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k a June Highlights and 2025 AoPS Online Class Information
jlacosta   0
Monday at 3:57 PM
Congratulations to all the mathletes who competed at National MATHCOUNTS! If you missed the exciting Countdown Round, you can watch the video at this link. Are you interested in training for MATHCOUNTS or AMC 10 contests? How would you like to train for these math competitions in half the time? We have accelerated sections which meet twice per week instead of once starting on July 8th (7:30pm ET). These sections fill quickly so enroll today!

[list][*]MATHCOUNTS/AMC 8 Basics
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[*]AMC 10 Problem Series[/list]
For those interested in Olympiad level training in math, computer science, physics, and chemistry, be sure to enroll in our WOOT courses before August 19th to take advantage of early bird pricing!

Summer camps are starting this 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 a transformative 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:
[list][*]June 5th, Thursday, 7:30pm ET: Open Discussion with Ben Kornell and Andrew Sutherland, Art of Problem Solving's incoming CEO Ben Kornell and CPO Andrew Sutherland host an Ask Me Anything-style chat. Come ask your questions and get to know our incoming CEO & CPO!
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0 replies
jlacosta
Monday at 3:57 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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[ELMO2] The Multiplication Table
v_Enhance   27
N 31 minutes ago by Mr.Sharkman
Source: ELMO 2015, Problem 2 (Shortlist N1)
Let $m$, $n$, and $x$ be positive integers. Prove that \[ \sum_{i = 1}^n \min\left(\left\lfloor \frac{x}{i} \right\rfloor, m \right) = \sum_{i = 1}^m \min\left(\left\lfloor \frac{x}{i} \right\rfloor, n \right). \]
Proposed by Yang Liu
27 replies
v_Enhance
Jun 27, 2015
Mr.Sharkman
31 minutes ago
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Problem 1
randomusername   74
N an hour ago by Mr.Sharkman
Source: IMO 2015, Problem 1
We say that a finite set $\mathcal{S}$ of points in the plane is balanced if, for any two different points $A$ and $B$ in $\mathcal{S}$, there is a point $C$ in $\mathcal{S}$ such that $AC=BC$. We say that $\mathcal{S}$ is centre-free if for any three different points $A$, $B$ and $C$ in $\mathcal{S}$, there is no points $P$ in $\mathcal{S}$ such that $PA=PB=PC$.

(a) Show that for all integers $n\ge 3$, there exists a balanced set consisting of $n$ points.

(b) Determine all integers $n\ge 3$ for which there exists a balanced centre-free set consisting of $n$ points.

Proposed by Netherlands
74 replies
randomusername
Jul 10, 2015
Mr.Sharkman
an hour ago
J
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Find Triples of Integers
termas   41
N an hour ago by ilikemath247365
Source: IMO 2015 problem 2
Find all positive integers $(a,b,c)$ such that
$$ab-c,\quad bc-a,\quad ca-b$$are all powers of $2$.

Proposed by Serbia
41 replies
termas
Jul 10, 2015
ilikemath247365
an hour ago
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DO NOT OVERSLEEP JOHN MACKEY’S CLASS
ike.chen   31
N an hour ago by Mr.Sharkman
Source: USA TSTST 2023/4
Let $n\ge 3$ be an integer and let $K_n$ be the complete graph on $n$ vertices. Each edge of $K_n$ is colored either red, green, or blue. Let $A$ denote the number of triangles in $K_n$ with all edges of the same color, and let $B$ denote the number of triangles in $K_n$ with all edges of different colors. Prove
\[ B\le 2A+\frac{n(n-1)}{3}.\](The complete graph on $n$ vertices is the graph on $n$ vertices with $\tbinom n2$ edges, with exactly one edge joining every pair of vertices. A triangle consists of the set of $\tbinom 32=3$ edges between $3$ of these $n$ vertices.)

Proposed by Ankan Bhattacharya
31 replies
ike.chen
Jun 26, 2023
Mr.Sharkman
an hour ago
J
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Grade IX - Problem I
icx   23
N 2 hours ago by shendrew7
Source: Romanian National Mathematical Olympiad 2007
Let $a, b, c, d \in \mathbb{N^{*}}$ such that the equation \[x^{2}-(a^{2}+b^{2}+c^{2}+d^{2}+1)x+ab+bc+cd+da=0 \] has an integer solution. Prove that the other solution is integer too and both solutions are perfect squares.
23 replies
icx
Apr 13, 2007
shendrew7
2 hours ago
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USAMO 2002 Problem 2
MithsApprentice   35
N 2 hours ago by sami1618
Let $ABC$ be a triangle such that
\[ \left( \cot \dfrac{A}{2} \right)^2 + \left( 2\cot \dfrac{B}{2} \right)^2 + \left( 3\cot \dfrac{C}{2} \right)^2 = \left( \dfrac{6s}{7r} \right)^2, \]
where $s$ and $r$ denote its semiperimeter and its inradius, respectively. Prove that triangle $ABC$ is similar to a triangle $T$ whose side lengths are all positive integers with no common divisors and determine these integers.
35 replies
MithsApprentice
Sep 30, 2005
sami1618
2 hours ago
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Center lies on altitude
plagueis   17
N 2 hours ago by bin_sherlo
Source: Mexico National Olympiad 2018 Problem 6
Let $ABC$ be an acute-angled triangle with circumference $\Omega$. Let the angle bisectors of $\angle B$ and $\angle C$ intersect $\Omega$ again at $M$ and $N$. Let $I$ be the intersection point of these angle bisectors. Let $M'$ and $N'$ be the respective reflections of $M$ and $N$ in $AC$ and $AB$. Prove that the center of the circle passing through $I$, $M'$, $N'$ lies on the altitude of triangle $ABC$ from $A$.

Proposed by Victor Domínguez and Ariel García
17 replies
plagueis
Nov 6, 2018
bin_sherlo
2 hours ago
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IMO Shortlist 2014 C6
hajimbrak   22
N 2 hours ago by awesomeming327.
We are given an infinite deck of cards, each with a real number on it. For every real number $x$, there is exactly one card in the deck that has $x$ written on it. Now two players draw disjoint sets $A$ and $B$ of $100$ cards each from this deck. We would like to define a rule that declares one of them a winner. This rule should satisfy the following conditions:
1. The winner only depends on the relative order of the $200$ cards: if the cards are laid down in increasing order face down and we are told which card belongs to which player, but not what numbers are written on them, we can still decide the winner.
2. If we write the elements of both sets in increasing order as $A =\{ a_1 , a_2 , \ldots, a_{100} \}$ and $B= \{ b_1 , b_2 , \ldots , b_{100} \}$, and $a_i > b_i$ for all $i$, then $A$ beats $B$.
3. If three players draw three disjoint sets $A, B, C$ from the deck, $A$ beats $B$ and $B$ beats $C$ then $A$ also beats $C$.
How many ways are there to define such a rule? Here, we consider two rules as different if there exist two sets $A$ and $B$ such that $A$ beats $B$ according to one rule, but $B$ beats $A$ according to the other.

Proposed by Ilya Bogdanov, Russia
22 replies
hajimbrak
Jul 11, 2015
awesomeming327.
2 hours ago
J
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annoying algebra with sequence :/
tabel   1
N 2 hours ago by L_.
Source: random 9th grade text book (section meant for contests)
Let \( a_1 = 1 \) and \( a_{n+1} = 1 + \frac{n}{a_n} \) for \( n \geq 1 \). Prove that the sequence \( (a_n)_{n \geq 1} \) is increasing.
1 reply
tabel
Yesterday at 4:55 PM
L_.
2 hours ago
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The Return of Triangle Geometry
peace09   16
N 2 hours ago by NO_SQUARES
Source: 2023 ISL A7
Let $N$ be a positive integer. Prove that there exist three permutations $a_1,\dots,a_N$, $b_1,\dots,b_N$, and $c_1,\dots,c_N$ of $1,\dots,N$ such that \[\left|\sqrt{a_k}+\sqrt{b_k}+\sqrt{c_k}-2\sqrt{N}\right|<2023\]for every $k=1,2,\dots,N$.
16 replies
peace09
Jul 17, 2024
NO_SQUARES
2 hours ago
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f(1)f(2)...f(n) has at most n prime factors
MarkBcc168   40
N 2 hours ago by shendrew7
Source: 2020 Cyberspace Mathematical Competition P2
Let $f(x) = 3x^2 + 1$. Prove that for any given positive integer $n$, the product
$$f(1)\cdot f(2)\cdot\dots\cdot f(n)$$has at most $n$ distinct prime divisors.

Proposed by Géza Kós
40 replies
MarkBcc168
Jul 15, 2020
shendrew7
2 hours ago
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smallest a so that S(n)-S(n+a) = 2018, where S(n)=sum of digits
parmenides51   3
N 2 hours ago by TheBaiano
Source: Lusophon 2018 CPLP P3
For each positive integer $n$, let $S(n)$ be the sum of the digits of $n$. Determines the smallest positive integer $a$ such that there are infinite positive integers $n$ for which you have $S (n) -S (n + a) = 2018$.
3 replies
parmenides51
Sep 13, 2018
TheBaiano
2 hours ago
J
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ABC is similar to XYZ
Amir Hossein   55
N 3 hours ago by Mr.Sharkman
Source: China TST 2011 - Quiz 2 - D2 - P1
Let $AA',BB',CC'$ be three diameters of the circumcircle of an acute triangle $ABC$. Let $P$ be an arbitrary point in the interior of $\triangle ABC$, and let $D,E,F$ be the orthogonal projection of $P$ on $BC,CA,AB$, respectively. Let $X$ be the point such that $D$ is the midpoint of $A'X$, let $Y$ be the point such that $E$ is the midpoint of $B'Y$, and similarly let $Z$ be the point such that $F$ is the midpoint of $C'Z$. Prove that triangle $XYZ$ is similar to triangle $ABC$.
55 replies
Amir Hossein
May 20, 2011
Mr.Sharkman
3 hours ago
J
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Russia 2001
sisioyus   25
N 3 hours ago by cubres
Find all odd positive integers $ n > 1$ such that if $ a$ and $ b$ are relatively prime divisors of $ n$, then $ a+b-1$ divides $ n$.
25 replies
sisioyus
Aug 18, 2007
cubres
3 hours ago
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Functional Equation
AnhQuang_67   5
N Apr 5, 2025 by jasperE3
Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $$2\cdot f\Big(\dfrac{-xy}{2}+f(x+y)\Big)=xf(y)+yf(x), \forall x, y \in \mathbb{R} $$
5 replies
AnhQuang_67
Apr 4, 2025
jasperE3
Apr 5, 2025
J
Functional Equation
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Reveal topic tags
function
algebra
functional equation
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AnhQuang_67
57 posts
AnhQuang_67
#1 h  Apr 4, 2025, 4:50 PM
Y by
Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $$2\cdot f\Big(\dfrac{-xy}{2}+f(x+y)\Big)=xf(y)+yf(x), \forall x, y \in \mathbb{R} $$
This post has been edited 1 time. Last edited by AnhQuang_67, Apr 5, 2025, 2:25 AM
Reason: oops my bad
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maromex
228 posts
maromex
#2 h  Apr 4, 2025, 4:59 PM • 1 Y
Y by eevee9406
Is that supposed to say $yf(x)$ instead of $y(x)$?
This post has been edited 1 time. Last edited by maromex, Apr 4, 2025, 4:59 PM
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jasperE3
11395 posts
jasperE3
#3 h  Apr 4, 2025, 6:05 PM
Y by
AnhQuang_67 wrote:
Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $$2\cdot f\Big(\dfrac{-xy}{2}+f(x+y)\Big)=xf(y)+y(x), \forall x, y \in \mathbb{R} $$

Let $P(x,y)$ be the assertion $f(-xy+f(2x+y))=xf(y)+xy$, which we get from substituting $x\mapsto2x$ in $2f\left(-\frac{xy}2+f(x+y)\right)=xf(y)+xy$.
$P(0,x)\Rightarrow f(f(x))=0\Rightarrow f(0)=f(f(f(x)))=0$
$P(-1,2)\Rightarrow f(2)=-1\Rightarrow f(-1)=f(f(2))=0$
$P(-1,1)\Rightarrow f(1)=-\frac12$
$P(1,-2)\Rightarrow f(-2)=1\Rightarrow\frac12=f(1)=f(f(-2))=0$
So no such function.
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GreekIdiot
277 posts
GreekIdiot
#4 h  Apr 4, 2025, 8:15 PM
Y by
maromex wrote:
Is that supposed to say $yf(x)$ instead of $y(x)$?

If $y(x)$ is a typo and supposed to be $yf(x)$ then $f(x)=0$ satisfies.
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AnhQuang_67
57 posts
AnhQuang_67
#5 h  Apr 5, 2025, 2:26 AM
Y by
maromex wrote:
Is that supposed to say $yf(x)$ instead of $y(x)$?
Sorry guys, my bad. This is $yf(x)$
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jasperE3
11395 posts
jasperE3
#6 h  Apr 5, 2025, 7:07 AM
Y by
Some progress here:
https://artofproblemsolving.com/community/c6h3348309p31047538
surprised that it's possible to solve the full equation
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