran2010   0
2 minutes ago
Source: Azerbaijan Junior National Olympiad 2021 P1
At least how many numbers must be deleted from the product $1 \times 2 \times \dots \times 22 \times 23$ in order to make it a perfect square?
0 replies
Nuran2010
2 minutes ago
0 replies
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Find the product
sqing   2
N 5 minutes ago by sqing
Source: Ecrin_eren
The roots of $ x^3 - 2x^2 - 11x + k=0 $ are $r_1, r_2, r_3 $ and $ r_1+2 r_2+3 r_3= 0.$ Find the product of all possible values of $ k .$
2 replies
sqing
3 hours ago
sqing
5 minutes ago
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Combinatorial Sum
P162008   1
N 23 minutes ago by Mathzeus1024
Evaluate $\sum_{n=0}^{\infty} \frac{2^n + 1}{(2n + 1) \binom{2n}{n}}$
1 reply
P162008
Apr 24, 2025
Mathzeus1024
23 minutes ago
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2xy is perfect square and x^2 + y^2 is prime
parmenides51   3
N 39 minutes ago by justaguy_69
Source: Dutch NMO 2020 p4
Determine all pairs of integers $(x, y)$ such that $2xy$ is a perfect square and $x^2 + y^2$ is a prime number.
3 replies
parmenides51
Nov 23, 2020
justaguy_69
39 minutes ago
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Mmo 9-10 graders P5
Bet667   10
N 39 minutes ago by User141208
Let $a,b,c,d$ be real numbers less than 2.Then prove that $\frac{a^3}{b^2+4}+\frac{b^3}{c^2+4}+\frac{c^3}{d^2+4}+\frac{d^3}{a^2+4}\le4$
10 replies
Bet667
Apr 3, 2025
User141208
39 minutes ago
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IMO Genre Predictions
ohiorizzler1434   13
N 44 minutes ago by nunoarala
Everybody, with IMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
13 replies
ohiorizzler1434
Today at 6:51 AM
nunoarala
44 minutes ago
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Number Theory Chain!
JetFire008   60
N an hour ago by whwlqkd
I will post a question and someone has to answer it. Then they have to post a question and someone else will answer it and so on. We can only post questions related to Number Theory and each problem should be more difficult than the previous. Let's start!

Question 1
60 replies
JetFire008
Apr 7, 2025
whwlqkd
an hour ago
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3 right-angled triangle area
NicoN9   1
N an hour ago by Mathzeus1024
Source: Japan Junior MO Preliminary 2020 P1
Right angled triangle $ABC$, and a square are drawn as shown below. Three numbers written below implies each of the area of shaded small right angled triangle. Find the value of $AB/AC$.

IMAGE
1 reply
NicoN9
Yesterday at 6:08 AM
Mathzeus1024
an hour ago
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two sequences of positive integers and inequalities
rmtf1111   50
N an hour ago by math-olympiad-clown
Source: EGMO 2019 P5
Let $n\ge 2$ be an integer, and let $a_1, a_2, \cdots , a_n$ be positive integers. Show that there exist positive integers $b_1, b_2, \cdots, b_n$ satisfying the following three conditions:

$\text{(A)} \ a_i\le b_i$ for $i=1, 2, \cdots , n;$

$\text{(B)} \ $ the remainders of $b_1, b_2, \cdots, b_n$ on division by $n$ are pairwise different; and

$\text{(C)} \ $ $b_1+b_2+\cdots b_n \le n\left(\frac{n-1}{2}+\left\lfloor \frac{a_1+a_2+\cdots a_n}{n}\right \rfloor \right)$

(Here, $\lfloor x \rfloor$ denotes the integer part of real number $x$, that is, the largest integer that does not exceed $x$.)
50 replies
rmtf1111
Apr 10, 2019
math-olympiad-clown
an hour ago
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inequalities
Tamako22   0
an hour ago
let $a,b,c> 1,\dfrac{1}{1+a}+\dfrac{1}{1+b}+\dfrac{1}{1+c}=1.$
prove that$$\sqrt{a}+\sqrt{b}+\sqrt{c}\ge \dfrac{2}{\sqrt{a}}+\dfrac{2}{\sqrt{b}}+\dfrac{2}{\sqrt{c}}$$
0 replies
Tamako22
an hour ago
0 replies
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Problem 6
SlovEcience   2
N an hour ago by mashumaro
Given two points A and B on the unit circle. The tangents to the circle at A and B intersect at point P. Then:
\[ p = \frac{2ab}{a + b} \], \[ p, a, b \in \mathbb{C} \]
2 replies
SlovEcience
4 hours ago
mashumaro
an hour ago
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A coincidence about triangles with common incenter
flower417477   3
N an hour ago by mashumaro
$\triangle ABC,\triangle ADE$ have the same incenter $I$.Prove that $BCDE$ is concyclic iff $BC,DE,AI$ is concurrent
3 replies
flower417477
Apr 30, 2025
mashumaro
an hour ago
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this geo is scarier than the omega variant
AwesomeYRY   11
N an hour ago by LuminousWolverine
Source: TSTST 2021/6
Triangles $ABC$ and $DEF$ share circumcircle $\Omega$ and incircle $\omega$ so that points $A,F,B,D,C,$ and $E$ occur in this order along $\Omega$. Let $\Delta_A$ be the triangle formed by lines $AB,AC,$ and $EF,$ and define triangles $\Delta_B, \Delta_C, \ldots, \Delta_F$ similarly. Furthermore, let $\Omega_A$ and $\omega_A$ be the circumcircle and incircle of triangle $\Delta_A$, respectively, and define circles $\Omega_B, \omega_B, \ldots, \Omega_F, \omega_F$ similarly.

(a) Prove that the two common external tangents to circles $\Omega_A$ and $\Omega_D$ and the two common external tangents to $\omega_A$ and $\omega_D$ are either concurrent or pairwise parallel.

(b) Suppose that these four lines meet at point $T_A$, and define points $T_B$ and $T_C$ similarly. Prove that points $T_A,T_B$, and $T_C$ are collinear.

Nikolai Beluhov
11 replies
AwesomeYRY
Dec 13, 2021
LuminousWolverine
an hour ago
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a combinatorial geometry problem
xyz123456   5
N Apr 18, 2025 by Fishheadtailbody
$A_i\left(x_i{,}y_i\right){,}0\le x_i{,}y_i\le 1{,}1\le i\le 6.prove\ that:\exists \ 1\le i<j\le 6\ {,}\left|A_iA_j\right|\le \frac{\sqrt{13}}{6}$
5 replies
xyz123456
Mar 3, 2025
Fishheadtailbody
Apr 18, 2025
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a combinatorial geometry problem
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Reveal topic tags
combinatorics
combinatorial geometry
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xyz123456
48 posts
xyz123456
#1 h  Mar 3, 2025, 12:41 PM
Y by
$A_i\left(x_i{,}y_i\right){,}0\le x_i{,}y_i\le 1{,}1\le i\le 6.prove\ that:\exists \ 1\le i<j\le 6\ {,}\left|A_iA_j\right|\le \frac{\sqrt{13}}{6}$
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kiyoras_2001
678 posts
kiyoras_2001
#2 h  Mar 3, 2025, 3:07 PM
Y by
I rushed...
This post has been edited 3 times. Last edited by kiyoras_2001, Mar 3, 2025, 5:36 PM
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xyz123456
48 posts
xyz123456
#5 h  Mar 4, 2025, 12:16 PM
Y by
bump this
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Math-Problem-Solving
66 posts
Math-Problem-Solving
#6 h  Apr 1, 2025, 5:06 AM
Y by
xyz123456 wrote:
bump this

Do you have the solution? Then please post.
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xyz123456
48 posts
xyz123456
#7 h  Apr 18, 2025, 12:34 PM
Y by
Math-Problem-Solving wrote:
xyz123456 wrote:
bump this

Do you have the solution? Then please post.

I don't have the solution.
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Fishheadtailbody
7 posts
Fishheadtailbody
#8 h  Apr 18, 2025, 1:45 PM • 1 Y
Y by kiyoras_2001
I can sense that \frac{\sqrt{13}}{6} = \sqrt{(\frac{1}{2})^2+(\frac{1}{3})^2}.

For example, by setting the unit square into 2 by 3 (6 rectangles), if any of the two points are lying in the same smaller rectangle, we are done.
Sadly, pigeon principle does gives anything here so there should be a better way to count.
Click to reveal hidden text

Well an archive here said that n = 6 was proved by Graham, but it is not a simple one at least in 1991.
https://archive.org/details/unsolvedproblems0000crof/page/108/mode/2up
Click to reveal hidden text
This post has been edited 1 time. Last edited by Fishheadtailbody, Apr 18, 2025, 4:09 PM
Reason: Added the archive.
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