3 var inquality

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

3 replies

+2 w

Blackhole.LightKing

3 hours ago

DottedCaculator

18 minutes ago

2 var inequalities

sqing 3

N 21 minutes ago by sqing

Source: Own

Let $a,b> 0$ and $a+b\leq 2ab .$ Prove that
$\frac{ a + b }{ a^2(1+ b^2)} \leq \sqrt 5-1$ $\frac{ a +ab+ b }{ a^2(1+ b^2)} \leq \frac{3(\sqrt5-1)}{2}$ $\frac{ a +a^2b^2+ b }{ a^2(1+ b^2)} \leq2$ Solution:
$a\ge\frac{b}{2b-1}, b>\frac12$ and $\frac{ a +a^2b^2+ b }{ a^2(1+ b^2)} \le\frac{2ab+a^2b^2}{a^2(1+b^2)}=1+\frac{2b-a}{a(1+b^2)} \le 1+\frac{4b-3}{b^2+1}$

Assume $u=4b-3>0$ then $\frac{ a +a^2b^2+ b }{ a^2(1+ b^2)} \le 1+\frac{16u}{u^2+6u+25} =2+ \frac{16}{6+u+\frac{25}u} \le 3$
Equalityholds when $a=\frac{2}{3},b=2.$

3 replies

sqing

Yesterday at 1:13 PM

sqing

21 minutes ago

hard problem

Cobedangiu 8

N 21 minutes ago by ReticulatedPython

Let $a,b,c>0$ and $a+b+c=3$ . Prove that:
$\dfrac{4}{a+b}+\dfrac{4}{b+c}+\dfrac{4}{c+a} \le \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+3$

8 replies

Cobedangiu

Apr 21, 2025

ReticulatedPython

21 minutes ago

Irrational equation

giangtruong13 3

N 26 minutes ago by navier3072

Solve the equation : $(\sqrt{x}+1)[2-(x-6)\sqrt{x-3}]=x+8$

3 replies

giangtruong13

2 hours ago

navier3072

26 minutes ago

2 var inequalities

sqing 0

31 minutes ago

Source: Own

Let $a,b> 0$ and $a+b\leq 3ab .$ Prove that
$\frac{ a + b }{ a^2(1+ 3b^2)} \leq \frac{3}{2}$ $\frac{ a - ab+ b }{ a^2(1+ 3b^2)} \leq 1$ $\frac{ a + 3ab+ b }{ a^2(1+ 3b^2)} \leq 3$ $\frac{ a -2ab+ b }{ a^2(1+ b^2)}\leq \sqrt{\frac{5}{2}}-\frac{1}{2}$ $\frac{ a +ab+ b }{ a^2(1+ b^2)} \leq 2(\sqrt{10}-1)$ $\frac{ a -2a^2b^2+ b }{ a^2(1+ b^2)}\leq \frac{\sqrt{82}-5}{2}$

0 replies

sqing

31 minutes ago

0 replies

Non-negative real variables inequality

KhuongTrang 0

33 minutes ago

Source: own

Problem. Let $a,b,c\ge 0: ab+bc+ca>0.$ Prove that $\color{blue}{\frac{\left(2ab+ca+cb\right)^{2}}{a^{2}+4ab+b^{2}}+\frac{\left(2bc+ab+ac\right)^{2}}{b^{2}+4bc+c^{2}}+\frac{\left(2ca+bc+ba\right)^{2}}{c^{2}+4ca+a^{2}}\ge \frac{8(ab+bc+ca)}{3}.}$

0 replies

KhuongTrang

33 minutes ago

0 replies

circle geometry showing perpendicularity

Kyj9981 4

N 44 minutes ago by cj13609517288

Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$ . A line through $B$ intersects $\omega_1$ and $\omega_2$ at points $C$ and $D$ , respectively. Line $AD$ intersects $\omega_1$ at point $E \neq A$ , and line $AC$ intersects $\omega_2$ at point $F \neq A$ . If $O$ is the circumcenter of $\triangle AEF$ , prove that $OB \perp CD$ .

4 replies

1 viewing

Kyj9981

Mar 18, 2025

cj13609517288

44 minutes ago

Prove excircle is tangent to circumcircle

sarjinius 8

N an hour ago by Lyzstudent

Source: Philippine Mathematical Olympiad 2025 P4

Let $ABC$ be a triangle with incenter $I$ , and let $D$ be a point on side $BC$ . Points $X$ and $Y$ are chosen on lines $BI$ and $CI$ respectively such that $DXIY$ is a parallelogram. Points $E$ and $F$ are chosen on side $BC$ such that $AX$ and $AY$ are the angle bisectors of angles $\angle BAE$ and $\angle CAF$ respectively. Let $\omega$ be the circle tangent to segment $EF$ , the extension of $AE$ past $E$ , and the extension of $AF$ past $F$ . Prove that $\omega$ is tangent to the circumcircle of triangle $ABC$ .

8 replies

sarjinius

Mar 9, 2025

Lyzstudent

an hour ago

IMO Shortlist 2014 N6

hajimbrak 28

N an hour ago by MajesticCheese

Let $a_1 < a_2 < \cdots <a_n$ be pairwise coprime positive integers with $a_1$ being prime and $a_1 \ge n + 2$ . On the segment $I = [0, a_1 a_2 \cdots a_n ]$ of the real line, mark all integers that are divisible by at least one of the numbers $a_1 , \ldots , a_n$ . These points split $I$ into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by $a_1$ .

Proposed by Serbia

28 replies

hajimbrak

Jul 11, 2015

MajesticCheese

an hour ago

3 knightlike moves is enough

sarjinius 3

N an hour ago by JollyEggsBanana

Source: Philippine Mathematical Olympiad 2025 P6

An ant is on the Cartesian plane. In a single move, the ant selects a positive integer $k$ , then either travels [list]
[*] $k$ units vertically (up or down) and $2k$ units horizontally (left or right); or
[*] $k$ units horizontally (left or right) and $2k$ units vertically (up or down).
[/list]
Thus, for any $k$ , the ant can choose to go to one of eight possible points.
Prove that, for any integers $a$ and $b$ , the ant can travel from $(0, 0)$ to $(a, b)$ using at most $3$ moves.

3 replies

sarjinius

Mar 9, 2025

JollyEggsBanana

an hour ago

3 var inquality

sqing 1

N Apr 8, 2025 by hashtagmath

Source: Own

Let $a,b,c>0$ and $\dfrac{a}{bc}+\dfrac{2b}{ca}+\dfrac{5c}{ab}\leq 12.$ Prove that $a^2+b^2+c^2\geq 1$

1 reply

sqing

Apr 6, 2025

hashtagmath

Apr 8, 2025

3 var inquality

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Source: Own

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sqing

41809 posts

sqing

#1 h Apr 6, 2025, 1:11 PM

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Let $a,b,c>0$ and $\dfrac{a}{bc}+\dfrac{2b}{ca}+\dfrac{5c}{ab}\leq 12.$ Prove that $a^2+b^2+c^2\geq 1$

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hashtagmath

1602 posts

hashtagmath

#2 h Apr 8, 2025, 7:41 PM

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Dear sqing,

I have noticed thousands upon thousands of inequalities that you have posted to HSO and was wondering where you get the inspiration, imagination, and even the validation that such inequalities are true? Also, what do you find particularly appealing and important about specifically inequalities rather than other branches of mathematics?

Thank you

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