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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.

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.
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[*]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
May 1, 2025
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
inequality thing
BinariouslyRandom   2
N 19 minutes ago by BinariouslyRandom
Source: Philippine MO 2025 P5
Find the largest real constant $k$ for which the inequality \[ (a^2+3)(b^2+3)(c^2+3)(d^2+3) + k(a-1)(b-1)(c-1)(d-1) \ge 0 \]holds for all real numbers $a$, $b$, $c$, and $d$.

answer
2 replies
1 viewing
BinariouslyRandom
2 hours ago
BinariouslyRandom
19 minutes ago
Geometry Concurrence
KHOMNYO2   0
35 minutes ago
Given triangle $XYZ$ such that $XY \neq XZ$. Let excircle-$X$ be tangent with $YZ, ZX, XY$ at points $U, V, W$ respectively. Let $R$ and $S$ be points on the segment $XZ, XY$ respectively such that $RS$ is parallel to $YZ$. Lastly, let $\gamma$ be the circle that is externally tangent with the excircle-$X$ on point $T$. Prove that $VW, UT$, and $RS$ concur at a point.
0 replies
KHOMNYO2
35 minutes ago
0 replies
inequality
mathematical-forest   1
N 35 minutes ago by lbh_qys
Positive real numbers $x_{1} ,x_{2} \cdots ,x_{n}$,satisfied $\sum_{i=1}^{n}x_{i} =1$
Proof:$$\sum_{i=1}^{n} \frac{\min  \left \{  x_{i-1},x_{i}\right \}\max \left \{  x_{i},x_{i+1}\right \}  }{x_{i}} \le 1$$
1 reply
mathematical-forest
an hour ago
lbh_qys
35 minutes ago
Some number theory
EeEeRUT   5
N an hour ago by shafikbara48593762
Source: Thailand MO 2025 P9
Let $p$ be an odd prime and $S = \{1,2,3,\dots, p\}$
Assume that $U: S \rightarrow S$ is a bijection and $B$ is an integer such that $$B\cdot U(U(a)) - a \: \text{ is a multiple of} \: p \: \text{for all} \: a \in S$$Show that $B^{\frac{p-1}{2}} -1$ is a multiple of $p$.
5 replies
EeEeRUT
May 14, 2025
shafikbara48593762
an hour ago
Sharygin 2025 CR P7
Gengar_in_Galar   5
N an hour ago by Blackbeam999
Source: Sharygin 2025
Let $I$, $I_{a}$ be the incenter and the $A$-excenter of a triangle $ABC$; $E$, $F$ be the touching points of the incircle with $AC$, $AB$ respectively; $G$ be the common point of $BE$ and $CF$. The perpendicular to $BC$ from $G$ meets $AI$ at point $J$. Prove that $E$, $F$, $J$, $I_{a}$ are concyclic.
Proposed by:Y.Shcherbatov
5 replies
Gengar_in_Galar
Mar 10, 2025
Blackbeam999
an hour ago
60^o angle wanted, equilateral on a square
parmenides51   5
N an hour ago by sunken rock
Source: 2019 Austrian Mathematical Olympiad Junior Regional Competition , Problem 2
A square $ABCD$ is given. Over the side $BC$ draw an equilateral triangle $BCS$ on the outside. The midpoint of the segment $AS$ is $N$ and the midpoint of the side $CD$ is $H$. Prove that $\angle NHC = 60^o$.
.
(Karl Czakler)
5 replies
parmenides51
Dec 18, 2020
sunken rock
an hour ago
Random walk
EthanWYX2009   0
an hour ago
As shown in the graph, an ant starts from $4$ and walks randomly. The probability of any point reaching all adjacent points is equal. Find the probability of the ant reaching $1$ without passing through $6.$
0 replies
EthanWYX2009
an hour ago
0 replies
Lemma on tangency involving a parallelogram with orthocenter
Gimbrint   0
2 hours ago
Source: Own
Let $ABC$ be an acute triangle ($AB<BC$) with circumcircle $\omega$ and orthocenter $H$. Let $M$ be the midpoint of $AC$. Line $BH$ intersects $\omega$ again at $L\neq B$, and line $ML$ intersects $\omega$ again at $P\neq L$. Points $D$ and $E$ lie on $AB$ and $BC$ respectively, such that $BEHD$ is a parallelogram.

Prove that $BP$ is tangent to the circumcircle of triangle $BDE$.
0 replies
Gimbrint
2 hours ago
0 replies
Consecutive squares are floors
ICE_CNME_4   10
N 2 hours ago by JARP091

Determine how many positive integers \( n \) have the property that both
\[
\left\lfloor \sqrt{2n - 1} \right\rfloor \quad \text{and} \quad \left\lfloor \sqrt{3n + 2} \right\rfloor
\]are consecutive perfect squares.
10 replies
ICE_CNME_4
Yesterday at 1:50 PM
JARP091
2 hours ago
Sharygin 2025 CR P10
Gengar_in_Galar   2
N 2 hours ago by Kappa_Beta_725
Source: Sharygin 2025
An acute-angled triangle with one side equal to the altitude from the opposite vertex is cut from paper. Construct a point inside this triangle such that the square of the distance from it to one of the vertices equals the sum of the squares of distances to to the remaining two vertices. No instruments are available, it is allowed only to fold the paper and to mark the common points of folding lines.
Proposed by: M.Evdokimov
2 replies
Gengar_in_Galar
Mar 10, 2025
Kappa_Beta_725
2 hours ago
Sharygin 2025 CR P12
Gengar_in_Galar   8
N 2 hours ago by Kappa_Beta_725
Source: Sharygin 2025
Circles $\omega_{1}$ and $\omega_{2}$ are given. Let $M$ be the midpoint of the segment joining their centers, $X$, $Y$ be arbitrary points on $\omega_{1}$, $\omega_{2}$ respectively such that $MX=MY$. Find the locus of the midpoints of segments $XY$.
Proposed by: L Shatunov
8 replies
Gengar_in_Galar
Mar 10, 2025
Kappa_Beta_725
2 hours ago
Sharygin 2025 CR P17
Gengar_in_Galar   6
N 3 hours ago by Kappa_Beta_725
Source: Sharygin 2025
Let $O$, $I$ be the circumcenter and the incenter of an acute-angled scalene triangle $ABC$; $D$, $E$, $F$ be the touching points of its excircle with the side $BC$ and the extensions of $AC$, $AB$ respectively. Prove that if the orthocenter of the triangle $DEF$ lies on the circumcircle of $ABC$, then it is symmetric to the midpoint of the arc $BC$ with respect to $OI$.
Proposed by: P.Puchkov,E.Utkin
6 replies
Gengar_in_Galar
Mar 10, 2025
Kappa_Beta_725
3 hours ago
Sharygin 2025 CR P21
Gengar_in_Galar   4
N 3 hours ago by Kappa_Beta_725
Source: Sharygin 2025
Let $P$ be a point inside a quadrilateral $ABCD$ such that $\angle APB+\angle CPD=180^{\circ}$. Points $P_{a}$, $P_{b}$, $P_{c},$ $P_{d}$ are isogonally conjugated to $P$ with respect to the triangles $BCD$, $CDA$, $DAB$, $ABC$ respectively. Prove that the diagonals of the quadrilaterals $ABCD$ and $P_{a}P_{b}P_{c}P_{d}$ concur.
Proposed by: G.Galyapin
4 replies
Gengar_in_Galar
Mar 10, 2025
Kappa_Beta_725
3 hours ago
Sharygin 2025 CR P18
Gengar_in_Galar   6
N 3 hours ago by Kappa_Beta_725
Source: Sharygin 2025
Let $ABCD$ be a quadrilateral such that the excircles $\omega_{1}$ and $\omega_{2}$ of triangles $ABC$ and $BCD$ touching their sides $AB$ and $BD$ respectively touch the extension of $BC$ at the same point $P$. The segment $AD$ meets $\omega_{2}$ at point $Q$, and the line $AD$ meets $\omega_{1}$ at $R$ and $S$. Prove that one of angles $RPQ$ and $SPQ$ is right
Proposed by: I.Kukharchuk
6 replies
Gengar_in_Galar
Mar 10, 2025
Kappa_Beta_725
3 hours ago
Obscure Set Problem
oVlad   1
N Apr 12, 2025 by kokcio
Source: Romania Junior TST 2025 Day 1 P5
Let $n\geqslant 3$ be a positive integer and $\mathcal F$ be a family of at most $n$ distinct subsets of the set $\{1,2,\ldots,n\}$ with the following property: we can consider $n$ distinct points in the plane, labelled $1,2,\ldots,n$ and draw segments connecting these points such that points $i$ and $j$ are connected if and only if $i{}$ belongs to $j$ subsets in $\mathcal F$ for any $i\neq j.$ Determine the maximal value that the sum of the cardinalities of the subsets in $\mathcal{F}$ can take.
1 reply
oVlad
Apr 12, 2025
kokcio
Apr 12, 2025
Obscure Set Problem
G H J
G H BBookmark kLocked kLocked NReply
Source: Romania Junior TST 2025 Day 1 P5
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oVlad
1746 posts
#1
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Let $n\geqslant 3$ be a positive integer and $\mathcal F$ be a family of at most $n$ distinct subsets of the set $\{1,2,\ldots,n\}$ with the following property: we can consider $n$ distinct points in the plane, labelled $1,2,\ldots,n$ and draw segments connecting these points such that points $i$ and $j$ are connected if and only if $i{}$ belongs to $j$ subsets in $\mathcal F$ for any $i\neq j.$ Determine the maximal value that the sum of the cardinalities of the subsets in $\mathcal{F}$ can take.
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kokcio
70 posts
#2
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We know that $i$ is in $j$ subsets if and only if $j$ is in $i$ subsets, because of there is segment between $i,j$ if and only if there is segment between $j,i$.
Let $a_k$ be a number of subsets, which contain $k$. Assume that $a_k=a_l=m$ for some integers $k,l,m$ and $m>0$. Then, we would have to have that $k=a_m=l$ from our first sentence. This means that all $a_i$, which are greater than $0$ have to pairwise different. This means that $a_1+a_2+...+a_n\leq1+2+...+n=\frac{n(n+1)}{2}$, but sum of cardinalities of subsets in $\mathcal{F}$ is equal to $a_1+a_2+...+a_n$, so we have upper bound. We can make $a_1+a_2+...+a_n=\frac{n(n+1)}{2}$ for subsets $\{1\},\{1,2\},...,\{1,2,...,n\}$ (because $a_i=n+1-i$), so maximal sum of cardinalities of subsets in $\mathcal{F}$ is $\frac{n(n+1)}{2}$.
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