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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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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…