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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:
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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
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
gcd nt from switzerland
AshAuktober   0
6 minutes ago
Source: Swiss 2025 Second Round
Let $a, b$ be positive integers. Prove that the expression
\[\frac{\gcd(a+b,ab)}{\gcd(a,b)}\]is always a positive integer, and determine all possible values it can take.
0 replies
AshAuktober
6 minutes ago
0 replies
set construction nt
top1vien   2
N 10 minutes ago by top1vien
Is there a set of 2025 positive integers $S$ that satisfies: for all different $a,b,c,d\in S$, we have $\gcd(ab+1000,cd+1000)=1$?
2 replies
1 viewing
top1vien
Yesterday at 10:04 AM
top1vien
10 minutes ago
strange geometry problem
Zavyk09   0
13 minutes ago
Source: own
Let $ABC$ be a triangle with circumcenter $O$ and internal bisector $AD$. Let $AD$ cuts $(O)$ again at $M$ and $MO$ cuts $(O)$ again at $N$. Point $L$ lie on $AD$ such that $(AD, LM) = -1$. The line pass through $L$ and perpendicular to $AD$ intersects $NC, NB$ at $P, Q$ respectively. Let circumcircle of $\triangle NPQ$ cuts $(O)$ at $G \ne N$. Prove that $\angle AGD = 90^{\circ}$.
0 replies
Zavyk09
13 minutes ago
0 replies
A sharp one with 3 var (3)
mihaig   3
N 16 minutes ago by JARP091
Source: Own
Let $a,b,c\geq0$ satisfying
$$\left(a+b+c-2\right)^2+8\leq3\left(ab+bc+ca\right).$$Prove
$$a^2+b^2+c^2+5abc\geq8.$$
3 replies
mihaig
Yesterday at 5:17 PM
JARP091
16 minutes ago
Dophantine equation
MENELAUSS   2
N 17 minutes ago by Assassino9931
Solve for $x;y \in \mathbb{Z}$ the following equation :
$$3^x-8^y =2xy+1 $$
2 replies
MENELAUSS
Yesterday at 11:35 PM
Assassino9931
17 minutes ago
Serbian selection contest for the IMO 2025 - P6
OgnjenTesic   11
N 20 minutes ago by JARP091
Source: Serbian selection contest for the IMO 2025
For an $n \times n$ table filled with natural numbers, we say it is a divisor table if:
- the numbers in the $i$-th row are exactly all the divisors of some natural number $r_i$,
- the numbers in the $j$-th column are exactly all the divisors of some natural number $c_j$,
- $r_i \ne r_j$ for every $i \ne j$.

A prime number $p$ is given. Determine the smallest natural number $n$, divisible by $p$, such that there exists an $n \times n$ divisor table, or prove that such $n$ does not exist.

Proposed by Pavle Martinović
11 replies
OgnjenTesic
May 22, 2025
JARP091
20 minutes ago
Shortest number theory you might've seen in your life
AlperenINAN   9
N 20 minutes ago by Assassino9931
Source: Turkey JBMO TST 2025 P4
Let $p$ and $q$ be prime numbers. Prove that if $pq(p+1)(q+1)+1$ is a perfect square, then $pq + 1$ is also a perfect square.
9 replies
AlperenINAN
May 11, 2025
Assassino9931
20 minutes ago
Inequality about number of spanning trees of graph
CBMaster   0
39 minutes ago
Let \( k(G) \) be the number of spanning trees in a graph \( G \), where \( G \) may have multiple edges and loops.

For two edges \( e \) and \( f \) of \( G \), let \( G/e \), \( G/f \), and \( G/\{e,f\} \) denote the graphs obtained by contracting the edges \( e \), \( f \), and both \( e \) and \( f \) in $G$, respectively.

Find a combinatorial proof of the following inequality:
\[
k(G/\{e,f\}) \cdot k(G) \leq k(G/e) \cdot k(G/f)
\]
0 replies
CBMaster
39 minutes ago
0 replies
1,2,...,2011 around circle such that 8 of 25 successive multiples of 5 and/or 7
parmenides51   1
N an hour ago by ririgggg
Source: 2011 Belarus TST 2.1
Is it possible to arrange the numbers $1,2,...,2011$ over the circle in some order so that among any $25$ successive numbers at least $8$ numbers are multiplies of $5$ or $7$ (or both $5$ and $7$) ?

I. Gorodnin
1 reply
parmenides51
Nov 8, 2020
ririgggg
an hour ago
Sipnayan JHS 2021 F-9
PikaVee   1
N an hour ago by PikaVee
Matt and Sai are playing a game of darts together. Matt has a slightly more accurate aim than Sai. In
fact, Matt can hit the bullseye 80% of the time while Sai can only hit it 60% of the time. They take turns
in playing and the first player is determined by a flip of a fair coin. If the probability that Sai scores the
first bullseye is given by $ \frac {a}{b} $ where a and b are relatively prime integers, what is b − a?
1 reply
PikaVee
an hour ago
PikaVee
an hour ago
Standart looking FE
Kimchiks926   13
N an hour ago by math-olympiad-clown
Source: Baltic Way 2022, Problem 5
Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(0)+1=f(1)$ and for any real numbers $x$ and $y$,
$$ f(xy-x)+f(x+f(y))=yf(x)+3 $$
13 replies
Kimchiks926
Nov 12, 2022
math-olympiad-clown
an hour ago
A sharp one with 3 var (2)
mihaig   4
N an hour ago by mihaig
Source: Own
Let $a,b,c\geq0$ satisfying
$$\left(a+b+c-2\right)^2+8\leq3\left(ab+bc+ca\right).$$Prove
$$a+b+c+\sqrt{abc}\geq4.$$
4 replies
mihaig
May 26, 2025
mihaig
an hour ago
3 var inequality
SunnyEvan   11
N an hour ago by mihaig
Let $ a,b,c \in R $ ,such that $ a^2+b^2+c^2=4(ab+bc+ca)$Prove that :$$ \frac{7-2\sqrt{14}}{48} \leq \frac{a^3b+b^3c+c^3a}{(a^2+b^2+c^2)^2} \leq \frac{7+2\sqrt{14}}{48} $$
11 replies
SunnyEvan
May 17, 2025
mihaig
an hour ago
trigonometric inequality
MATH1945   12
N an hour ago by mihaig
Source: ?
In triangle $ABC$, prove that $$sin^2(A)+sin^2(B)+sin^2(C) \leq \frac{9}{4}$$
12 replies
MATH1945
May 26, 2016
mihaig
an hour ago
Prove that DP=DR
WakeUp   8
N Jul 25, 2020 by dchenmathcounts
Source: Polish Second Round 2001
Points $A,B,C$ with $AB<BC$ lie in this order on a line. Let $ABDE$ be a square. The circle with diameter $AC$ intersects the line $DE$ at points $P$ and $Q$ with $P$ between $D$ and $E$. The lines $AQ$ and $BD$ intersect at $R$. Prove that $DP=DR$.
8 replies
WakeUp
Mar 6, 2012
dchenmathcounts
Jul 25, 2020
Prove that DP=DR
G H J
G H BBookmark kLocked kLocked NReply
Source: Polish Second Round 2001
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WakeUp
1347 posts
#1 • 2 Y
Y by Adventure10, Mango247
Points $A,B,C$ with $AB<BC$ lie in this order on a line. Let $ABDE$ be a square. The circle with diameter $AC$ intersects the line $DE$ at points $P$ and $Q$ with $P$ between $D$ and $E$. The lines $AQ$ and $BD$ intersect at $R$. Prove that $DP=DR$.
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WakeUp
1347 posts
#2 • 1 Y
Y by Adventure10
Since $EA\perp AC$ clearly $EA$ is a tangent to the circle with diameter $AC$. Then $\angle PAE=\angle PQA=\angle QAC$. Clearly this makes $\triangle PAE$ and $\triangle BAR$ congruent since they're already both right-angled and $BA=AE$. Thus $PE=BR$, implying $DP=DR$.
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vanu1996
607 posts
#3 • 2 Y
Y by Adventure10, Mango247
$\angle PAE=\angle PQA=\angle RAB$,so $\angle DAP=\angle DAR=45-\angle RAB$,also $\angle PDA=\angle RDA=45$,hence $DPA$ and $DRA$ are congruent,so $DP=DR$.
Z K Y
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jayme
9801 posts
#4 • 2 Y
Y by Adventure10, Mango247
Dear Mathlinkers,
see
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=468077
and also my proof on
http://perso.orange.fr/jl.ayme vol. 7 Miniatures geometriques p. 87-89.
Sincerely
Jean-Louis
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armpist
527 posts
#5 • 2 Y
Y by Adventure10, Mango247
jayme wrote:
Dear Mathlinkers,
see
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=468077
and also my proof on
http://perso.orange.fr/jl.ayme vol. 7 Miniatures geometriques p. 87-89.
Sincerely
Jean-Louis

Dear J-L,

There is smth wrong with your reference: your paper has only 83 pages.

I wrote to you about it before, and hopefully this time you will make
needed corrections.


Friendly,

M.T.
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jayme
9801 posts
#6 • 2 Y
Y by Adventure10, Mango247
Dear Mathlinkers,
ops, I have not charge the new version... done
Sorry and sincerely
Jean-Louis
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AlastorMoody
2125 posts
#7 • 4 Y
Y by lolmanc123, Adventure10, Mango247, Mango247
$$\angle CAP=x=\angle ACQ=\angle ARB=\angle APE \implies \Delta APE \cong \Delta ARB \Longrightarrow PE=BR \implies DP=DR$$
Z K Y
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WolfusA
1900 posts
#8
Y by
Consider the construction on cartesian plane. Let $$A=(0,0),\ B=(1,0),\ D=(1,1),\ E=(0,1),\ C=(2c,0)$$for some $c>1$. Then
$$P=\left(c-\sqrt{c^2-1},1\right),\ Q =\left(c+\sqrt{c^2-1},1\right),\ R =\left(1,\frac{1}{c+\sqrt{c^2-1}}\right)$$and finally
$$|DP|=1-c+\sqrt{c^2-1}= 1-\frac{1}{c+\sqrt{c^2-1}}=|DR|$$QED
This post has been edited 4 times. Last edited by WolfusA, Jul 26, 2020, 8:40 PM
Reason: post #9 from a troll deleted
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dchenmathcounts
2443 posts
#10 • 3 Y
Y by Mango247, Mango247, Mango247
Boring length bash.

Notice this is equivalent to proving $EP=BR.$ Let $EP=x,$ $BR=y,$ $AC=2r,$ and $AB=h.$

By Power of a Point $EA^2=h^2=EP\cdot EQ=x(x+2\sqrt{r^2-h^2})=(x+\sqrt{r^2-h^2})^2+h^2-r^2,$ implying $r^2=(x+\sqrt{r^2-h^2}^2)$ or $x=r-\sqrt{r^2-h^2}.$

Now note $\frac{AC}{QC}=\frac{r+\sqrt{r^2+h^2}}{h}=\frac{AB}{RB}=\frac{h}{y},$ implying $y=\frac{h^2}{r+\sqrt{r^2+h^2}}=r-\sqrt{r^2-h^2}.$ Thus $x=y.$
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