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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)!

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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
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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
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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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Weird locus problem
Sedro   6
N 15 minutes ago by Sedro
Points $A$ and $B$ are in the coordinate plane such that $AB=2$. Let $\mathcal{H}$ denote the locus of all points $P$ in the coordinate plane satisfying $PA\cdot PB=2$, and let $M$ be the midpoint of $AB$. Points $X$ and $Y$ are on $\mathcal{H}$ such that $\angle XMY = 45^\circ$ and $MX\cdot MY=\sqrt{2}$. The value of $MX^4 + MY^4$ can be expressed in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
6 replies
Sedro
May 11, 2025
Sedro
15 minutes ago
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Prove angles are equal
BigSams   51
N 20 minutes ago by Fly_into_the_sky
Source: Canadian Mathematical Olympiad - 1994 - Problem 5.
Let $ABC$ be an acute triangle. Let $AD$ be the altitude on $BC$, and let $H$ be any interior point on $AD$. Lines $BH,CH$, when extended, intersect $AC,AB$ at $E,F$ respectively. Prove that $\angle EDH=\angle FDH$.
51 replies
BigSams
May 13, 2011
Fly_into_the_sky
20 minutes ago
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incircle and excircles
micliva   5
N 26 minutes ago by Double07
Source: 2013 All-Russian Olympiad Final Round Grade 10 Day 2 P7
The incircle of triangle $ ABC $ has centre $I$ and touches the sides $ BC $, $ CA $, $ AB $ at points $ A_1 $, $ B_1 $, $ C_1 $, respectively. Let $ I_a $, $ I_b $, $ I_c $ be excentres of triangle $ ABC $, touching the sides $ BC $, $ CA $, $ AB $ respectively. The segments $ I_aB_1 $ and $ I_bA_1 $ intersect at $ C_2 $. Similarly, segments $ I_bC_1 $ and $ I_cB_1 $ intersect at $ A_2 $, and the segments $ I_cA_1 $ and $ I_aC_1 $ at $ B_2 $. Prove that $ I $ is the center of the circumcircle of the triangle $ A_2B_2C_2 $.

L. Emelyanov, A. Polyansky
5 replies
micliva
May 16, 2014
Double07
26 minutes ago
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4-vars inequality
xytunghoanh   2
N 33 minutes ago by JARP091
For $a,b,c,d \ge 0$ and $a\ge c$, $b \ge d$. Prove that
$$a+b+c+d+ac+bd+8 \ge 2(\sqrt{ab}+\sqrt{bc}+\sqrt{cd}+\sqrt{da}+\sqrt{ac}+\sqrt{bd})$$.
2 replies
xytunghoanh
4 hours ago
JARP091
33 minutes ago
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Iranian TST 2019, first exam, day1, problem 2
Hamper.r   14
N an hour ago by bin_sherlo
$a, a_1,a_2,\dots ,a_n$ are natural numbers. We know that for any natural number $k$ which $ak+1$ is square, at least one of $a_1k+1,\dots ,a_n k+1$ is also square.
Prove $a$ is one of $a_1,\dots ,a_n$
Proposed by Mohsen Jamali
14 replies
1 viewing
Hamper.r
Apr 7, 2019
bin_sherlo
an hour ago
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IOQM P23 2024
SomeonecoolLovesMaths   3
N an hour ago by lakshya2009
Consider the fourteen numbers, $1^4,2^4,...,14^4$. The smallest natural numebr $n$ such that they leave distinct remainders when divided by $n$ is:
3 replies
SomeonecoolLovesMaths
Sep 8, 2024
lakshya2009
an hour ago
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Symmetric squares wrt centre of 4x4 square add to 17
Goutham   1
N an hour ago by Orange_Quail_9
Numbers $1, 2,\cdots, 16$ are written in a $4\times 4$ square matrix so that the sum of the numbers in every row, every column, and every diagonal is the same and furthermore that the numbers $1$ and $16$ lie in opposite corners. Prove that the sum of any two numbers symmetric with respect to the center of the square equals $17$.
1 reply
Goutham
Dec 6, 2010
Orange_Quail_9
an hour ago
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Find all p(x) such that p(p) is a power of 2
truongphatt2668   3
N an hour ago by waterbottle432
Source: ???
Find all polynomial $P(x) \in \mathbb{R}[x]$ such that:
$$P(p_i) = 2^{a_i}$$with $p_i$ is an $i$ th prime and $a_i$ is an arbitrary positive integer.
3 replies
truongphatt2668
5 hours ago
waterbottle432
an hour ago
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functional equation
uaua   4
N an hour ago by jasperE3
f:R--R
f(f(x)+xy) = xf(x) + f(x)
4 replies
uaua
Jan 3, 2023
jasperE3
an hour ago
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Sixth smallest divisor
sevket12   3
N an hour ago by MITDragon
Source: 2025 Turkey EGMO TST P4
Find all positive integers $n$ such that the number
\[ \frac{3 + \sqrt{4n + 9}}{2} \]is the sixth smallest positive divisor of $n$.
3 replies
sevket12
Feb 8, 2025
MITDragon
an hour ago
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Inequalities
sqing   4
N an hour ago by Entrepreneur
Let $ a,b,c>0 , a+b+c +abc=4$. Prove that
$$ \frac {a}{a^2+2}+\frac {b}{b^2+2}+\frac {c}{c^2+2} \leq 1$$Let $ a,b,c>0 , ab+bc+ca+abc=4$. Prove that
$$ \frac {a}{a^2+2}+\frac {b}{b^2+2}+\frac {c}{c^2+2} \leq 1$$
4 replies
sqing
5 hours ago
Entrepreneur
an hour ago
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Points on the sides of cyclic quadrilateral satisfy the angle conditions
AlperenINAN   4
N an hour ago by Draq
Source: Turkey JBMO TST 2025 P1
Let $ABCD$ be a cyclic quadrilateral and let the intersection point of lines $AB$ and $CD$ be $E$. Let the points $K$ and $L$ be arbitrary points on sides $CD$ and $AB$ respectively, which satisfy the conditions
$$\angle KAD = \angle KBC \quad \text{and} \quad \angle LDA = \angle LCB.$$Prove that $EK = EL$.
4 replies
AlperenINAN
May 11, 2025
Draq
an hour ago
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Minimum number of points
Ecrin_eren   1
N an hour ago by rachelcassano
There are 18 teams in a football league. Each team plays against every other team twice in a season—once at home and once away. A win gives 3 points, a draw gives 1 point, and a loss gives 0 points. One team became the champion by earning more points than every other team. What is the minimum number of points this team could have?

1 reply
Ecrin_eren
2 hours ago
rachelcassano
an hour ago
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IMO 2010 Problem 1
canada   121
N 2 hours ago by maromex
Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[ f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$

Proposed by Pierre Bornsztein, France
121 replies
canada
Jul 7, 2010
maromex
2 hours ago
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Dot product
SomeonecoolLovesMaths   4
N Apr 29, 2025 by quasar_lord
How to prove that dot product is distributive?
4 replies
SomeonecoolLovesMaths
Apr 28, 2025
quasar_lord
Apr 29, 2025
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Dot product
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SomeonecoolLovesMaths
3256 posts
SomeonecoolLovesMaths
#1 h  Apr 28, 2025, 6:06 PM
Y by
How to prove that dot product is distributive?
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Rohit-2006
243 posts
Rohit-2006
#2 h  Apr 28, 2025, 6:58 PM
Y by
Because $\cos x=\cos(-x)$ and we define $\vec{a}\cdot\vec{b}$ as $\mid a\mid\mid b\mid \cos \theta$....assume that you are taking the angle in anticlockwise manner .... But taking clockwise manner doesn't change sign of the expression
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greenturtle3141
3559 posts
greenturtle3141
#3 h  Apr 28, 2025, 7:26 PM
Y by
SomeonecoolLovesMaths wrote:
How to prove that dot product is distributive?

write it out componentwise (x = (x1,x2,...), x.y = x1y1 + ...) and it is quite clear.
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SomeonecoolLovesMaths
3256 posts
SomeonecoolLovesMaths
#4 h  Apr 29, 2025, 5:28 AM
Y by
greenturtle3141 wrote:
SomeonecoolLovesMaths wrote:
How to prove that dot product is distributive?

write it out componentwise (x = (x1,x2,...), x.y = x1y1 + ...) and it is quite clear.

While multiplying component wise we multiply every term with every term, hence does it not in a sense already assume the distributivity?
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quasar_lord
299 posts
quasar_lord
#5 h  Apr 29, 2025, 11:25 AM
Y by
Defining the dot product $\mathbf{a \cdot b} = \sum_{i=1}^n(a_ib_i)$
$$(\mathbf a + \mathbf b) \cdot \mathbf c = \sum (a_i + b_i) c_i = \sum (a_i c_i) + \sum(b_i c_i) = \mathbf{a \cdot c} + \mathbf{b\cdot c}$$
For a purely geometric proof see this or this
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