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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
J
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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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What is the likelihood the last card left in the deck is black?
BEHZOD_UZ   1
N 16 minutes ago by sami1618
Source: Yandex Uzbekistan Coding and Math Contest 2025
You have a deck of cards containing $26$ black and $13$ red cards. You pull out $2$ cards, one after another, and check their colour. If both cards are the same colour, then a black card is added to the deck. However, if the cards are of different colours, then a red card is used to replace them. Once the cards are taken out of the deck, they are not returned to the deck, and thus the number of cards keeps reducing. What is the likelihood the last card left in the deck is black?
1 reply
BEHZOD_UZ
31 minutes ago
sami1618
16 minutes ago
J
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abc(a+b+c)=3, show that prod(a+b)>=8 [Indian RMO 2012(b) Q4]
Potla   31
N 20 minutes ago by sqing
Let $a,b,c$ be positive real numbers such that $abc(a+b+c)=3.$ Prove that we have
\[(a+b)(b+c)(c+a)\geq 8.\]
Also determine the case of equality.
31 replies
Potla
Dec 2, 2012
sqing
20 minutes ago
J
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AGI-Origin Solves Full IMO 2020–2024 Benchmark Without Solver (30/30) beat Alpha
AGI-Origin   10
N 31 minutes ago by TestX01
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
10 replies
AGI-Origin
5 hours ago
TestX01
31 minutes ago
J
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Incredible combinatorics problem
A_E_R   1
N an hour ago by CerealCipher
Source: Turkmenistan Math Olympiad - 2025
For any integer n, prove that there are exactly 18 integer whose sum and the sum of the fifth powers of each are equal to the integer n.
1 reply
A_E_R
2 hours ago
CerealCipher
an hour ago
J
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FE solution too simple?
Yiyj1   6
N an hour ago by Primeniyazidayi
Source: 101 Algebra Problems from the AMSP
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that the equality $$f(f(x)+y) = f(x^2-y)+4f(x)y$$holds for all pairs of real numbers $(x,y)$.

My solution

I feel like my solution is too simple. Is there something I did wrong or something I missed?
6 replies
Yiyj1
Apr 9, 2025
Primeniyazidayi
an hour ago
J
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Combo problem
soryn   1
N an hour ago by soryn
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.
1 reply
soryn
6 hours ago
soryn
an hour ago
J
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Two very hard parallel
jayme   5
N an hour 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
5 replies
jayme
Yesterday at 12:46 PM
jayme
an hour ago
J
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Number theory
XAN4   1
N 2 hours ago by NTstrucker
Source: own
Prove that there exists infinitely many positive integers $x,y,z$ such that $x,y,z\ne1$ and $x^x\cdot y^y=z^z$.
1 reply
XAN4
Apr 20, 2025
NTstrucker
2 hours ago
J
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R+ FE with arbitrary constant
CyclicISLscelesTrapezoid   25
N 2 hours ago by DeathIsAwe
Source: APMO 2023/4
Let $c>0$ be a given positive real and $\mathbb{R}_{>0}$ be the set of all positive reals. Find all functions $f \colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that \[f((c+1)x+f(y))=f(x+2y)+2cx \quad \textrm{for all } x,y \in \mathbb{R}_{>0}.\]
25 replies
CyclicISLscelesTrapezoid
Jul 5, 2023
DeathIsAwe
2 hours ago
J
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Combo with cyclic sums
oVlad   1
N 3 hours ago by ja.
Source: Romania EGMO TST 2017 Day 1 P4
In $p{}$ of the vertices of the regular polygon $A_0A_1\ldots A_{2016}$ we write the number $1{}$ and in the remaining ones we write the number $-1.{}$ Let $x_i{}$ be the number written on the vertex $A_i{}.$ A vertex is good if \[x_i+x_{i+1}+\cdots+x_j>0\quad\text{and}\quad x_i+x_{i-1}+\cdots+x_k>0,\]for any integers $j{}$ and $k{}$ such that $k\leqslant i\leqslant j.$ Note that the indices are taken modulo $2017.$ Determine the greatest possible value of $p{}$ such that, regardless of numbering, there always exists a good vertex.
1 reply
oVlad
Yesterday at 1:41 PM
ja.
3 hours ago
J
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Stronger inequality than an old result
KhuongTrang   20
N 3 hours ago by KhuongTrang
Source: own, inspired
Problem. Find the best constant $k$ satisfying $$(ab+bc+ca)\left[\frac{1}{(a+b)^{2}}+\frac{1}{(b+c)^{2}}+\frac{1}{(c+a)^{2}}\right]\ge \frac{9}{4}+k\cdot\frac{a(a-b)(a-c)+b(b-a)(b-c)+c(c-a)(c-b)}{(a+b+c)^{3}}$$holds for all $a,b,c\ge 0: ab+bc+ca>0.$
20 replies
KhuongTrang
Aug 1, 2024
KhuongTrang
3 hours ago
J
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Incircle of a triangle is tangent to (ABC)
amar_04   11
N 4 hours ago by Nari_Tom
Source: XVII Sharygin Correspondence Round P18
Let $ABC$ be a scalene triangle, $AM$ be the median through $A$, and $\omega$ be the incircle. Let $\omega$ touch $BC$ at point $T$ and segment $AT$ meet $\omega$ for the second time at point $S$. Let $\delta$ be the triangle formed by lines $AM$ and $BC$ and the tangent to $\omega$ at $S$. Prove that the incircle of triangle $\delta$ is tangent to the circumcircle of triangle $ABC$.
11 replies
amar_04
Mar 2, 2021
Nari_Tom
4 hours ago
J
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Inspired by hlminh
sqing   1
N 4 hours ago by sqing
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 $0< k\leq 1.$
1 reply
sqing
4 hours ago
sqing
4 hours ago
J
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Inequality with n-gon sides
mihaig   3
N 4 hours ago by mihaig
Source: VL
If $a_1,a_2,\ldots, a_n~(n\geq3)$ are are the lengths of the sides of a $n-$gon such that
$$\sum_{i=1}^{n}{a_i}=1,$$then
$$(n-2)\left[\sum_{i=1}^{n}{\frac{a_i^2}{(1-a_i)^2}}-\frac n{(n-1)^2}\right]\geq(2n-1)\left(\sum_{i=1}^{n}{\frac{a_i}{1-a_i}}-\frac n{n-1}\right)^2.$$
When do we have equality?

(V. Cîrtoaje and L. Giugiuc, 2021)
3 replies
mihaig
Feb 25, 2022
mihaig
4 hours ago
J
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J
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The locus of P with supplementary angles condition
WakeUp   3
N Apr 4, 2025 by Nari_Tom
Source: Baltic Way 2001
Given a rhombus $ABCD$, find the locus of the points $P$ lying inside the rhombus and satisfying $\angle APD+\angle BPC=180^{\circ}$.
3 replies
WakeUp
Nov 17, 2010
Nari_Tom
Apr 4, 2025
J
The locus of P with supplementary angles condition
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Reveal topic tags
geometry
rhombus
parallelogram
circumcircle
conics
hyperbola
cyclic quadrilateral
L
G H BBookmark kLocked kLocked NReply
Source: Baltic Way 2001
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WakeUp
1347 posts
WakeUp
#1 h  Nov 17, 2010, 7:19 PM • 2 Y
Y by Adventure10, Mango247
Given a rhombus $ABCD$, find the locus of the points $P$ lying inside the rhombus and satisfying $\angle APD+\angle BPC=180^{\circ}$.
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yetti
2643 posts
yetti
#2 h  Nov 17, 2010, 8:29 PM • 2 Y
Y by Adventure10, Mango247
Let $ABCD$ be just a parallelogram and $\angle APD + \angle CPB = 180^\circ$ $\Longrightarrow$ circumcircles $(O_1), (O_3)$ of $\triangle APD, \triangle CPB$ are congruent. Translating $\triangle APD$ by $\overrightarrow{AB}$ into a $\triangle CP'B$ creates a cyclic quadrilateral $PBP'C$ and parallelogram $PABP'$ $\Longrightarrow$ circumcircles $(O_2), (O_3)$ of $\triangle BPA, \triangle CPB$ are also congruent. It follows that $\angle PCD = \angle DAP,$ which means that isogonal conjugate $P^*$ of $P$ WRT $\triangle ACD$ is on perpendicular bisector of $AC.$ As a result, the locus of $P$ is a rectangular circum-hyperbola of the parallelogram $ABCD,$ centered at its diagonal intersection $E.$ If $ABCD$ is a rhombus, this hyperbola degenerates to its diagonals $AC, BD.$
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isaacmeng
113 posts
isaacmeng
#3 h  May 5, 2020, 10:45 AM
Y by
What if P is not necessarily in the plane ABCD?
Z K Y
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Nari_Tom
116 posts
Nari_Tom
#4 h  Apr 4, 2025, 10:01 AM
Y by
There is a theorem states that if $P$ is a point lies in the plane of $ABCD$ and $\angle APD+\angle CPD=180^{\circ}$, then there exists a isogonal conjugate of $P$. Since $ABCD$ is a rhombus we can easily conclude that $P$ lies on $AC$ or $BD$.
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