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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)!

Be sure to mark your calendars for the following events:
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[*]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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Transformation of a cross product when multiplied by matrix A
Math-lover1   0
15 minutes ago
I was working through AoPS Volume 2 and this statement from Chapter 11: Cross Products/Determinants confused me.
[quote=AoPS Volume 2]A quick comparison of $|\underline{A}|$ to the cross product $(\underline{A}\vec{i}) \times (\underline{A}\vec{j})$ reveals that a negative determinant [of $\underline{A}$] corresponds to a matrix which reverses the direction of the cross product of two vectors.[/quote]
I understand that this is true for the unit vectors $\vec{i} = (1 \ 0)$ and $\vec{j} = (0 \ 1)$, but am confused on how to prove this statement for general vectors $\vec{v}$ and $\vec{w}$ although its supposed to be a quick comparison.

How do I prove this statement easily with any two 2D vectors?
0 replies
Math-lover1
15 minutes ago
0 replies
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Find points with sames integer distances as given
nAalniaOMliO   2
N 33 minutes ago 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
33 minutes ago
J
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trigonometric functions
VivaanKam   3
N 36 minutes ago by aok
Hi could someone explain the basic trigonometric functions to me like sin, cos, tan etc.
Thank you!
3 replies
VivaanKam
2 hours ago
aok
36 minutes ago
J
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Geometry tangent circles
Stefan4024   68
N an hour ago by zuat.e
Source: EGMO 2016 Day 2 Problem 4
Two circles $\omega_1$ and $\omega_2$, of equal radius intersect at different points $X_1$ and $X_2$. Consider a circle $\omega$ externally tangent to $\omega_1$ at $T_1$ and internally tangent to $\omega_2$ at point $T_2$. Prove that lines $X_1T_1$ and $X_2T_2$ intersect at a point lying on $\omega$.
68 replies
Stefan4024
Apr 13, 2016
zuat.e
an hour ago
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My Unsolved Problem
MinhDucDangCHL2000   2
N an hour ago by hukilau17
Source: 2024 HSGS Olympiad
Let triangle $ABC$ be inscribed in the circle $(O)$. A line through point $O$ intersects $AC$ and $AB$ at points $E$ and $F$, respectively. Let $P$ be the reflection of $E$ across the midpoint of $AC$, and $Q$ be the reflection of $F$ across the midpoint of $AB$. Prove that:
a) the reflection of the orthocenter $H$ of triangle $ABC$ across line $PQ$ lies on the circle $(O)$.
b) the orthocenters of triangles $AEF$ and $HPQ$ coincide.

Im looking for a solution used complex bashing :(
2 replies
MinhDucDangCHL2000
6 hours ago
hukilau17
an hour ago
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Easy Combinatorics
MuradSafarli   2
N an hour ago by Sadigly
A student firstly wrote $x=3$ on the board. For each procces, the stutent deletes the number x and replaces it with either $(2x+4)$ or $(3x+8)$ or $(x^2+5x)$. Is this possible to make the number $(20^{25}+2024)$ on the board?
2 replies
MuradSafarli
4 hours ago
Sadigly
an hour ago
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4 variables with quadrilateral sides 2
mihaig   0
2 hours ago
Source: Own
Let $a,b,c,d\geq0$ satisfying
$$\frac1{a+1}+\frac1{b+1}+\frac1{c+1}+\frac1{d+1}=2.$$Prove
$$\left(a+b+c+d-2\right)^2+8\geq3\left(abc+abd+acd+bcd\right).$$
0 replies
mihaig
2 hours ago
0 replies
J
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Number theory
MuradSafarli   1
N 2 hours ago by Sadigly
Prove that for any natural number \( n \) :

\[ 1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n + 1) \mid (4n + 3)(4n + 5) \cdot \ldots \cdot (8n + 3). \]
1 reply
MuradSafarli
3 hours ago
Sadigly
2 hours ago
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D1025 : Can you do that?
Dattier   0
2 hours ago
Source: les dattes à Dattier
Let $x_{n+1}=x_n^3$ and $x_0=3$.

Can you calculate $\sum\limits_{i=1}^{2^{2025}} x_i \mod 10^{30}$?
0 replies
Dattier
2 hours ago
0 replies
J
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Perpendicularity
April   32
N 3 hours ago by zuat.e
Source: CGMO 2007 P5
Point $D$ lies inside triangle $ABC$ such that $\angle DAC = \angle DCA = 30^{\circ}$ and $\angle DBA = 60^{\circ}$. Point $E$ is the midpoint of segment $BC$. Point $F$ lies on segment $AC$ with $AF = 2FC$. Prove that $DE \perp EF$.
32 replies
April
Dec 28, 2008
zuat.e
3 hours ago
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Geometry books
T.Mousavidin   2
N 3 hours ago by VivaanKam
Hello, I wanted to ask if anybody knows some good books for geometry that has these topics in:
Desargues's Theorem, Projective geometry, 3D geometry,
2 replies
T.Mousavidin
Today at 4:25 PM
VivaanKam
3 hours ago
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The number of integers
Fang-jh   16
N 4 hours ago by ihategeo_1969
Source: ChInese TST 2009 P3
Prove that for any odd prime number $ p,$ the number of positive integer $ n$ satisfying $ p|n! + 1$ is less than or equal to $ cp^\frac{2}{3}.$ where $ c$ is a constant independent of $ p.$
16 replies
Fang-jh
Apr 4, 2009
ihategeo_1969
4 hours ago
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functional equation interesting
skellyrah   12
N 4 hours ago by jasperE3
find all functions IR->IR such that $$xf(x+yf(xy)) + f(f(x)) = f(xf(y))^2 + (x+1)f(x)$$
12 replies
skellyrah
Apr 24, 2025
jasperE3
4 hours ago
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Inequalities
sqing   16
N 5 hours ago by martianrunner
Let $ a,b \in [0 ,1] . $ Prove that
$$\frac{a}{ 1-ab+b }+\frac{b }{ 1-ab+a } \leq 2$$$$ \frac{a}{ 1+ab+b^2 }+\frac{b }{ 1+ab+a^2 }+\frac{ab }{2+ab } \leq 1$$$$\frac{a}{ 1-ab+b^2 }+\frac{b }{ 1-ab+a^2 }+\frac{1 }{1+ab }\leq \frac{5}{2}$$$$\frac{a}{ 1-ab+b^2 }+\frac{b }{ 1-ab+a^2 }+\frac{1 }{1+2ab }\leq \frac{7}{3}$$$$\frac{a}{ 1+ab+b^2 }+\frac{b }{ 1+ab+a^2 } +\frac{ab }{1+ab }\leq \frac{7}{6 }$$
16 replies
sqing
Apr 25, 2025
martianrunner
5 hours ago
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Base 9 Repunits = Triangular Numbers
MCrawford   4
N May 9, 2006 by Iron Fist
Prove that every positive integer that can be written using only 1 as each digit in base 9 is a triangular number:
\begin{eqnarray*} 1_9 &=& 1 \\ 11_9 &=& 1 + 2 + 3 + 4 \\ 111_9 &=& 1 + 2 + \cdots + 13 \\ 1111_9 &=& 1 + 2 + \cdots + 40 \\ & \vdots & \end{eqnarray*}
4 replies
MCrawford
May 8, 2006
Iron Fist
May 9, 2006
J
Base 9 Repunits = Triangular Numbers
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Reveal topic tags
modular arithmetic
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MCrawford
6325 posts
MCrawford
#1 h  May 8, 2006, 9:24 PM • 1 Y
Y by Adventure10
Prove that every positive integer that can be written using only 1 as each digit in base 9 is a triangular number:
\begin{eqnarray*} 1_9 &=& 1 \\ 11_9 &=& 1 + 2 + 3 + 4 \\ 111_9 &=& 1 + 2 + \cdots + 13 \\ 1111_9 &=& 1 + 2 + \cdots + 40 \\ & \vdots & \end{eqnarray*}
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chess64
4794 posts
chess64
#2 h  May 8, 2006, 10:00 PM • 3 Y
Y by Adventure10, Adventure10, Mango247
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towersfreak2006
3012 posts
towersfreak2006
#3 h  May 8, 2006, 10:06 PM • 2 Y
Y by Adventure10, Mango247
MCrawford wrote:
Prove that every positive integer that can be written using only 1 as each digit in base 9 is a triangular number:
\begin{eqnarray*} 1_9 &=& 1 \\ 11_9 &=& 1 + 2 + 3 + 4 \\ 111_9 &=& 1 + 2 + \cdots + 13 \\ 1111_9 &=& 1 + 2 + \cdots + 40 \\ & \vdots & \end{eqnarray*}
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maniacman384
282 posts
maniacman384
#4 h  May 9, 2006, 12:16 AM • 2 Y
Y by Adventure10, Mango247
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Iron Fist
98 posts
Iron Fist
#5 h  May 9, 2006, 12:20 AM • 2 Y
Y by Adventure10, Mango247
maniacman384 wrote:
I just noticed something, any number that starts with a 1 and has the rest of its digits as zeros is a triangular number.

Last time I checked, 1000 was a triangle number, not in base 10 nor base 9...
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