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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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hard problem
Cobedangiu   5
N a few seconds ago by arqady
Let $a,b,c>0$ and $a+b+c=3$. Prove that:
$\dfrac{4}{a+b}+\dfrac{4}{b+c}+\dfrac{4}{c+a} \le \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+3$
5 replies
Cobedangiu
Yesterday at 1:51 PM
arqady
a few seconds ago
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AGI-Origin Solves Full IMO 2020–2024 Benchmark Without Solver (30/30) beat Alpha
AGI-Origin   1
N 8 minutes ago by mannshah1211
Hello IMO community,

I’m sharing here a full 30-problem solution set to all IMO problems from 2020 to 2024.

Standard AI: Prompt --> Symbolic Solver (SymPy, Geometry API, etc.)

Unlike AlphaGeometry or symbolic math tools that solve through direct symbolic computation, AGI-Origin operates via recursive symbolic cognition.

AGI-Origin:
Prompt --> Internal symbolic mapping --> Recursive contradiction/repair --> Structural reasoning --> Human-style proof

It builds human-readable logic paths by recursively tracing contradictions, repairing structure, and collapsing ambiguity — not by invoking any external symbolic solver.

These results were produced by a recursive symbolic cognition framework called AGI-Origin, designed to simulate semi-AGI through contradiction collapse, symbolic feedback, and recursion-based error repair.

These were solved without using any symbolic computation engine or solver.
Instead, the solutions were derived using a recursive symbolic framework called AGI-Origin, based on:
- Contradiction collapse
- Self-correcting recursion
- Symbolic anchoring and logical repair

Full PDF: [Upload to Dropbox/Google Drive/Notion or arXiv link when ready]

This effort surpasses AlphaGeometry’s previous 25/30 mark by covering:
- Algebra
- Combinatorics
- Geometry
- Functional Equations

Each solution follows a rigorous logical path and is written in fully human-readable format — no machine code or symbolic solvers were used.

I would greatly appreciate any feedback on the solution structure, logic clarity, or symbolic methodology.

Thank you!

— AGI-Origin Team
AGI-Origin.com
1 reply
1 viewing
AGI-Origin
an hour ago
mannshah1211
8 minutes ago
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Inspired by hlminh
sqing   0
11 minutes ago
Source: Own
Let $ a,b,c $ be real numbers such that $ a^2+b^2+c^2=1. $ Prove that$$ |a-kb|+|kb-c|+|c-a|\leq 2\sqrt {k^2+1}$$Where $ k\geq 1.$
$$ |a-kb|+|kb-c|+|c-a|\leq 2\sqrt {2}$$Where $0< k\leq 1.$
0 replies
sqing
11 minutes ago
0 replies
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Two very hard parallel
jayme   1
N 22 minutes ago by jayme
Source: own inspired by EGMO
Dear Mathlinkers,

1. ABC a triangle
2. D, E two point on the segment BC so that BD = DE= EC
3. M, N the midpoint of ED, AE
4. H the orthocenter of the acutangle triangle ADE
5. 1, 2 the circumcircle of the triangle DHM, EHN
6. P, Q the second point of intersection of 1 and BM, 2 and CN
7. U, V the second points of intersection of 2 and MN, PQ.

Prove : UV is parallel to PM.

Sincerely
Jean-Louis
1 reply
+1 w
jayme
Yesterday at 12:46 PM
jayme
22 minutes ago
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Diophantine equation !
ComplexPhi   5
N an hour ago by aops.c.c.
Source: Romania JBMO TST 2015 Day 1 Problem 4
Solve in nonnegative integers the following equation :
$$21^x+4^y=z^2$$
5 replies
ComplexPhi
May 14, 2015
aops.c.c.
an hour ago
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Combo problem
soryn   0
2 hours ago
The school A has m1 boys and m2 girls, and ,the school B has n1 boys and n2 girls. Each school is represented by one team formed by p students,boys and girls. If f(k) is the number of cases for which,the twice schools has,togheter k girls, fund f(k) and the valute of k, for which f(k) is maximum.
0 replies
soryn
2 hours ago
0 replies
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Parity and sets
betongblander   7
N 2 hours ago by ihategeo_1969
Source: Brazil National Olympiad 2020 5 Level 3
Let $n$ and $k$ be positive integers with $k$ $\le$ $n$. In a group of $n$ people, each one or always
speak the truth or always lie. Arnaldo can ask questions for any of these people
provided these questions are of the type: “In set $A$, what is the parity of people who speak to
true? ”, where $A$ is a subset of size $ k$ of the set of $n$ people. The answer can only
be “$even$” or “$odd$”.
a) For which values of $n$ and $k$ is it possible to determine which people speak the truth and
which people always lie?
b) What is the minimum number of questions required to determine which people
speak the truth and which people always lie, when that number is finite?
7 replies
betongblander
Mar 18, 2021
ihategeo_1969
2 hours ago
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Mount Inequality erupts on a sequence :o
GrantStar   88
N 2 hours ago by Nari_Tom
Source: 2023 IMO P4
Let $x_1,x_2,\dots,x_{2023}$ be pairwise different positive real numbers such that
\[a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}\]is an integer for every $n=1,2,\dots,2023.$ Prove that $a_{2023} \geqslant 3034.$
88 replies
GrantStar
Jul 9, 2023
Nari_Tom
2 hours ago
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JBMO Shortlist 2022 N1
Lukaluce   8
N 2 hours ago by godchunguus
Source: JBMO Shortlist 2022
Determine all pairs $(k, n)$ of positive integers that satisfy
$$1! + 2! + ... + k! = 1 + 2 + ... + n.$$
8 replies
Lukaluce
Jun 26, 2023
godchunguus
2 hours ago
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P(x) | P(x^2-2)
GreenTea2593   4
N 2 hours ago by GreenTea2593
Source: Valentio Iverson
Let $P(x)$ be a monic polynomial with complex coefficients such that there exist a polynomial $Q(x)$ with complex coefficients for which \[P(x^2-2)=P(x)Q(x).\]Determine all complex numbers that could be the root of $P(x)$.

Proposed by Valentio Iverson, Indonesia
4 replies
GreenTea2593
5 hours ago
GreenTea2593
2 hours ago
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USEMO P6 (Idk what to say here)
franzliszt   16
N 3 hours ago by MathLuis
Source: USEMO 2020/6
Prove that for every odd integer $n > 1$, there exist integers $a, b > 0$ such that, if we let $Q(x) = (x + a)^ 2 + b$, then the following conditions hold:
$\bullet$ we have $\gcd(a, n) = gcd(b, n) = 1$;
$\bullet$ the number $Q(0)$ is divisible by $n$; and
$\bullet$ the numbers $Q(1), Q(2), Q(3), \dots$ each have a prime factor not dividing $n$.
16 replies
franzliszt
Oct 25, 2020
MathLuis
3 hours ago
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Prove that the fraction (21n + 4)/(14n + 3) is irreducible
DPopov   110
N 3 hours ago by Shenhax
Source: IMO 1959 #1
Prove that the fraction $ \dfrac{21n + 4}{14n + 3}$ is irreducible for every natural number $ n$.
110 replies
DPopov
Oct 5, 2005
Shenhax
3 hours ago
J
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Let \( a, b, c \) be positive real numbers satisfying \[ a^2 + c^2 = b(a + c). \
Jackson0423   3
N 3 hours ago by Mathzeus1024
Let \( a, b, c \) be positive real numbers satisfying
\[ a^2 + c^2 = b(a + c). \]Let
\[ m = \min \left( \frac{a^2 + ab + b^2}{ab + bc + ca} \right). \]Find the value of \( 2024m \).
3 replies
Jackson0423
Apr 16, 2025
Mathzeus1024
3 hours ago
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real+ FE
pomodor_ap   3
N 3 hours ago by MathLuis
Source: Own, PDC001-P7
Let $f : \mathbb{R}^+ \to \mathbb{R}^+$ be a function such that
$$f(x)f(x^2 + y f(y)) = f(x)f(y^2) + x^3$$for all $x, y \in \mathbb{R}^+$. Determine all such functions $f$.
3 replies
pomodor_ap
Yesterday at 11:24 AM
MathLuis
3 hours ago
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Modular NT
oVlad   3
N Apr 3, 2025 by EVKV
Source: Romania JBMO TST 2024 Day 1 P1
Find all the positive integers $a{}$ and $b{}$ such that $(7^a-5^b)/8$ is a prime number.

Cosmin Manea and Dragoș Petrică
3 replies
oVlad
Jul 31, 2024
EVKV
Apr 3, 2025
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Modular NT
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Reveal topic tags
number theory
modular arithmetic
prime numbers
L
G H BBookmark kLocked kLocked NReply
Source: Romania JBMO TST 2024 Day 1 P1
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oVlad
1742 posts
oVlad
#1 h  Jul 31, 2024, 12:29 PM • 1 Y
Y by ItsBesi
Find all the positive integers $a{}$ and $b{}$ such that $(7^a-5^b)/8$ is a prime number.

Cosmin Manea and Dragoș Petrică
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sami1618
895 posts
sami1618
#2 h  Jul 31, 2024, 5:30 PM
Y by
Answer: $(a,b)=(2,2)$

Notice that for $7^a\equiv 5^b$ we must have both $a$ and $b$ being even. Let $a=2a'$ and $b=2b'$ then we must have that $$\frac{49^{a'}-25^{b'}}{8}=p$$But as $3$ divides the top we must have that $p=3$ or that $(7^{a'}+5^{b'})(7^{a'}-5^{b'})=24$. But $7^{a'}-5^{b'}$ is at least $2$ and $7^{a'}+5^{b'}$ is at least $12$ so both must be equalities leading to the above solution.
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wizixez
168 posts
wizixez
#3 h  Dec 15, 2024, 3:28 PM
Y by
$49^x-25^y=24$
Since $A(x,y)=49^x-25^y=(7^x-5^y)(7^x+5^y)=24$
checking the cases gives $(a,b)=(2,2)$
Z K Y
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EVKV
49 posts
EVKV
#4 h  Apr 3, 2025, 1:50 AM
Y by
mod 8 gives both a,b as even
So, $(7^{\frac{a}{2}} - 5^{\frac{b}{2}})(7^{\frac{a}{2}} + 5^{\frac{b}{2}}) = 8p$
now checking mod 3 leads to the fact that the prime is 3
So,$(7^{\frac{a}{2}} - 5^{\frac{b}{2}})(7^{\frac{a}{2}} + 5^{\frac{b}{2}}) = 24$
now just casework and u are done

not deserving of a jbmo tst
but follows pattern of random difficulty in jbmo tbh
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