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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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0 replies
jlacosta
Apr 2, 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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Sharing My Solutions for IMO 2023 (All 6 Problems) – Feedback Welcome
Blackhole.LightKing   0
6 minutes ago
Hi everyone,

I worked through all six problems from the 2023 International Mathematical Olympiad (IMO 2023), and I would like to share my solutions here.

I tried to make the explanations clear and complete, and I also added diagrams for the geometry problems to make them easier to follow.

Topics covered:

Number Theory

Geometry

Algebra

Combinatorics

You can find the full solution PDF attached (or linked below).

https://agi-origin.ai/assets/pdf/AGI-Origin_IMO_2023_Solution.pdf

I’m still learning, so if you see any mistakes, unclear parts, or better ways to explain things, I would really appreciate your feedback!

Thank you for reading, and I hope this can be helpful to others who are studying IMO problems too
0 replies
+2 w
Blackhole.LightKing
6 minutes ago
0 replies
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Inspired by old results
sqing   8
N 6 minutes ago by SunnyEvan
Source: Own
Let $ a,b>0. $ Prove that
$$\frac{(a+1)^2}{b}+\frac{(b+k)^2}{a} \geq4(k+1) $$Where $ k\geq 0. $
$$\frac{a^2}{b}+\frac{(b+1)^2}{a} \geq4$$
8 replies
sqing
Apr 19, 2025
SunnyEvan
6 minutes ago
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A bash problem
kjhgyuio   0
7 minutes ago
........
0 replies
kjhgyuio
7 minutes ago
0 replies
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Connecting chaos in a grid
Assassino9931   3
N 16 minutes ago by dgrozev
Source: Bulgaria National Olympiad 2025, Day 1, Problem 2
Exactly \( n \) cells of an \( n \times n \) square grid are colored black, and the remaining cells are white. The cost of such a coloring is the minimum number of white cells that need to be recolored black so that from any black cell \( c_0 \), one can reach any other black cell \( c_k \) through a sequence \( c_0, c_1, \ldots, c_k \) of black cells where each consecutive pair \( c_i, c_{i+1} \) are adjacent (sharing a common side) for every \( i = 0, 1, \ldots, k-1 \). Let \( f(n) \) denote the maximum possible cost over all initial colorings with exactly \( n \) black cells. Determine a constant $\alpha$ such that
\[ \frac{1}{3}n^{\alpha} \leq f(n) \leq 3n^{\alpha} \]for any $n\geq 100$.
3 replies
Assassino9931
Apr 8, 2025
dgrozev
16 minutes ago
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Balanced Tournaments
anantmudgal09   7
N 17 minutes ago by Mathgloggers
Source: The 1st India-Iran Friendly Competition Problem 1
A league consists of $2024$ players. A round involves splitting the players into two different teams and having every member of one team play with every member of the other team. A round is called balanced if both teams have an equal number of players. A tournament consists of several rounds at the end of which any two players have played each other. The committee organised a tournament last year which consisted of $N$ rounds. Prove that the committee can organise a tournament this year with $N$ balanced rounds.

Proposed by Anant Mudgal and Navilarekallu Tejaswi
7 replies
anantmudgal09
Jun 12, 2024
Mathgloggers
17 minutes ago
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2 var inequalities
sqing   1
N 17 minutes ago by sqing
Source: Own
Let $ a,b \in [0 ,1] . $ Prove that
$$ \frac{a}{ 1+a+b^2 }+\frac{b }{ 1+b+a^2 }\leq \frac{2}{3}$$$$ \frac{a}{ 1+a^2+b }+\frac{b }{ 1+b^2+a }\leq \frac{2}{3}$$$$ \frac{a}{ 1+a^2+b }+\frac{b }{ 1+b^2+a }+\frac{ab }{1+ab }\leq \frac{7}{6}$$$$ \frac{a}{ 1+a^2+b }+\frac{b }{ 1+b^2+a }+\frac{ab }{2+ab }\leq1$$$$ \frac{a}{ 1+a^2+b }+\frac{b }{ 1+b^2+a }+\frac{ab }{1+2ab }\leq1$$
1 reply
+1 w
sqing
25 minutes ago
sqing
17 minutes ago
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nice [symmedians in a triangle, < ABM = < BAN]
grodij   10
N 21 minutes ago by Lemmas
Source: IMO Shortlist 2000, G5
The tangents at $B$ and $A$ to the circumcircle of an acute angled triangle $ABC$ meet the tangent at $C$ at $T$ and $U$ respectively. $AT$ meets $BC$ at $P$, and $Q$ is the midpoint of $AP$; $BU$ meets $CA$ at $R$, and $S$ is the midpoint of $BR$. Prove that $\angle ABQ=\angle BAS$. Determine, in terms of ratios of side lengths, the triangles for which this angle is a maximum.
10 replies
grodij
Nov 14, 2004
Lemmas
21 minutes ago
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Brute force in diophantine equation?
primemystic   0
26 minutes ago
As the title. Are there any method like brute force in solving diophantine equation? Thanks!
0 replies
primemystic
26 minutes ago
0 replies
J
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4 variables with quadrilateral sides
mihaig   2
N 40 minutes ago by removablesingularity
Source: VL
Let $a,b,c,d\geq0$ satisfying
$$\frac1{a+1}+\frac1{b+1}+\frac1{c+1}+\frac1{d+1}=2.$$Prove
$$4\left(abc+abd+acd+bcd\right)\geq3\left(a+b+c+d\right)+4.$$
2 replies
mihaig
Today at 5:11 AM
removablesingularity
40 minutes ago
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Killer NT that nobody solved (also my hardest NT ever created)
mshtand1   7
N an hour ago by SimplisticFormulas
Source: Ukraine IMO 2025 TST P8
A positive integer number \( a \) is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence \( \{b_k\}_{k=1}^{\infty} \), where
\[ b_k = a^{k^k} \cdot 2^{2^k - k} + 1. \]
Proposed by Arsenii Nikolaev and Mykhailo Shtandenko
7 replies
mshtand1
Apr 19, 2025
SimplisticFormulas
an hour ago
J
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4-var cyclic ineq
RainbowNeos   0
an hour ago
For nonnegative $a,b,c,d$, show that
\[\frac{2}{3}\left(\sqrt{a+b+c}+\sqrt{b+c+d}+\sqrt{c+d+a}+\sqrt{d+a+b}\right)\leq\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+d}+\sqrt{d+a}\leq 2(\sqrt{2}-1)\left(\sqrt{a+b+c}+\sqrt{b+c+d}+\sqrt{c+d+a}+\sqrt{d+a+b}\right).\]
0 replies
RainbowNeos
an hour ago
0 replies
J
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Functional equation from R to R-[INMO 2011]
Potla   36
N an hour ago by Adywastaken
Find all functions $f:\mathbb{R}\to \mathbb R$ satisfying
\[f(x+y)f(x-y)=\left(f(x)+f(y)\right)^2-4x^2f(y),\]For all $x,y\in\mathbb R$.
36 replies
+1 w
Potla
Feb 6, 2011
Adywastaken
an hour ago
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Deduction card battle
anantmudgal09   54
N an hour ago by anudeep
Source: INMO 2021 Problem 4
A Magician and a Detective play a game. The Magician lays down cards numbered from $1$ to $52$ face-down on a table. On each move, the Detective can point to two cards and inquire if the numbers on them are consecutive. The Magician replies truthfully. After a finite number of moves, the Detective points to two cards. She wins if the numbers on these two cards are consecutive, and loses otherwise.

Prove that the Detective can guarantee a win if and only if she is allowed to ask at least $50$ questions.

Proposed by Anant Mudgal
54 replies
anantmudgal09
Mar 7, 2021
anudeep
an hour ago
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2016 SMO Open Geometry
vlwk   5
N 2 hours ago by mqoi_KOLA
Let $D$ be a point in the interior of $\triangle{ABC}$ such that $AB=ab$, $AC=ac$, $BC=bc$, $AD=ad$, $BD=bd$, $CD=cd$. Show that $\angle{ABD}+\angle{ACD}=60^{\circ}$.

Source: 2016 Singapore Mathematical Olympiad (Open) Round 2, Problem 1
5 replies
vlwk
Jul 5, 2016
mqoi_KOLA
2 hours ago
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n=y^2+108
Havu   6
N Apr 3, 2025 by ektorasmiliotis
Given the positive integer $n = y^2 + 108$ where $y \in \mathbb{N}$.
Prove that $n$ cannot be a perfect cube of a positive integer.
6 replies
Havu
Apr 3, 2025
ektorasmiliotis
Apr 3, 2025
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n=y^2+108
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Reveal topic tags
number theory
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Havu
41 posts
Havu
#1 h  Apr 3, 2025, 4:30 PM • 1 Y
Y by MR.1
Given the positive integer $n = y^2 + 108$ where $y \in \mathbb{N}$.
Prove that $n$ cannot be a perfect cube of a positive integer.
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MuradSafarli
86 posts
MuradSafarli
#2 h  Apr 3, 2025, 4:52 PM
Y by
this problem is greece 2025 p4
This post has been edited 1 time. Last edited by MuradSafarli, Apr 3, 2025, 4:54 PM
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MuradSafarli
86 posts
MuradSafarli
#3 h  Apr 3, 2025, 4:53 PM
Y by
https://artofproblemsolving.com/community/c4242999_2025_greece_national_olympiad
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MR.1
102 posts
MR.1
#4 h  Apr 3, 2025, 5:28 PM
Y by
wow what a great time to shine. use mordell's equation which has no solution for $k=108$ :D
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MathPerson12321
3725 posts
MathPerson12321
#5 h  Apr 3, 2025, 5:30 PM
Y by
does mod 9 work
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GreekIdiot
179 posts
GreekIdiot
#6 h  Apr 3, 2025, 8:33 PM • 1 Y
Y by InftyByond
yeah mod 9 works. also FCT works too, so does legendre's symbol.
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ektorasmiliotis
102 posts
ektorasmiliotis
#7 h  Apr 3, 2025, 8:45 PM
Y by
can you post your solution with mod 9?FCT works,you can express 100 as 10^2 +2^3
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