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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.

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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
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
Sadigly   3
N an hour ago by pooh123
Source: Azerbaijan Junior MO 2025 P5
For positive real numbers $x;y;z$ satisfying $0<x,y,z<2$, find the biggest value the following equation could acquire:


$$(2x-yz)(2y-zx)(2z-xy)$$
3 replies
Sadigly
Friday at 7:59 AM
pooh123
an hour ago
Calculus
youochange   2
N an hour ago by youochange
Find the area enclosed by the curves $e^x,e^{-x},x^2+y^2=1$
2 replies
youochange
Yesterday at 2:38 PM
youochange
an hour ago
A strong inequality problem
hn111009   0
an hour ago
Source: Somewhere
Let $a,b,c$ be the positive number satisfied $a^2+b^2+c^2=3.$ Find the minimum of $$P=\dfrac{a^2}{b+c}+\dfrac{b^2}{c+a}+\dfrac{c^2}{a+b}+\dfrac{3abc}{2(ab+bc+ca)}.$$
0 replies
hn111009
an hour ago
0 replies
help me please,thanks
tnhan.129   0
an hour ago
find f: R+ -> R such that:
f(x)/x + f(y)/y = (1/x + 1/y).f(sqrt(xy))
0 replies
tnhan.129
an hour ago
0 replies
Easy divisibility
a_507_bc   2
N an hour ago by TUAN2k8
Source: ARO Regional stage 2023 9.4~10.4
Let $a, b, c$ be positive integers such that no number divides some other number. If $ab-b+1 \mid abc+1$, prove that $c \geq b$.
2 replies
a_507_bc
Feb 16, 2023
TUAN2k8
an hour ago
Inspired by old results
sqing   0
an hour ago
Source: Own
Let $a,b,c,d$ be real numbers such that $a^2+b^2+c^2 =3$. Prove that$$\frac{9}{5}>(a-b)(b-c)(2a-1)(2c-1)\geq -16$$
0 replies
sqing
an hour ago
0 replies
integer functional equation
ABCDE   149
N an hour ago by ezpotd
Source: 2015 IMO Shortlist A2
Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\]holds for all $x,y\in\mathbb{Z}$.
149 replies
ABCDE
Jul 7, 2016
ezpotd
an hour ago
A geometry problem involving 2 circles
Ujiandsd   0
an hour ago
Source: L
Point M is the midpoint of side BC of triangle ABC. The length of the radius of the outer circle of triangle ABM, triangle ACM
is 5 and 7 respectively find the distance between the center of their outer circles
0 replies
Ujiandsd
an hour ago
0 replies
Inequality, inequality, inequality...
Assassino9931   10
N 2 hours ago by sqing
Source: Al-Khwarizmi Junior International Olympiad 2025 P6
Let $a,b,c$ be real numbers such that \[ab^2+bc^2+ca^2=6\sqrt{3}+ac^2+cb^2+ba^2.\]Find the smallest possible value of $a^2 + b^2 + c^2$.

Binh Luan and Nhan Xet, Vietnam
10 replies
Assassino9931
Yesterday at 9:38 AM
sqing
2 hours ago
Grid with rooks
a_507_bc   3
N 2 hours ago by TUAN2k8
Source: ARO Regional stage 2022 9.3
Given is a positive integer $n$. There are $2n$ mutually non-attacking rooks placed on a grid $2n \times 2n$. The grid is splitted into two connected parts, symmetric with respect to the center of the grid. What is the largest number of rooks that could lie in the same part?
3 replies
a_507_bc
Feb 16, 2023
TUAN2k8
2 hours ago
IMO Shortlist 2013, Number Theory #3
lyukhson   47
N 2 hours ago by cursed_tangent1434
Source: IMO Shortlist 2013, Number Theory #3
Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.
47 replies
lyukhson
Jul 10, 2014
cursed_tangent1434
2 hours ago
Darboux cubic
srirampanchapakesan   1
N 2 hours ago by srirampanchapakesan
Source: Own
Let P be a point on the Darboux cubic (or the McCay Cubic ) of triangle ABC.

P1P2P3 is the circumcevian or pedal triangle of P wrt ABC.

Prove that P also lie on the Darboux cubic ( or the McCay Cubic) of P1P2P3 .
1 reply
srirampanchapakesan
May 7, 2025
srirampanchapakesan
2 hours ago
IMO Shortlist 2011, Algebra 2
orl   43
N 2 hours ago by ezpotd
Source: IMO Shortlist 2011, Algebra 2
Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with \[\sum^{2011}_{j=1} j  x^n_j = a^{n+1} + 1\]

Proposed by Warut Suksompong, Thailand
43 replies
1 viewing
orl
Jul 11, 2012
ezpotd
2 hours ago
Sequence inequality
BR1F1SZ   1
N 2 hours ago by IndoMathXdZ
Source: 2025 Francophone MO Seniors P1
Let $a_1, a_2, a_3, \ldots$ be a sequence of positive integers satisfying the following property: for all positive integers $k < \ell$, for all distinct integers $m_1, m_2, \ldots, m_k$ and for all distinct integers $n_1, n_2, \ldots, n_\ell$,
\[
a_{m_1} + a_{m_2} + \cdots + a_{m_k} \leqslant a_{n_1} + a_{n_2} + \cdots + a_{n_\ell}.
\]Prove that there exist two integers $N$ and $b$ such that $a_n = b$ for all $n \geqslant N$.
1 reply
BR1F1SZ
4 hours ago
IndoMathXdZ
2 hours ago
number theory
Levieee   7
N Apr 19, 2025 by g0USinsane777
Idk where it went wrong, marks was deducted for this solution
$\textbf{Question}$
Show that for a fixed pair of distinct positive integers \( a \) and \( b \), there cannot exist infinitely many \( n \in \mathbb{Z} \) such that
\[
\sqrt{n + a} + \sqrt{n + b} \in \mathbb{Z}.
\]
$\textbf{Solution}$

Let
\[
x = \sqrt{n + a} + \sqrt{n + b} \in \mathbb{N}.
\]
Then,
\[
x^2 = (\sqrt{n + a} + \sqrt{n + b})^2 = (n + a) + (n + b) + 2\sqrt{(n + a)(n + b)}.
\]So:
\[
x^2 = 2n + a + b + 2\sqrt{(n + a)(n + b)}.
\]
Therefore,
\[
\sqrt{(n + a)(n + b)} \in \mathbb{N}.
\]
Let
\[
(n + a)(n + b) = k^2.
\]Assume \( n + a \neq n + b \). Then we have:
\[
n + a \mid k \quad \text{and} \quad k \mid n + b,
\]or it could also be that \( k \mid n + a \quad \text{and} \quad n + b \mid k \).

Without loss of generality, we take the first case:
\[
(n + a)k_1 = k \quad \text{and} \quad kk_2 = n + b.
\]
Thus,
\[
k_1 k_2 = \frac{n + b}{n + a}.
\]
Since \( k_1 k_2 \in \mathbb{N} \), we have:
\[
k_1 k_2 = 1 + \frac{b - a}{n + a}.
\]
For infinitely many \( n \), \( \frac{b - a}{n + a} \) must be an integer, which is not possible.

Therefore, there cannot be infinitely many such \( n \).
7 replies
Levieee
Apr 18, 2025
g0USinsane777
Apr 19, 2025
number theory
G H J
G H BBookmark kLocked kLocked NReply
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Levieee
233 posts
#1
Y by
Idk where it went wrong, marks was deducted for this solution
$\textbf{Question}$
Show that for a fixed pair of distinct positive integers \( a \) and \( b \), there cannot exist infinitely many \( n \in \mathbb{Z} \) such that
\[
\sqrt{n + a} + \sqrt{n + b} \in \mathbb{Z}.
\]
$\textbf{Solution}$

Let
\[
x = \sqrt{n + a} + \sqrt{n + b} \in \mathbb{N}.
\]
Then,
\[
x^2 = (\sqrt{n + a} + \sqrt{n + b})^2 = (n + a) + (n + b) + 2\sqrt{(n + a)(n + b)}.
\]So:
\[
x^2 = 2n + a + b + 2\sqrt{(n + a)(n + b)}.
\]
Therefore,
\[
\sqrt{(n + a)(n + b)} \in \mathbb{N}.
\]
Let
\[
(n + a)(n + b) = k^2.
\]Assume \( n + a \neq n + b \). Then we have:
\[
n + a \mid k \quad \text{and} \quad k \mid n + b,
\]or it could also be that \( k \mid n + a \quad \text{and} \quad n + b \mid k \).

Without loss of generality, we take the first case:
\[
(n + a)k_1 = k \quad \text{and} \quad kk_2 = n + b.
\]
Thus,
\[
k_1 k_2 = \frac{n + b}{n + a}.
\]
Since \( k_1 k_2 \in \mathbb{N} \), we have:
\[
k_1 k_2 = 1 + \frac{b - a}{n + a}.
\]
For infinitely many \( n \), \( \frac{b - a}{n + a} \) must be an integer, which is not possible.

Therefore, there cannot be infinitely many such \( n \).
This post has been edited 2 times. Last edited by Levieee, Apr 18, 2025, 7:47 PM
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Lil_flip38
56 posts
#2
Y by
I might be missing something, but why does \((n+a)(n+b)=k^2\) imply \(n+a\mid k\) and \(k\mid n+b\)? Why cant for example \(n+a=25\) and \(n+b=4\) and \(k=10\). The factors of \(k\) do not have to be distributed nicely between \(n+a\) and \(n+b\).
Z K Y
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DTforever
5 posts
#3
Y by
I think the condition implies both $n+a$ and $n+b$ are perfect squares.
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MathLuis
1525 posts
#4 • 1 Y
Y by Levieee
You need a slightly more sophisitcated finish after realising $(n+a)(n+b)$ is a square for infinitely many $n$ and it is that we have $n^2+(a+b)n+ab$ is a square for infinitely many $n$ and now setting some arbitrarily large $n$ and considering some $k$ for which $2k \ge a+b >2k-2$ we could have that it is a square bounded between $(n+k)^2$ and $(n+k-1)^2$ so it has to be $(n+k)^2$ which shows that $(a+b)^2=4ab$ by coefficient checking and matching and thus $(a-b)^2=0$ so $a=b$ which is a contradiction!.
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Safal
169 posts
#5
Y by
Levieee wrote:
Idk where it went wrong, marks was deducted for this solution
$\textbf{Question}$
Show that for a fixed pair of distinct positive integers \( a \) and \( b \), there cannot exist infinitely many \( n \in \mathbb{Z} \) such that
\[
\sqrt{n + a} + \sqrt{n + b} \in \mathbb{Z}.
\]
$\textbf{Solution}$

Let
\[
x = \sqrt{n + a} + \sqrt{n + b} \in \mathbb{N}.
\]
Then,
\[
x^2 = (\sqrt{n + a} + \sqrt{n + b})^2 = (n + a) + (n + b) + 2\sqrt{(n + a)(n + b)}.
\]So:
\[
x^2 = 2n + a + b + 2\sqrt{(n + a)(n + b)}.
\]
Therefore,
\[
\sqrt{(n + a)(n + b)} \in \mathbb{N}.
\]
Let
\[
(n + a)(n + b) = k^2.
\]Assume \( n + a \neq n + b \). Then we have:
\[
n + a \mid k \quad \text{and} \quad k \mid n + b,
\]or it could also be that \( k \mid n + a \quad \text{and} \quad n + b \mid k \).

Without loss of generality, we take the first case:
\[
(n + a)k_1 = k \quad \text{and} \quad kk_2 = n + b.
\]
Thus,
\[
k_1 k_2 = \frac{n + b}{n + a}.
\]
Since \( k_1 k_2 \in \mathbb{N} \), we have:
\[
k_1 k_2 = 1 + \frac{b - a}{n + a}.
\]
For infinitely many \( n \), \( \frac{b - a}{n + a} \) must be an integer, which is not possible.

Therefore, there cannot be infinitely many such \( n \).

$k_1$ and $k_2$ need not be integers, they can be rational beacuse we can have that the denominator ot that rational divide $n+a$
This post has been edited 1 time. Last edited by Safal, Apr 18, 2025, 9:19 PM
Z K Y
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Levieee
233 posts
#6
Y by
Safal wrote:
Levieee wrote:
Idk where it went wrong, marks was deducted for this solution
$\textbf{Question}$
Show that for a fixed pair of distinct positive integers \( a \) and \( b \), there cannot exist infinitely many \( n \in \mathbb{Z} \) such that
\[
\sqrt{n + a} + \sqrt{n + b} \in \mathbb{Z}.
\]
$\textbf{Solution}$

Let
\[
x = \sqrt{n + a} + \sqrt{n + b} \in \mathbb{N}.
\]
Then,
\[
x^2 = (\sqrt{n + a} + \sqrt{n + b})^2 = (n + a) + (n + b) + 2\sqrt{(n + a)(n + b)}.
\]So:
\[
x^2 = 2n + a + b + 2\sqrt{(n + a)(n + b)}.
\]
Therefore,
\[
\sqrt{(n + a)(n + b)} \in \mathbb{N}.
\]
Let
\[
(n + a)(n + b) = k^2.
\]Assume \( n + a \neq n + b \). Then we have:
\[
n + a \mid k \quad \text{and} \quad k \mid n + b,
\]or it could also be that \( k \mid n + a \quad \text{and} \quad n + b \mid k \).

Without loss of generality, we take the first case:
\[
(n + a)k_1 = k \quad \text{and} \quad kk_2 = n + b.
\]
Thus,
\[
k_1 k_2 = \frac{n + b}{n + a}.
\]
Since \( k_1 k_2 \in \mathbb{N} \), we have:
\[
k_1 k_2 = 1 + \frac{b - a}{n + a}.
\]
For infinitely many \( n \), \( \frac{b - a}{n + a} \) must be an integer, which is not possible.

Therefore, there cannot be infinitely many such \( n \).

$k_1$ and $k_2$ need not be integers, they can be rational beacuse we can have that the denominator ot that rational divide $n+a$

yea but after it's reduced we can get it to be an integer , doesnt $ a \mid b$ imply for some $k$ $ak=b$ , k is an integer

@2above how do u say that $n^2 + ab + (a+b)n$
This post has been edited 2 times. Last edited by Levieee, Apr 19, 2025, 7:41 AM
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Levieee
233 posts
#7
Y by
DTforever wrote:
I think the condition implies both $n+a$ and $n+b$ are perfect squares.

does the counter example provided by @Lil_flip38 work? or that counter example can be invalidated because $n = 4-b$, $b \ge 1$ so as n increases we can just show it cant happen
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g0USinsane777
48 posts
#8 • 1 Y
Y by Levieee
Levieee wrote:
Assume \( n + a \neq n + b \). Then we have:
\[
n + a \mid k \quad \text{and} \quad k \mid n + b,
\]or it could also be that \( k \mid n + a \quad \text{and} \quad n + b \mid k \).
This part is certainly wrong, since $(n+a)(n+b)=k^2$ can imply $n+a \leq k$ and $n+b \geq k$ or vice-versa but not the divisibility condition above. You can find many counter examples to this.

Another finish from this part can be that let $\gcd(n+a,n+b)=d$. (of course, $n+a \neq n+b$)
So, $n+a = dx$ and $n+b=dy$, $\gcd(x,y)=1$
This gives us that, $d^2xy = k^2$
Which means that $xy=l^2$ and $x=u^2$ and $y=v^2$, $\gcd(u,v)=1$.
This means that, $b-a=d(v^2 - u^2)$ for a particular $n$.
Since, the original condition needs to be true for infinitely many $n$, by infinite PHP we will have infinitely many $n$ for which we get the same value of $d$ for a particular value of $k$.
So, for infinitely many $n$, which in turn give infinitely many value of the pair $(u,v)$, the value $\frac{b-a}{d}$ is constant, say $t$.
From the above argument, since infinitely many pairs $(u,v)$ satisfy $t = v^2-u^2 = (v-u)(v+u)$, where $v,u$ are integers, this means that $t$ has infinitely many factors $\implies$ $t=0$, i.e., $b=a$ which contradicts that $a$ and $b$ are distinct.
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