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Inspired by 2024 Fall LMT Guts
sqing   1
N 30 minutes ago by sqing
Source: Own
Let $x$, $y$, $z$ are pairwise distinct real numbers satisfying $x^2+y =y^2 +z = z^2+x. $ Prove that
$$(x+y)(y+z)(z+x)=-1$$Let $x$, $y$, $z$ are pairwise distinct real numbers satisfying $x^2+2y =y^2 +2z = z^2+2x. $ Prove that
$$(x+y)(y+z)(z+x)=-8$$
1 reply
sqing
37 minutes ago
sqing
30 minutes ago
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How many non-attacking pawns can be placed on a $n \times n$ chessboard?
DylanN   2
N 32 minutes ago by zRevenant
Source: 2019 Pan-African Shortlist - C1
A pawn is a chess piece which attacks the two squares diagonally in front if it. What is the maximum number of pawns which can be placed on an $n \times n$ chessboard such that no two pawns attack each other?
2 replies
DylanN
Jan 18, 2021
zRevenant
32 minutes ago
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Solve All 6 IMO 2024 Problems (42/42), New Framework Looking for Feedback
Blackhole.LightKing   0
an hour ago
Hi everyone,

I’ve been experimenting with a different way of approaching mathematical problem solving — a framework that emphasizes recursive structures and symbolic alignment rather than conventional step-by-step strategies.

Using this method, I recently attempted all six problems from IMO 2024 and was able to arrive at what I believe are valid full-mark solutions across the board (42/42 total score, by standard grading).

However, I don’t come from a formal competition background, so I’m sure there are gaps in clarity, communication, or even logic that I’m not fully aware of.

If anyone here is willing to take a look and provide feedback, I’d appreciate it — especially regarding:

The correctness and completeness of the proofs

Suggestions on how to make the ideas clearer or more elegant

Whether this approach has any broader potential or known parallels

I'm here to learn more and improve the presentation and thinking behind the work.

You can download the Solution here.

https://agi-origin.com/assets/pdf/AGI-Origin_IMO_2024_Solution.pdf


Thanks in advance,
— BlackholeLight0


0 replies
Blackhole.LightKing
an hour ago
0 replies
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Russian NT with a Ceiling
naman12   45
N an hour ago by InterLoop
Source: 2019 ISL N8
Let $a$ and $b$ be two positive integers. Prove that the integer
\[a^2+\left\lceil\frac{4a^2}b\right\rceil\]is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.)

Russia
45 replies
+1 w
naman12
Sep 22, 2020
InterLoop
an hour ago
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Excircle Tangency Points Concyclic with A
tastymath75025   35
N an hour ago by bin_sherlo
Source: USA Winter TST for IMO 2019, Problem 6, by Ankan Bhattacharya
Let $ABC$ be a triangle with incenter $I$, and let $D$ be a point on line $BC$ satisfying $\angle AID=90^{\circ}$. Let the excircle of triangle $ABC$ opposite the vertex $A$ be tangent to $\overline{BC}$ at $A_1$. Define points $B_1$ on $\overline{CA}$ and $C_1$ on $\overline{AB}$ analogously, using the excircles opposite $B$ and $C$, respectively.

Prove that if quadrilateral $AB_1A_1C_1$ is cyclic, then $\overline{AD}$ is tangent to the circumcircle of $\triangle DB_1C_1$.

Ankan Bhattacharya
35 replies
tastymath75025
Jan 21, 2019
bin_sherlo
an hour ago
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Inspired by SXTX (4)2025 Q712
sqing   0
an hour ago
Source: Own
Let $ a ,b,c>0 $ and $ (a+b)^2+2(b+c)^2+(c+a)^2=12. $ Prove that$$ abc(a+b+c) \leq \frac{9}{5} $$Let $ a ,b,c>0 $ and $ 2(a+b)^2+ (b+c)^2+2(c+a)^2=12. $ Prove that$$ abc(a+b+c) \leq \frac{9}{8} $$
0 replies
sqing
an hour ago
0 replies
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Domain swept by a parabola
Kunihiko_Chikaya   1
N an hour ago by Mathzeus1024
Source: 2015 The University of Tokyo entrance exam for Medicine, BS
For a positive real number $a$, consider the following parabola on the coordinate plane.
$C:\ y=ax^2+\frac{1-4a^2}{4a}$
When $a$ ranges over all positive real numbers, draw the domain of the set swept out by $C$.
1 reply
Kunihiko_Chikaya
Feb 25, 2015
Mathzeus1024
an hour ago
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AZE JBMO TST
IstekOlympiadTeam   5
N an hour ago by wh0nix
Source: AZE JBMO TST
Find all non-negative solutions to the equation $2013^x+2014^y=2015^z$
5 replies
IstekOlympiadTeam
May 2, 2015
wh0nix
an hour ago
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Find the minimum
sqing   1
N an hour ago by sqing
Source: SXTX Q616
In acute triangle $ABC$, Find the minimum of $ 2\tan A +9\tan B +17 \tan C .$
h h
In acute triangle $ABC$, Find the minimum of $ 4\tan A +7\tan B +14 \tan C .$
In acute triangle $ABC$. Prove that$$ 2\tan A +9\tan B +17 \tan C \geq 40 $$
1 reply
sqing
Jul 25, 2023
sqing
an hour ago
J
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Show that XD and AM meet on Gamma
MathStudent2002   91
N 2 hours ago by IndexLibrorumProhibitorum
Source: IMO Shortlist 2016, Geometry 2
Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$. The points $D$, $E$, $F$ are selected on sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$, $\overline{IE}\perp \overline{AI}$, and $\overline{IF}\perp \overline{AI}$. Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $\Gamma$.

Proposed by Evan Chen, Taiwan
91 replies
MathStudent2002
Jul 19, 2017
IndexLibrorumProhibitorum
2 hours ago
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a