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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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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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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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Inspired by Jackson0423
sqing   0
8 minutes ago
Source: Own
Let $ a, b, c>0 $ and $ a^2 + b^2 =c(a + b). $ Prove that
$$ \frac{b^2 +bc+ c^2}{ a(a +b+ c)} \geq 2\sqrt 3-3$$
0 replies
1 viewing
sqing
8 minutes ago
0 replies
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What is the likelihood the last card left in the deck is black?
BEHZOD_UZ   1
N 40 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
an hour ago
sami1618
40 minutes ago
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abc(a+b+c)=3, show that prod(a+b)>=8 [Indian RMO 2012(b) Q4]
Potla   31
N an hour 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
an hour ago
J
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AGI-Origin Solves Full IMO 2020–2024 Benchmark Without Solver (30/30) beat Alpha
AGI-Origin   10
N an hour 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
an hour ago
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Medium geometry with AH diameter circle
v_Enhance   93
N Yesterday at 10:36 AM by waterbottle432
Source: USA TSTST 2016 Problem 2, by Evan Chen
Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Denote by $M$, $N$ the midpoints of $\overline{AH}$, $\overline{BC}$. Suppose the circle $\gamma$ with diameter $\overline{AH}$ meets the circumcircle of $ABC$ at $G \neq A$, and meets line $AN$ at a point $Q \neq A$. The tangent to $\gamma$ at $G$ meets line $OM$ at $P$. Show that the circumcircles of $\triangle GNQ$ and $\triangle MBC$ intersect at a point $T$ on $\overline{PN}$.

Proposed by Evan Chen
93 replies
v_Enhance
Jun 28, 2016
waterbottle432
Yesterday at 10:36 AM
J
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Reflected point lies on radical axis
Mahdi_Mashayekhi   3
N Apr 19, 2025 by Mahdi_Mashayekhi
Source: Iran 2025 second round P4
Given is an acute and scalene triangle $ABC$ with circumcenter $O$. $BO$ and $CO$ intersect the altitude from $A$ to $BC$ at points $P$ and $Q$ respectively. $X$ is the circumcenter of triangle $OPQ$ and $O'$ is the reflection of $O$ over $BC$. $Y$ is the second intersection of circumcircles of triangles $BXP$ and $CXQ$. Show that $X,Y,O'$ are collinear.
3 replies
Mahdi_Mashayekhi
Apr 19, 2025
Mahdi_Mashayekhi
Apr 19, 2025
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A mediane as a radical axis
breloje17fr   0
Apr 19, 2025
Hello, ladies and gentlemen
Let ABC be a triangle, and D, E and F the middles of the sides BC, CA and AB respectively. The perpendicular bissector of CA intersects the line AB at E' and the bissector of the A angle at K, and the perpendicular bissector of AB intersects the line AC at F' and the bissector of the A angle at J. The two circles passing through J, F and E' and through K, E and F' intersect each other at P and Q.
Show that the radical axis of these circles is the A-mediane of ABC.
0 replies
breloje17fr
Apr 19, 2025
0 replies
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Concurrence in Cyclic Quadrilateral
GrantStar   38
N Apr 17, 2025 by wu2481632
Source: IMO Shortlist 2023 G3
Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$.

Prove that lines $AD, PM$, and $BC$ are concurrent.
38 replies
GrantStar
Jul 17, 2024
wu2481632
Apr 17, 2025
J
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Circles with same radical axis
Jalil_Huseynov   9
N Apr 17, 2025 by Nari_Tom
Source: DGO 2021, Individual stage, Day2 P3
Let $O$ be the circumcenter of triangle $ABC$. The altitudes from $A, B, C$ of triangle $ABC$ intersects the circumcircle of the triangle $ABC$ at $A_1, B_1, C_1$ respectively. $AO, BO, CO$ meets $BC, CA, AB$ at $A_2, B_2, C_2$ respectively. Prove that the circumcircles of triangles $AA_1A_2, BB_1B_2, CC_1C_2$ share two common points.

Proporsed by wassupevery1
9 replies
Jalil_Huseynov
Dec 26, 2021
Nari_Tom
Apr 17, 2025
J
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2011 Japan Mathematical Olympiad Finals Problem 1
Kunihiko_Chikaya   20
N Apr 14, 2025 by zhoujef000
Source: Japanese MO Finals 2011
Given an acute triangle $ABC$ with the midpoint $M$ of $BC$. Draw the perpendicular $HP$ from the orthocenter $H$ of $ABC$ to $AM$.
Show that $AM\cdot PM=BM^2$.
20 replies
Kunihiko_Chikaya
Feb 11, 2011
zhoujef000
Apr 14, 2025
J
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IMO 2008, Question 1
orl   154
N Apr 8, 2025 by eg4334
Source: IMO Shortlist 2008, G1
Let $ H$ be the orthocenter of an acute-angled triangle $ ABC$. The circle $ \Gamma_{A}$ centered at the midpoint of $ BC$ and passing through $ H$ intersects the sideline $ BC$ at points $ A_{1}$ and $ A_{2}$. Similarly, define the points $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$.

Prove that the six points $ A_{1}$, $ A_{2}$, $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$ are concyclic.

Author: Andrey Gavrilyuk, Russia
154 replies
orl
Jul 16, 2008
eg4334
Apr 8, 2025
J
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Lines pass through a common point
April   4
N Apr 7, 2025 by Nari_Tom
Source: Baltic Way 2008, Problem 18
Let $ AB$ be a diameter of a circle $ S$, and let $ L$ be the tangent at $ A$. Furthermore, let $ c$ be a fixed, positive real, and consider all pairs of points $ X$ and $ Y$ lying on $ L$, on opposite sides of $ A$, such that $ |AX|\cdot |AY| = c$. The lines $ BX$ and $ BY$ intersect $ S$ at points $ P$ and $ Q$, respectively. Show that all the lines $ PQ$ pass through a common point.
4 replies
April
Nov 23, 2008
Nari_Tom
Apr 7, 2025
J
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perpendicularity involving ex and incenter
Erken   19
N Apr 6, 2025 by Primeniyazidayi
Source: Kazakhstan NO 2008 problem 2
Suppose that $ B_1$ is the midpoint of the arc $ AC$, containing $ B$, in the circumcircle of $ \triangle ABC$, and let $ I_b$ be the $ B$-excircle's center. Assume that the external angle bisector of $ \angle ABC$ intersects $ AC$ at $ B_2$. Prove that $ B_2I$ is perpendicular to $ B_1I_B$, where $ I$ is the incenter of $ \triangle ABC$.
19 replies
Erken
Dec 24, 2008
Primeniyazidayi
Apr 6, 2025
J
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Geo with unnecessary condition
egxa   8
N Apr 4, 2025 by ErTeeEs06
Source: Turkey Olympic Revenge 2024 P4
Let the circumcircle of a triangle $ABC$ be $\Gamma$. The tangents to $\Gamma$ at $B,C$ meet at point $E$. For a point $F$ on line $BC$ which is not on the segment $BC$, let the midpoint of $EF$ be $G$. Lines $GB,GC$ meet $\Gamma$ again at points $I,H$ respectively. Let $M$ be the midpoint of $BC$. Prove that the points $F,I,H,M$ lie on a circle.

Proposed by Mehmet Can Baştemir
8 replies
egxa
Aug 6, 2024
ErTeeEs06
Apr 4, 2025
J
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J
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four variables inequality
JK1603JK   1
N Apr 5, 2025 by arqady
Source: unknown?
Prove that $$27(a^4+b^4+c^4+d^4)+148abcd\ge (a+b+c+d)^4,\ \ \forall a,b,c,d\ge 0.$$
1 reply
JK1603JK
Apr 4, 2025
arqady
Apr 5, 2025
J
four variables inequality
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Reveal topic tags
inequalities
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JK1603JK
49 posts
JK1603JK
#1 h  Apr 4, 2025, 4:26 PM
Y by
Prove that $$27(a^4+b^4+c^4+d^4)+148abcd\ge (a+b+c+d)^4,\ \ \forall a,b,c,d\ge 0.$$
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arqady
30207 posts
arqady
#2 h  Apr 5, 2025, 1:38 PM
Y by
JK1603JK wrote:
Prove that $$27(a^4+b^4+c^4+d^4)+148abcd\ge (a+b+c+d)^4,\ \ \forall a,b,c,d\ge 0.$$
Use SOS.
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