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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)!

Be sure to mark your calendars for the following upcoming events:
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[*]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
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
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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Diophantine Equation with prime numbers and bonus conditions
p.lazarov06   10
N 9 minutes ago by mathbetter
Source: 2023 Bulgaria JBMO TST Problem 3
Find all natural numbers $a$, $b$, $c$ and prime numbers $p$ and $q$, such that:

$\blacksquare$ $4\nmid c$
$\blacksquare$ $p\not\equiv 11\pmod{16}$
$\blacksquare$ $p^aq^b-1=(p+4)^c$
10 replies
p.lazarov06
May 7, 2023
mathbetter
9 minutes ago
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Concurrence in Cyclic Quadrilateral
GrantStar   39
N 11 minutes ago by ItsBesi
Source: IMO Shortlist 2023 G3
Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$.

Prove that lines $AD, PM$, and $BC$ are concurrent.
39 replies
GrantStar
Jul 17, 2024
ItsBesi
11 minutes ago
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IMO Genre Predictions
ohiorizzler1434   22
N 17 minutes ago by rhydon516
Everybody, with IMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
22 replies
ohiorizzler1434
Today at 6:51 AM
rhydon516
17 minutes ago
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Inequality
MathsII-enjoy   1
N an hour ago by arqady
A interesting problem generalized :-D
1 reply
MathsII-enjoy
2 hours ago
arqady
an hour ago
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Inequality
lgx57   2
N an hour ago by mashumaro
Source: Own
$a,b,c>0,ab+bc+ca=1$. Prove that

$$\sum \sqrt{8ab+1} \ge 5$$
(I don't know whether the equality holds)
2 replies
lgx57
an hour ago
mashumaro
an hour ago
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Find min
lgx57   1
N an hour ago by arqady
Source: Own
Find min of $\dfrac{a^2}{ab+1}+\dfrac{b^2+2}{a+b}$
1 reply
lgx57
an hour ago
arqady
an hour ago
J
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Product is a perfect square( very easy)
Nuran2010   1
N an hour ago by SomeonecoolLovesMaths
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?
1 reply
Nuran2010
3 hours ago
SomeonecoolLovesMaths
an hour ago
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smo 2018 open 2nd round q2
dominicleejun   7
N an hour ago by Kyj9981
Let O be a point inside triangle ABC such that $\angle BOC$ is $90^\circ$ and $\angle BAO = \angle BCO$. Prove that $\angle OMN$ is $90$ degrees, where $M$ and $N$ are the midpoints of $\overline{AC}$ and $\overline{BC}$, respectively.
7 replies
dominicleejun
Aug 15, 2019
Kyj9981
an hour ago
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inequalities
Cobedangiu   4
N an hour ago by Nguyenhuyen_AG
$a,b,c>0$ and $\sum ab=\dfrac{1}{3}$. Prove that:
$\sum \dfrac{1}{a^2-bc+1}\le 3$
4 replies
Cobedangiu
Today at 4:06 AM
Nguyenhuyen_AG
an hour ago
J
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Two equal angles
jayme   2
N 2 hours ago by jayme
Dear Mathlinkers,

1. ABCD a square
2. I the midpoint of AB
3. 1 the circle center at A passing through B
4. Q the point of intersection of 1 with the segment IC
5. X the foot of the perpendicular to BC from Q
6. Y the point of intersection of 1 with the segment AX
7. M the point of intersection of CY and AB.

Prove : <ACI = <IYM.

Sincerely
Jean-Louis
2 replies
jayme
Yesterday at 6:52 AM
jayme
2 hours ago
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minimum of \sqrt{\frac{a}{b(3a+2)}}+\sqrt{\frac{b}{a(3b+2)}}
parmenides51   11
N 2 hours ago by sqing
Source: JBMO Shortlist 2017 A2
Let $a$ and $b$ be positive real numbers such that $3a^2 + 2b^2 = 3a + 2b$. Find the minimum value of $A =\sqrt{\frac{a}{b(3a+2)}} + \sqrt{\frac{b}{a(2b+3)}} $
11 replies
parmenides51
Jul 25, 2018
sqing
2 hours ago
J
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Interesting inequalities
sqing   1
N 3 hours ago by sqing
Source: Own
Let $ a,b,c>0 $ and $ (a+b)^2 (a+c)^2=16abc. $ Prove that
$$ 2a+b+c\leq \frac{128}{27}$$$$ \frac{9}{2}a+b+c\leq \frac{864}{125}$$$$3a+b+c\leq 24\sqrt{3}-36$$$$5a+b+c\leq \frac{4(8\sqrt{6}-3)}{9}$$
1 reply
sqing
Yesterday at 2:35 PM
sqing
3 hours ago
J
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Geometry Problem
Euler_Gauss   0
3 hours ago
Given that $D$ is the midpoint of $BC$, $DM$ bisects $\angle ADB$ and intersects $AB$ at $M$, $I$ is the incenter of $\triangle {}{}{}ABD$, $AT$ bisects $\angle BAC$ and intersects the circumcircle of \(\triangle {}{}ABC\) at $T$, $MS$ is parallel to $BC$ and intersects $AT$ at $S$. Prove that $\angle MIS + \angle BIT = \pi.$
0 replies
Euler_Gauss
3 hours ago
0 replies
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Tricky invariant with 3 numbers on the board
Nuran2010   0
3 hours ago
Source: Azerbaijan Junior National Olympiad 2021
Initially, the numbers $1,1,-1$ written on the board.At every step,Mikail chooses the two numbers $a,b$ and substitutes them with $2a+c$ and $\frac{b-c}{2}$ where $c$ is the unchosen number on the board. Prove that at least $1$ negative number must be remained on the board at any step.
0 replies
Nuran2010
3 hours ago
0 replies
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J
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Intersections are concyclic
Pompombojam   0
Apr 26, 2025
Source: Xueersi Grade 9 Program
In parallelogram $ABCD$, $\angle BAD \neq 90^{\circ}$. Construct a circle with center $B$ and radius $BA$, intersecting the extensions of $AB$ and $CB$ at $E$ and $F$ respectively. Construct a circle with center $D$ and radius $DA$, intersecting the extensions of $AD$ and $CD$ at $M$ and $N$. Let $G$ be the intersection of $EN$ and $FM$, $T$ be the intersection of $AG$ and $ME$, $P$ ($\neq N$) be the intersection of $EN$ and the circle with center $D$, and $Q$ ($\neq F$) be the intersection of $MF$ and the circle with center $B$. Show that $G$, $P$, $T$ and $Q$ are concyclic.
0 replies
Pompombojam
Apr 26, 2025
0 replies
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Intersections are concyclic
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Reveal topic tags
geometry
cyclic quadrilateral
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Pompombojam
4 posts
Pompombojam
#1 h  Apr 26, 2025, 11:15 AM
Y by
In parallelogram $ABCD$, $\angle BAD \neq 90^{\circ}$. Construct a circle with center $B$ and radius $BA$, intersecting the extensions of $AB$ and $CB$ at $E$ and $F$ respectively. Construct a circle with center $D$ and radius $DA$, intersecting the extensions of $AD$ and $CD$ at $M$ and $N$. Let $G$ be the intersection of $EN$ and $FM$, $T$ be the intersection of $AG$ and $ME$, $P$ ($\neq N$) be the intersection of $EN$ and the circle with center $D$, and $Q$ ($\neq F$) be the intersection of $MF$ and the circle with center $B$. Show that $G$, $P$, $T$ and $Q$ are concyclic.
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