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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
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0 replies
jwelsh
Jul 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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Next term is sum of three largest proper divisors
vsamc   19
N 26 minutes ago by v_Enhance
Source: 2025 IMO P4
A proper divisor of a positive integer $N$ is a positive divisor of $N$ other than $N$ itself.

The infinite sequence $a_1, a_2, \cdots$ consists of positive integers, each of which has at least three proper divisors. For each $n\geq 1$, the integer $a_{n+1}$ is the sum of the three largest proper divisors of $a_n$.

Determine all possible values of $a_1$.

Proposed by Paulius Aleknavičius, Lithuania
19 replies
vsamc
Jul 16, 2025
v_Enhance
26 minutes ago
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[PMO25 Areas I.18] P Divides Everything
kae_3   2
N 34 minutes ago by tapilyoca
Suppose that $p$ is a prime number which divides infinitely many numbers of the form $10^{n!}+2023$ where $n$ is a positive integer. What is the sum of all possible values of $p$?

Answer Confirmation
2 replies
kae_3
Feb 23, 2025
tapilyoca
34 minutes ago
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IMO online scoreboard
Shayan-TayefehIR   106
N an hour ago by ephone
Is there still an active link for IMO's online scoreboard?, I guess the scoring process is not over yet and that old link doesn't work...
106 replies
Shayan-TayefehIR
Yesterday at 5:31 AM
ephone
an hour ago
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IMO 2025 Medal Cutoffs Prediction
GreenTea2593   96
N an hour ago by ephone
What are your prediction for IMO 2025 medal cutoffs?
96 replies
GreenTea2593
Jul 16, 2025
ephone
an hour ago
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Find the least value of k
Math5000   2
N an hour ago by mijail
Source: ONEM, 4th (final) round, Level 3, 2018
4) A $100\times 200$ board has $k$ black cells. An operations consists of choosing a $2\times 3$ or $3\times 2$ sub-board having exactly $5$ black cells and painting of black the remaining cell. Find the least value of $k$ for which exists an initial distribution of the black cells such that after some operations the board is completely black.
2 replies
1 viewing
Math5000
Jan 6, 2020
mijail
an hour ago
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a^2-3a-19 not divisible by 289
keyree10   21
N an hour ago by brainfertiIzer
Source: (Indian) RMO 2009 Problem 2
Show that there is no integer $ a$ such that $ a^2 - 3a - 19$ is divisible by $ 289$.
21 replies
keyree10
Nov 29, 2009
brainfertiIzer
an hour ago
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ISL 2025 A8
gaussiemann144   1
N an hour ago by Bonime
Let $p \neq q$ be coprime positive integers. Determine all infinite sequences $a_1, a_2, ...$ of
positive integers such that the following conditions hold for all $n \geq 1$:
\[ \max(a_n, a_{n+1}, \ldots, a_{n+p}) - \min(a_n, a_{n+1}, \ldots, a_{n+p}) = p \]and
\[ \max(a_n, a_{n+1}, \ldots, a_{n+q}) - \min(a_n, a_{n+1}, \ldots, a_{n+q}) = q \]
1 reply
gaussiemann144
an hour ago
Bonime
an hour ago
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IMO LongList 1967, Poland 6
orl   6
N an hour ago by Plane_geometry_youtuber
Source: IMO LongList 1967, Poland 6
A line $l$ is drawn through the intersection point $H$ of altitudes of acute-angle triangles. Prove that symmetric images $l_a, l_b, l_c$ of $l$ with respect to the sides $BC,CA,AB$ have one point in common, which lies on the circumcircle of $ABC.$
6 replies
orl
Dec 16, 2004
Plane_geometry_youtuber
an hour ago
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A game of digits and seventh powers
v_Enhance   29
N 2 hours ago by chenghaohu
Source: Taiwan 2014 TST3 Quiz 1, P2
Alice and Bob play the following game. They alternate selecting distinct nonzero digits (from $1$ to $9$) until they have chosen seven such digits, and then consider the resulting seven-digit number by concatenating the digits in the order selected, with the seventh digit appearing last (i.e. $\overline{A_1B_2A_3B_4A_6B_6A_7}$). Alice wins if and only if the resulting number is the last seven decimal digits of some perfect seventh power. Please determine which player has the winning strategy.
29 replies
v_Enhance
Jul 18, 2014
chenghaohu
2 hours ago
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2015 mathtastic Mock AIME #3 min sum (1 - x)/(1 + x) for x+y+z=1
parmenides51   5
N 2 hours ago by mathprodigy2011
The positive $x,y,z$ satisfy $x + y + z = 1$. If the minimum possible value of $\frac{1 - x}{1 + x}+ \frac{1-y}{1 + y} + \frac{1 - z}{1 + z}$ equals $\frac{m}{n}$, find $10m + n$.

Proposed by vincenthuang75025
5 replies
parmenides51
Dec 11, 2023
mathprodigy2011
2 hours ago
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no 4 black squares exist that form a 2x2 subboard
parmenides51   2
N 2 hours ago by mijail
Source: 2019 Peru MO (ONEM) L3 p4 - finals
A board that has some of its squares painted black is called acceptable if there are no four black squares that form a $2 \times 2$ subboard. Find the largest real number $\lambda$ such that for every positive integer $n$ the following proposition holds: mercy: if an $n \times n$ board is acceptable and has fewer than $\lambda n^2$ dark squares, then an additional square black can be painted so that the board is still acceptable.
2 replies
parmenides51
Nov 24, 2022
mijail
2 hours ago
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A symmetric inequality in n variables
Nguyenhuyen_AG   5
N 2 hours ago by Nguyenhuyen_AG
Let $a_1, \,a_2,\ldots,a_n (n \geqslant 2)$ be non-negative real numbers. Prove that
\[\sum_{i=1}^n \frac{\displaystyle \sum_{j=1}^n a_j^2 - a_i^2}{\displaystyle \sum_{j=1}^n a_j - a_i} \leq n \cdot \frac{\displaystyle \sum_{i=1}^n a_i^2}{\displaystyle \sum_{i=1}^n a_i}.\](Assume all denominators are non-zero).
5 replies
Nguyenhuyen_AG
Yesterday at 6:56 AM
Nguyenhuyen_AG
2 hours ago
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Inequalities
sqing   0
3 hours ago
Let $ a,b,c\geq 0 . $ Prove that
$$ \sqrt{ a^3+b^3+c^3+\frac{1}{4}} + \frac{9}{5}abc+\frac{1}{2} \geq a+b+c$$O706
0 replies
sqing
3 hours ago
0 replies
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Minimal of xy subjected to a constraint!
persamaankuadrat   9
N 4 hours ago by ChickensEatGrass
Let $x,y$ be positive real numbers such that

$$x+y^{2}+x^{3} = 1481$$
Find the minimal value of $xy$
9 replies
persamaankuadrat
Thursday at 1:44 PM
ChickensEatGrass
4 hours ago
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Minimum value of 2 variable function
girishpimoli   6
N May 11, 2025 by Mathzeus1024
Minimum value of $x^2+y^2-xy+3x-3y+4$ , Where $x,y\in\mathbb{R}$
6 replies
girishpimoli
Apr 1, 2024
Mathzeus1024
May 11, 2025
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Minimum value of 2 variable function
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girishpimoli
219 posts
girishpimoli
#1 h  Apr 1, 2024, 2:03 AM
Y by
Minimum value of $x^2+y^2-xy+3x-3y+4$ , Where $x,y\in\mathbb{R}$
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icecream1234
360 posts
icecream1234
#2 h  Apr 1, 2024, 2:26 AM • 2 Y
Y by AndrewTom, girishpimoli
Discriminant Sol
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girishpimoli
219 posts
girishpimoli
#5 h  Apr 3, 2024, 4:24 AM
Y by
Thanks icecream1234, Can we solve it using Completing squrae method.
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Maheshwarananda
259 posts
Maheshwarananda
#6 h  Apr 3, 2024, 5:01 AM
Y by
girishpimoli wrote:
Thanks icecream1234, Can we solve it using Completing squrae method.

No, because ${{x}^{2}}+{{y}^{2}}-xy+3x-3y+4=\frac{1}{2}\left( 2{{x}^{2}}+2{{y}^{2}}-2xy+6x-6y+8 \right)=\frac{1}{2}\left( {{\left( x-y \right)}^{2}}+{{(x-3)}^{2}}+{{(y+3)}^{2}}-10 \right)\ge -5$ but the minimum can not be achieved because result $x=y,x=3,y=-3$ contradiction
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alexheinis
10747 posts
alexheinis
#7 h  Apr 3, 2024, 3:18 PM • 1 Y
Y by AndrewTom
@girish: yes, that's possible. We have $x^2+y^2-xy+3x-3y+4=({{2x-y+3}\over 2})^2+3({{y-1}\over 2})^2+1$, from which the minimal value 1 can be read immediately.
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Countmath1
180 posts
Countmath1
#8 h  Apr 6, 2024, 2:18 AM
Y by
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Mathzeus1024
1064 posts
Mathzeus1024
#9 h  May 11, 2025, 10:14 AM
Y by
Let $f(x,y) = x^2+y^2-xy+3x-3y+4$. Taking $\nabla f = 0$ yields the set of equations:

$f_{x} = 2x-y+3=0$ (i);

$f_{y}=2y-x-3=0$ (ii)

which adding (i) to (ii) yields $x+y=0 \Rightarrow y=-x \Rightarrow (x,y)=(-1,1)$ as our critical point. A check of the Hessian Matrix $F(x,y)$ shows that:

$F(x,y) = \begin{bmatrix} f_{xx} && f_{xy} \\ f_{yx} && f_{yy}\end{bmatrix} = \begin{bmatrix} 2 && -1 \\ -1 && 2\end{bmatrix}$ (iii);

and $detF(-1,1) = 3 > 0 \Rightarrow$ positive definite (hence, a minimum). Thus, $f_{MIN}=f(-1,1) =\textcolor{red}{1}$.
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