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Prove that \[4 \sin (x) \sin (\omega x) \sin (\omega^2 x) = - (\sin (2x) + \sin (2 \omega x) + \sin (2 \omega^2 x))\]

Prove that

\[4 \sin (x) \sin (\omega x) \sin (\omega^2 x) = - (\sin (2x) + \sin (2 \omega x) + \sin (2 \omega^2 x))\]

\[\] Notation: \(\omega\) denotes a primitive cube root ...

I studied something after making that crazy thing. This time I will use the Euler's theorem which states that, if two positive integers \(a\) and \(b\) are relatively prime ...

Is the set of all positive integers which cannot be expressed as a sum of distinct triangular numbers finite?

Proposition:

Prove (or disprove) that for every integer \(n \ge 4\), there exists at least one ordered triplet \( (p_1 , n , p_2) \) where \(p_1\), \(n\) and \(p_2\) are in arithmetic progression ...

Topics covered are :

1) Number Theory (Elementary)

2) Combinatorics

3) Geometry

4) Calculus (Higher integration,Special Functions,partial differentiation,optimisation)

5) Algebra

6) Series & sequences ...

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