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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.

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Be sure to mark your calendars for the following upcoming events:
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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
Yesterday 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
J
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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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Triangles in dissections
Assassino9931   0
a few seconds ago
Source: RMM Shortlist 2024 C2
Fix an integer $n\geq 3$ and let $A_1A_2\ldots A_n$ be a convex polygon in the plane. Let $\mathcal{M}$ be the set of all midpoints $M_{i,j}$ of segments $A_iA_j$ where $i\neq j$. Assume that all of these midpoints are distinct, i.e. $\mathcal{M}$ consists of $\frac{n(n-1)}{2}$ elements. Dissect the polygon $M_{1,2}M_{2,3}\ldots M_{n,1}$ into triangles so that the following hold:

(1) The intersection of every two triangles (interior and boundary) is either empty or a common
vertex or a common side.
(2) The vertices of all triangles lie in M (not all points in M are necessarily used).
(3) Each side of every triangle is of the form $M_{i,j}M_{i,k}$ for some pairwise distinct indices $i,j,k$.

Prove that the total number of triangles in such a dissection is $3n-8$.
0 replies
Assassino9931
a few seconds ago
0 replies
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IMO Shortlist Problems
ABCD1728   1
N 3 minutes ago by mrtheory
Source: IMO official website
Where can I get the official solution for ISL before 2005? The official website only has solutions after 2006. Thanks :)
1 reply
ABCD1728
Today at 12:44 PM
mrtheory
3 minutes ago
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Process on scalar products and permutations
Assassino9931   0
5 minutes ago
Source: RMM Shortlist 2024 C1
Fix an integer $n\geq 2$. Consider $2n$ real numbers $a_1,\ldots,a_n$ and $b_1,\ldots, b_n$. Let $S$ be the set of all pairs $(x, y)$ of real numbers for which $M_i = a_ix + b_iy$, $i=1,2,\ldots,n$ are pairwise distinct. For every such pair sort the corresponding values $M_1, M_2, \ldots, M_n$ increasingly and let $M(i)$ be the $i$-th term in the list thus sorted. This denes an index permutation of $1,2,\ldots,n$. Let $N$ be the number of all such permutations, as the pairs run through all of $S$. In terms of $n$, determine the largest value $N$ may achieve over all possible choices of $a_1,\ldots,a_n,b_1,\ldots,b_n$.
0 replies
Assassino9931
5 minutes ago
0 replies
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Tangency geo
Assassino9931   0
9 minutes ago
Source: RMM Shortlist 2024 G1
Let $ABC$ be an acute triangle with $\angle ABC > 45^{\circ}$ and $\angle ACB > 45^{\circ}$. Let $M$ be the midpoint of the side $BC$. The circumcircle of triangle $ABM$ intersects the side $AC$ again at $X\neq A$ and the circumcircle of triangle $ACM$ intersects the side $AB$ again at $Y\neq A$. The point $P$ lies on the perpendicular bisector of the segment $BC$ so that the points $P$ and $A$ lie on the same side of $XY$ and $\angle XPY = 90^{\circ} + \angle BAC$. Prove that the circumcircles of triangles $BPY$ and $CPX$ are tangent.
0 replies
Assassino9931
9 minutes ago
0 replies
J
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Inequalities in real math research
Assassino9931   0
13 minutes ago
Source: RMM Shortlist 2024 A3
For a positive integer $n$ denote $F_n(x_1,x_2,\ldots,x_n) = 1 + x_1 + x_1x_2 + \cdots +x_1x_2\ldots x_n$. For any real numbers $x_1\geq x_2 \geq \ldots \geq x_k \geq 0$ prove that
\[ \prod_{i=1}^k F_i(x_{k-i+1},x_{k-i+2},\ldots,x_k) \geq \prod_{i=1}^k F_i(x_i,x_i,\ldots,x_i)\]
0 replies
Assassino9931
13 minutes ago
0 replies
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A folklore polynomial game
Assassino9931   0
15 minutes ago
Source: RMM Shortlist 2024 A1, also Bulgaria Regional Round 2016, Grade 12
Fix a positive integer $d$. Yael and Ziad play a game as follows, involving a monic polynomial of degree $2d$. With Yael going first, they take turns to choose a strictly positive real number as the value of one of the coecients of the polynomial. Once a coefficient is assigned a value, it cannot be chosen again later in the game. So the game
lasts for $2d$ rounds, until Ziad assigns the final coefficient. Yael wins if $P(x) = 0$ for some real
number $x$. Otherwise, Ziad wins. Decide who has the winning strategy.
0 replies
Assassino9931
15 minutes ago
0 replies
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Two circles, a tangent line and a parallel
Valentin Vornicu   104
N an hour ago by cubres
Source: IMO 2000, Problem 1, IMO Shortlist 2000, G2
Two circles $ G_1$ and $ G_2$ intersect at two points $ M$ and $ N$. Let $ AB$ be the line tangent to these circles at $ A$ and $ B$, respectively, so that $ M$ lies closer to $ AB$ than $ N$. Let $ CD$ be the line parallel to $ AB$ and passing through the point $ M$, with $ C$ on $ G_1$ and $ D$ on $ G_2$. Lines $ AC$ and $ BD$ meet at $ E$; lines $ AN$ and $ CD$ meet at $ P$; lines $ BN$ and $ CD$ meet at $ Q$. Show that $ EP = EQ$.
104 replies
Valentin Vornicu
Oct 24, 2005
cubres
an hour ago
J
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Austrian Regional MO 2025 P4
BR1F1SZ   2
N 2 hours ago by NumberzAndStuff
Source: Austrian Regional MO
Let $z$ be a positive integer that is not divisible by $8$. Furthermore, let $n \geqslant 2$ be a positive integer. Prove that none of the numbers of the form $z^n + z + 1$ is a square number.

(Walther Janous)
2 replies
BR1F1SZ
Apr 18, 2025
NumberzAndStuff
2 hours ago
J
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Austrian Regional MO 2025 P3
BR1F1SZ   1
N 2 hours ago by NumberzAndStuff
Source: Austrian Regional MO
There are $6$ different bus lines in a city, each stopping at exactly $5$ stations and running in both directions. Nevertheless, for every two different stations there is always a bus line connecting these two stations. Determine the maximum number of stations in this city.

(Karl Czakler)
1 reply
BR1F1SZ
Apr 18, 2025
NumberzAndStuff
2 hours ago
J
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Austrian Regional MO 2025 P2
BR1F1SZ   2
N 2 hours ago by NumberzAndStuff
Source: Austrian Regional MO
Let $\triangle{ABC}$ be an isosceles triangle with $AC = BC$ and circumcircle $\omega$. The line through $B$ perpendicular to $BC$ is denoted by $\ell$. Furthermore, let $M$ be any point on $\ell$. The circle $\gamma$ with center $M$ and radius $BM$ intersects $AB$ once more at point $P$ and the circumcircle $\omega$ once more at point $Q$. Prove that the points $P,Q$ and $C$ lie on a straight line.

(Karl Czakler)
2 replies
BR1F1SZ
Apr 18, 2025
NumberzAndStuff
2 hours ago
J
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Austrian Regional MO 2025 P1
BR1F1SZ   2
N 2 hours ago by NumberzAndStuff
Source: Austrian Regional MO
Let $n \geqslant 3$ be a positive integer. Furthermore, let $x_1, x_2,\ldots, x_n \in [0, 2]$ be real numbers subject to $x_1 + x_2 +\cdots + x_n = 5$. Prove the inequality$$x_1^2 + x_2^2 + \cdots + x_n^2 \leqslant 9.$$When does equality hold?

(Walther Janous)
2 replies
BR1F1SZ
Apr 18, 2025
NumberzAndStuff
2 hours ago
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positive integers forming a perfect square
cielblue   0
3 hours ago
Find all positive integers $n$ such that $2^n-n^2+1$ is a perfect square.
0 replies
cielblue
3 hours ago
0 replies
J
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Function equation
LeDuonggg   6
N 3 hours ago by MathLuis
Find all functions $f: \mathbb{R^+} \rightarrow \mathbb{R^+}$ , such that for all $x,y>0$:
\[ f(x+f(y))=\dfrac{f(x)}{1+f(xy)}\]
6 replies
LeDuonggg
Yesterday at 2:59 PM
MathLuis
3 hours ago
J
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A nice and easy gem off of StackExchange
NamelyOrange   0
3 hours ago
Source: https://math.stackexchange.com/questions/3818796/
Define $S$ as the set of all numbers of the form $2^i5^j$ for some nonnegative $i$ and $j$. Find (with proof) all pairs $(m,n)$ such that $m,n\in S$ and $m-n=1$.
0 replies
NamelyOrange
3 hours ago
0 replies
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J
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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
J
a combinatorial geometry problem
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Reveal topic tags
combinatorics
combinatorial geometry
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.
Z K Y
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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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