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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
Estimate on number of progressions
Assassino9931   0
20 minutes ago
Source: RMM Shortlist 2024 C4
Let $n$ be a positive integer. For a set $S$ of $n$ real numbers, let $f(S)$ denote the number of increasing arithmetic progressions of length at least two all of whose terms are in $S$. Prove that, if $S$ is a set of $n$ real numbers, then
\[ f(S) \leq \frac{n^2}{4} + f(\{1,2,\ldots,n\})\]
0 replies
1 viewing
Assassino9931
20 minutes ago
0 replies
Popular children at camp with algebra and geometry
Assassino9931   0
23 minutes ago
Source: RMM Shortlist 2024 C3
Fix an odd integer $n\geq 3$. At a maths camp, there are $n^2$ children, each of whom selects
either algebra or geometry as their favourite topic. At lunch, they sit at $n$ tables, with $n$ children
on each table, and start talking about mathematics. A child is said to be popular if their favourite
topic has a majority at their table. For dinner, the students again sit at $n$ tables, with $n$ children
on each table, such that no two children share a table at both lunch and dinner. Determine the
minimal number of young mathematicians who are popular at both mealtimes. (The minimum is across all sets of topic preferences and seating arrangements.)
0 replies
Assassino9931
23 minutes ago
0 replies
Triangles in dissections
Assassino9931   0
25 minutes 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
25 minutes ago
0 replies
IMO Shortlist Problems
ABCD1728   1
N 27 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
27 minutes ago
Process on scalar products and permutations
Assassino9931   0
29 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
1 viewing
Assassino9931
29 minutes ago
0 replies
Tangency geo
Assassino9931   0
33 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
+1 w
Assassino9931
33 minutes ago
0 replies
Inequalities in real math research
Assassino9931   0
37 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
1 viewing
Assassino9931
37 minutes ago
0 replies
A folklore polynomial game
Assassino9931   0
39 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
39 minutes ago
0 replies
Two circles, a tangent line and a parallel
Valentin Vornicu   104
N 2 hours 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
2 hours ago
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
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
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
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
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
Find points with sames integer distances as given
nAalniaOMliO   2
N Apr 29, 2025 by nAalniaOMliO
Source: Belarus TST 2024
Points $A_1, \ldots A_n$ with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points $B_1, \ldots ,B_n$ with integer coordinates such that $A_iA_j=B_iB_j$ for every pair $1 \leq i \leq j \leq n$
N. Sheshko, D. Zmiaikou
2 replies
nAalniaOMliO
Jul 17, 2024
nAalniaOMliO
Apr 29, 2025
Find points with sames integer distances as given
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G H BBookmark kLocked kLocked NReply
Source: Belarus TST 2024
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nAalniaOMliO
296 posts
#1
Y by
Points $A_1, \ldots A_n$ with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points $B_1, \ldots ,B_n$ with integer coordinates such that $A_iA_j=B_iB_j$ for every pair $1 \leq i \leq j \leq n$
N. Sheshko, D. Zmiaikou
This post has been edited 1 time. Last edited by nAalniaOMliO, Oct 31, 2024, 10:12 AM
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Rohit-2006
238 posts
#2
Y by
If I can move only one point to a lattice point then all the others points must be in lattice points why? Because distance between a rational coordinate points($\mathbb{Q-Z}$) to a lattice point can never be an integer. So choose $A_1$ and say it's rational coordinates are (p,q) then move the point to ([p],[q]) where [•] denote the box function and we are done with all points with integral coordinates.

Remark:
Since distance between the points are all integers so we can move the system of points with rational coordinates to integer coordinates.
This post has been edited 2 times. Last edited by Rohit-2006, Apr 29, 2025, 7:27 AM
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nAalniaOMliO
296 posts
#3
Y by
The distance between points $(0,0)$ and $(\frac{3}{5},\frac{4}{5})$ is clearly 1, where the first point is a lattice point and the second one has rational coordinates.
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