My name is Ziv. I am a high school Chinese student who love exploring mathematics! Currently, my favorite fields include complex analysis and abstract algebra!

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1. Argument by Contradiction

Problem 1: Determine all functions satisfying     for all positive integers and . idea and solution: We could attempt some functions. We find that constant function works, does not work. does not work. This prompt us, maybe the only possible solution is ? This is because the left and right hand side has different…

April 2025
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  • 1. Argument by Contradiction

    Problem 1: Determine all functions satisfying     for all positive integers and . idea and solution: We could attempt some functions. We find that constant function works, does not work. does not work. This prompt us, maybe the only possible solution is ? This is because the left and right hand side has different […]

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  • How to understand Weierstrass infinite product?

    One of the famous expression of is     Here, we are writing as a some product of polynomials with zeros apparently shown. Remember that the right hand side has to be carefully chose to ensure its convergence. We are remained to consider: Can be factorize every entire function in the form of product of […]

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  • Some thoughts on proving Fourier Inversion(Complex analysis method)

    In complex analysis, we have the following well-known theorem: Theorem If , then the Fourier inversion holds, namely, given     We have     The proof is not easy for a beginner who just touched this theorem. As for myself, I spent a tons of time on managing the proof. So I would like […]

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  • Harmonic Number(2)

    Our goal is to prove the following theorem by Carlo Sanna. The proof is adapted and changed slightly from his paper. Theorem. For any prime number and any , we have     Lemma 1 If is prime, , and , then     and     Proof. . Each of the sum is by […]

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  • Harmonic Number (Starting from an AMC problem) (1)

    The -th Harmonic number, denoted is defined by the sum of the reciprocals of the first  natural numbers:     This number has quite interesting properties. On thing we could consider when is finite is the quotient of the sum. If we write it as     Then many problems raise. Here, we have a problem […]

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  • An inequality problem

    Let be a continuous integrable real function on , then     Solution 1 By Cauchy-Schwarz inequality,     Solution 2 By Jensen’s inequality, since is a convex function, we have     So     Solution 3 By Hölder’s Inequality, we get     Choose , we get the case of solution 1. We […]

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  • Abstract Algebra – Sylow Theorems and Its applications

    In abstract algebra, finite group theory is in its core. Now from Lagrange theorem, we have Lemma 1 Let be a subgroup of a finite group . Then the order of is a divisor of the order of . This theorem states the necessary condition for a finite group to be a subgroup of a […]

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  • An approximation problem

    Problem: Let us call an irrational number well approximated by rational numbers if for any natural numbers there exists a rational number such that     Solution: 1. Pick     Why does it work ? Let’s say we let . Then we have     and     We now proceed in proving that […]

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  • On IMO2017 P2

    I did IMO 2017 P2 with my friends yesterday. Now I would like to share my thoughts on this problem. Problem: Let be the set of real numbers. Determine all functions such that, for any real numbers and ,     The initial step must be taking certain values, testing its properties and then start […]

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