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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
Yesterday 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)!

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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
Yesterday at 11:16 PM
0 replies
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
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
Inequality with 3 variables and a special condition
Nuran2010   8
N 6 minutes ago by sqing
Source: Azerbaijan Al-Khwarizmi IJMO TST 2024
For positive real numbers $a,b,c$ we have $3abc \geq ab+bc+ca$.
Prove that:

$\frac{1}{a^3+b^3+c}+\frac{1}{b^3+c^3+a}+\frac{1}{c^3+a^3+b} \leq \frac{3}{a+b+c}$.

Determine the equality case.
8 replies
Nuran2010
Apr 29, 2025
sqing
6 minutes ago
D1024 : Can you do that?
Dattier   2
N 13 minutes ago by sansgankrsngupta
Source: les dattes à Dattier
Let $x_{n+1}=x_n^2+1$ and $x_0=1$.

Can you calculate $\sum\limits_{i=1}^{2^{2025}} x_i \mod 10^{30}$?
2 replies
Dattier
Apr 29, 2025
sansgankrsngupta
13 minutes ago
4-var inequality
RainbowNeos   3
N 33 minutes ago by RainbowNeos
Given $a,b,c,d>0$, show that
\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]
3 replies
1 viewing
RainbowNeos
Yesterday at 9:31 AM
RainbowNeos
33 minutes ago
Hard inequality
ys33   0
35 minutes ago
Let $a, b, c, d>0$. Prove that
$\sqrt[3]{ab}+ \sqrt[3]{cd} < \sqrt[3]{(a+b+c)(b+c+d)}$.
0 replies
ys33
35 minutes ago
0 replies
4 lines concurrent
Zavyk09   7
N 39 minutes ago by bin_sherlo
Source: Homework
Let $ABC$ be triangle with circumcenter $(O)$ and orthocenter $H$. $BH, CH$ intersect $(O)$ again at $K, L$ respectively. Lines through $H$ parallel to $AB, AC$ intersects $AC, AB$ at $E, F$ respectively. Point $D$ such that $HKDL$ is a parallelogram. Prove that lines $KE, LF$ and $AD$ are concurrent at a point on $OH$.
7 replies
+1 w
Zavyk09
Apr 9, 2025
bin_sherlo
39 minutes ago
Generalized mirror problem
Taha1381   8
N 44 minutes ago by Lemmas
Source: Iranian second round/day1/problem1
We have a rectangle with it sides being a mirror.A light Ray enters from one of the corners of the rectangle and after being reflected several times enters to the opposite corner it started.Prove that at some time the light Ray passed the center of rectangle(Intersection of diagonals.)
8 replies
Taha1381
May 2, 2019
Lemmas
44 minutes ago
4 variables with quadrilateral sides 2
mihaig   5
N an hour ago by mihaig
Source: Own
Let $a,b,c,d\geq0$ satisfying
$$\frac1{a+1}+\frac1{b+1}+\frac1{c+1}+\frac1{d+1}=2.$$Prove
$$\left(a+b+c+d-2\right)^2+8\geq3\left(abc+abd+acd+bcd\right).$$
5 replies
mihaig
Apr 29, 2025
mihaig
an hour ago
Consecutive sum of integers sum up to 2020
NicoN9   1
N an hour ago by Mathzeus1024
Source: Japan Junior MO Preliminary 2020 P2
Let $a$ and $b$ be positive integers. Suppose that the sum of integers between $a$ and $b$, including $a$ and $b$, are equal to $2020$.
All among those pairs $(a, b)$, find the pair such that $a$ achieves the minimum.
1 reply
NicoN9
4 hours ago
Mathzeus1024
an hour ago
IMO 2023 P2
799786   91
N an hour ago by ND_
Source: IMO 2023 P2
Let $ABC$ be an acute-angled triangle with $AB < AC$. Let $\Omega$ be the circumcircle of $ABC$. Let $S$ be the midpoint of the arc $CB$ of $\Omega$ containing $A$. The perpendicular from $A$ to $BC$ meets $BS$ at $D$ and meets $\Omega$ again at $E \neq A$. The line through $D$ parallel to $BC$ meets line $BE$ at $L$. Denote the circumcircle of triangle $BDL$ by $\omega$. Let $\omega$ meet $\Omega$ again at $P \neq B$. Prove that the line tangent to $\omega$ at $P$ meets line $BS$ on the internal angle bisector of $\angle BAC$.
91 replies
799786
Jul 8, 2023
ND_
an hour ago
Geometry
VicKmath7   9
N an hour ago by tilya_TASh
Source: 8th European Mathematical Cup 2019 Junior Q3
Let $ABC$ be a triangle with circumcircle $\omega$. Let $l_B$ and $l_C$ be two lines through the points $B$ and $C$, respectively, such that $l_B  \parallel  l_C$. The second intersections of $l_B$ and $l_C$ with $\omega$ are $D$ and $E$, respectively. Assume that $D$ and $E$ are on the same side of $BC$ as $A$. Let $DA$ intersect $l_C$ at $F$ and let $EA$ intersect $l_B$ at $G$. If $O$, $O_1$ and $O_2$ are circumcenters of the triangles $ABC$, $ADG$ and $AEF$, respectively, and $P$ is the circumcenter of the triangle $OO_1O_2$, prove that $l_B  \parallel  OP \parallel l_C$.

Proposed by Stefan Lozanovski, Macedonia
9 replies
VicKmath7
Dec 26, 2019
tilya_TASh
an hour ago
2017 CNMO Grade 11 P5
minecraftfaq   3
N an hour ago by CovertQED
Source: 2017 China Northern MO, Grade 11, Problem 5
Length of sides of regular hexagon $ABCDEF$ is $a$. Two moving points $M,N$ moves on sides $BC,DE$, satisfy that $\angle MAN=\frac{\pi}{3}$. Prove that $AM\cdot AN-BM\cdot DN$ is a definite value.
3 replies
minecraftfaq
Feb 24, 2020
CovertQED
an hour ago
Inspired by Nice inequality
sqing   0
an hour ago
Source: Own
Let $  a,b,c >0  $. Show that
$$\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+1\right)^2 \geq \frac{16}{k+3}\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}+k\right)$$Where $ k\geq 1.$
$$\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+1\right)^2 \geq  \frac{b}{a}+\frac{c}{b}+\frac{a}{c}+13$$$$\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+1\right)^2 \geq \frac{16}{5}\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}+2\right)$$
0 replies
sqing
an hour ago
0 replies
Mmo 9-10 graders P5
Bet667   5
N 2 hours ago by User141208
Let $a,b,c,d$ be real numbers less than 2.Then prove that $\frac{a^3}{b^2+4}+\frac{b^3}{c^2+4}+\frac{c^3}{d^2+4}+\frac{d^3}{a^2+4}\le4$
5 replies
Bet667
Apr 3, 2025
User141208
2 hours ago
3 var inequality
sqing   2
N 2 hours ago by sqing
Source: Own
Let $ a,b,c>0 ,\frac{a}{b} +\frac{b}{c} +\frac{c}{a} \leq 2\left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right). $ Prove that
$$a+b+c+2\geq abc$$Let $ a,b,c>0 , a^3+b^3+c^3\leq 2(ab+bc+ca). $ Prove that
$$a+b+c+2\geq abc$$
2 replies
sqing
Apr 30, 2025
sqing
2 hours ago
Or statement function
ItzsleepyXD   2
N Yesterday at 6:42 AM by cursed_tangent1434
Source: Own , Mock Thailand Mathematic Olympiad P2
Find all $f: \mathbb{R} \to \mathbb{Z^+}$ such that $$f(x+f(y))=f(x)+f(y)+1\quad\text{ or }\quad f(x)+f(y)-1$$for all real number $x$ and $y$
2 replies
ItzsleepyXD
Apr 30, 2025
cursed_tangent1434
Yesterday at 6:42 AM
Or statement function
G H J
G H BBookmark kLocked kLocked NReply
Source: Own , Mock Thailand Mathematic Olympiad P2
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ItzsleepyXD
129 posts
#1
Y by
Find all $f: \mathbb{R} \to \mathbb{Z^+}$ such that $$f(x+f(y))=f(x)+f(y)+1\quad\text{ or }\quad f(x)+f(y)-1$$for all real number $x$ and $y$
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Haris1
76 posts
#2
Y by
$P(-f(0),0)$ gives $f(a)=1$ for some $a$.
$P(x,a)$ gives $f(x+1)=f(x)$ or $f(x)+2$
If there is some $b$ such that $f(b+1)=f(b)$
Then $f(b+f(y))$=$f(b+1+f(y))+2$ or $f(b+1+f(y))$
If at some point we get $f(x+f(y))$=$f(x+1+f(y))+2$ then taking $y=a$ gives contradiction.
So $f$-constant for big enough values, so $1=0$ which is not possible.
Otherwise when $f(x+1)=f(x)+2$ for all $x$.
Doing induction gives $f(n)=c+2n$ for natural numbers $n$
Which gives contradiction when trying natural values for $x$ and $y$ gives contradiction.
So there exist no functions.
Edit: After seeing @down solution i saw that the constant solution $f(x)=1$ worked.
This post has been edited 2 times. Last edited by Haris1, Yesterday at 3:09 PM
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cursed_tangent1434
611 posts
#3
Y by
The answer is $f(x) = 1$ for all $x\in \mathbb{R}$. It’s easy to see that these functions satisfy the given equation. We now show these are the only solutions. Let $P(x,y)$ be the assertion that $f(x+f(y))=f(x)+f(y)+1 \text{ or } f(x)+f(y)-1$ for real numbers $x$ and $y$.

Since the range of $f$ is the positive integers the Well-Ordering principle states that there exists a real number $\alpha$ such that $f(x) \ge f(\alpha)$ for all $x \in \mathbb{R}$. But then, $P(\alpha-f(x),x)$ yields,
\begin{align*}
f\left((\alpha - f(x))+f(x)\right) &= f(\alpha - f(x)) + f(x) \pm 1\\
f(\alpha - f(x)) & = f(\alpha) - f(x) \pm 1
\end{align*}However,
\begin{align*}
f(\alpha) \le f(\alpha - f(x))  &= f(\alpha) - f(x) \pm 1\\
f(x) & \le \pm 1
\end{align*}which since the range of $f$ is the positive integers implies that we must have $+1$ in $P(\alpha-f(x),x)$ and further, that $f(x) \le 1$ for all $x \in \mathbb{R}$ which immediately allows us to conclude that $f(x)=1$ for all real numbers $x$ as desired.
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