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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. 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)!

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April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
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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D is incenter
Layaliya   5
N a few seconds ago by Layaliya
Source: From my friend in Indonesia
Given an acute triangle \( ABC \) where \( AB > AC \). Point \( O \) is the circumcenter of triangle \( ABC \), and \( P \) is the projection of point \( A \) onto line \( BC \). The midpoints of \( BC \), \( CA \), and \( AB \) are \( D \), \( E \), and \( F \), respectively. The line \( AO \) intersects \( DE \) and \( DF \) at points \( Q \) and \( R \), respectively. Prove that \( D \) is the incenter of triangle \( PQR \).
5 replies
Layaliya
Apr 3, 2025
Layaliya
a few seconds ago
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k pupils move on a 3x3 grid only up and right , at least 2 same path
parmenides51   2
N 8 minutes ago by Congruence
Source: 2009 Greek Junior p4
In the figure we see the paths connecting the square of a city (point $P$) with the school (point $S$). In the square there are $k$ pupils starting to go to the school. They have the ability to move only to the right and up. If the pupils are free to choose any allowed path (in order to get to school), determine the minimum value of $k$ so that in any case at least two pupils follow the same path.
IMAGE
2 replies
parmenides51
Mar 17, 2020
Congruence
8 minutes ago
J
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inequalities -070425
pennypc123456789   2
N 12 minutes ago by KevinKV01
Let \( x,y,z \) be real numbers satisfying
\[x^2 +y^2+z^2 = 1 \]Prove that
\[ Q = xy + yz +2xz \le \dfrac{1+\sqrt{3}}{2}\]
2 replies
pennypc123456789
Yesterday at 3:52 AM
KevinKV01
12 minutes ago
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Positive reals FE
VicKmath7   7
N 16 minutes ago by bin_sherlo
Source: Bulgaria NMO 2024, Problem 3
Find all functions $f:\mathbb {R}^{+} \rightarrow \mathbb{R}^{+}$, such that $$f(af(b)+a)(f(bf(a))+a)=1$$for any positive reals $a, b$.
7 replies
VicKmath7
Apr 15, 2024
bin_sherlo
16 minutes ago
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partitioning 1 to p-1 into several a+b=c (mod p)
capoouo   2
N 18 minutes ago by DTforever
Source: own
Given a prime number $p$, a set is said to be $p$-good if the set contains exactly three elements $a, b, c$ and $a + b \equiv c \pmod{p}$.
Find all prime number $p$ such that $\{ 1, 2, \cdots, p-1 \}$ can be partitioned into several $p$-good sets.

Proposed by capoouo
2 replies
capoouo
Apr 21, 2024
DTforever
18 minutes ago
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APMO 2023 Problem 1
carefully   19
N 23 minutes ago by sadat465
Let $n \geq 5$ be an integer. Consider $n$ squares with side lengths $1, 2, \dots , n$, respectively. The squares are arranged in the plane with their sides parallel to the $x$ and $y$ axes. Suppose that no two squares touch, except possibly at their vertices. Show that it is possible to arrange these squares in a way such that every square touches exactly two other squares.
19 replies
carefully
Jul 5, 2023
sadat465
23 minutes ago
J
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Infinite Riemann-like sum, without using Riemann
MathsZ   4
N 30 minutes ago by KevinKV01
The goal is to compute $$\lim_{n\to\infty}\sum_{k=1}^{n}\frac{n}{(n+k)^2}$$without using the following method. I know that you can solve it using $$\lim_{n\to\infty}\sum_{k=1}^{n}\frac{n}{(n+k)^2}=\lim_{n\to\infty}\dfrac1n\sum_{k=1}^{n}\dfrac1{(1+\frac{k}{n})^2}=\int_0^1\dfrac1{(1+x)^2}\ \mathrm{d}x=\ldots=\dfrac12$$but I'm searching for an algebraic proof.
4 replies
MathsZ
2 hours ago
KevinKV01
30 minutes ago
J
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Remainder problem
girishpimoli   3
N 36 minutes ago by sadat465
Finding remainder when $\displaystyle 3^{107}$ divided by $23$
3 replies
girishpimoli
Feb 6, 2025
sadat465
36 minutes ago
J
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Easy Number Theory
VicKmath7   3
N 43 minutes ago by Congruence
Source: Archimedes Junior 2009
If the number $K = \frac{9n^2+31}{n^2+7}$ is integer, find the possible values of $n \in Z$.
3 replies
VicKmath7
Mar 17, 2020
Congruence
43 minutes ago
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Help me to understand 8
youochange   2
N an hour ago by sadat465
∆$ABC$ is a triangle, $M$,$N$ are the midpoints of $AB$, $AC$. $H$ be the orthocenter of $(ABC)$. $NH\cap (ABC)={Q,T} $and $MH \cap (ABC)={P,S}$. Prove that $AHBS$ and $CHAT$ are parallelograms.
2 replies
youochange
Feb 6, 2025
sadat465
an hour ago
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inequality ( 4 var
SunnyEvan   9
N an hour ago by SunnyEvan
Let $ a,b,c,d \in R $ , such that $ a+b+c+d=4 . $ Prove that :
$$ a^4+b^4+c^4+d^4+3 \geq \frac{7}{4}(a^3+b^3+c^3+d^3) $$$$ a^4+b^4+c^4+d^4+ \frac{76}{25} \geq \frac{44}{25}(a^3+b^3+c^3+d^3) $$
9 replies
SunnyEvan
Apr 4, 2025
SunnyEvan
an hour ago
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FIGO is perpendicular
USJL   6
N an hour ago by WLOGQED1729
Source: 2018 Taiwan TST Round 3
Let $I,G,O$ be the incenter, centroid and the circumcenter of triangle $ABC$, respectively. Let $X,Y,Z$ be on the rays $BC, CA, AB$ respectively so that $BX=CY=AZ$. Let $F$ be the centroid of $XYZ$.

Show that $FG$ is perpendicular to $IO$.
6 replies
1 viewing
USJL
Apr 2, 2020
WLOGQED1729
an hour ago
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Pigeon holle- one more !
stergiu   6
N an hour ago by Congruence
Source: Greek Olympiad 2006 , pr- 3
Prove that between every $27$ different positive integers , less than $100$, there exist some two which are NOT relative prime.

babis
6 replies
stergiu
Feb 25, 2006
Congruence
an hour ago
J
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Binomial coefficients are a power of 2
MathsZ   2
N an hour ago by MathsZ
Source: 3Blue1Brown's video https://www.youtube.com/watch?v=YtkIWDE36qU
Find all positive integers $n$ such that $$1+\binom n2+\binom n4$$is a power of $2$.
2 replies
MathsZ
Aug 15, 2023
MathsZ
an hour ago
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J
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2006 BMO1 P2 (Geometry)
maplestory   6
N Oct 27, 2012 by maplestory
In the convex quadrilateral $ABCD$, points $M,N$ lie on $\overline{AB}$ such that $AM = MN = NB$, and points $P,Q$ lie on the $\overline{CD}$ such that $CP = PQ = QD$. Prove that $[AMCP] = [MNPQ] =\frac{1}{3}[ABCD]$.
6 replies
maplestory
Oct 27, 2012
maplestory
Oct 27, 2012
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2006 BMO1 P2 (Geometry)
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Reveal topic tags
geometry
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maplestory
1458 posts
maplestory
#1 h  Oct 27, 2012, 10:05 PM • 2 Y
Y by Adventure10, Mango247
In the convex quadrilateral $ABCD$, points $M,N$ lie on $\overline{AB}$ such that $AM = MN = NB$, and points $P,Q$ lie on the $\overline{CD}$ such that $CP = PQ = QD$. Prove that $[AMCP] = [MNPQ] =\frac{1}{3}[ABCD]$.
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Farenhajt
5167 posts
Farenhajt
#2 h  Oct 27, 2012, 10:29 PM • 1 Y
Y by Adventure10
$[ACP]={1\over 3}[ACD]\land [AMC]={1\over 3}[ABC]$. Add them up to get $[AMCP]={1\over 3}[ABCD]$

Similar proof for $[BQDN]={1\over 3}[ABCD]$.

And what remains at the end is $[MNPQ]$.
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maplestory
1458 posts
maplestory
#3 h  Oct 27, 2012, 10:34 PM • 1 Y
Y by Adventure10
What does the arrow symbol pointing upwards represent?

Thanks
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Farenhajt
5167 posts
Farenhajt
#4 h  Oct 27, 2012, 10:38 PM • 1 Y
Y by Adventure10
You mean this: $\land$?
Z K Y
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maplestory
1458 posts
maplestory
#5 h  Oct 27, 2012, 10:41 PM • 1 Y
Y by Adventure10
Yep that one
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Farenhajt
5167 posts
Farenhajt
#6 h  Oct 27, 2012, 10:43 PM • 2 Y
Y by maplestory, Adventure10
It means "and". This one: $\lor$ - means "or".
Z K Y
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maplestory
1458 posts
maplestory
#7 h  Oct 27, 2012, 10:48 PM • 1 Y
Y by Adventure10
Thanks I understand the solution now.
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