Problem: Let , .Find the largest that satisfies:

T702. Let and Prove that S
T703. Let be real numbers such that . Find the minimum of
T704. Let and Prove that S

Let be an odd prime number and a sequence satisfy and Prove that always exists positive integer satisfying for all positive integers , then .

P/s: is "not congruence"

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

Let and Prove that
Solution:
and

Assume then
Equalityholds when

Let and . Prove that:

Solve the equation :

Let and Prove that

Problem. Let Prove that

Two circles and intersect at points and . A line through intersects and at points and , respectively. Line intersects at point , and line intersects at point . If is the circumcenter of , prove that .

Let and Prove that

Let be a triangle with incenter , and let be a point on side . Points and are chosen on lines and respectively such that is a parallelogram. Points and are chosen on side such that and are the angle bisectors of angles and respectively. Let be the circle tangent to segment , the extension of past , and the extension of past . Prove that is tangent to the circumcircle of triangle .