MathDB

3

Part of 2004 Nicolae Coculescu

Problems(4)

Fractional part Hermite-like identity

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12/14/2019
Prove the identity n−12=∑k=1n{m+k−1n}, \frac{n-1}{2}=\sum_{k=1}^n \left\{ \frac{m+k-1}{n} \right\} , where n≥2,m n\ge 2, m are natural numbers, and {} \{\} denotes the fractional part.
equationsHermitealgebrafractionalFloor
Complex modulus property

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12/14/2019
Let be three nonzero complex numbers a,b,c a,b,c satisfying ∣a∣=∣b∣=∣c∣=∣a+b+c−abcab+bc+ca−1∣. |a|=|b|=|c|=\left| \frac{a+b+c-abc}{ab+bc+ca-1} \right| . Prove that these three numbers have all modulus 1 1 or there are two distinct numbers among them whose sum is 0. 0.
Costel Anghel
complex numbersalgebra
X³+X+2I=0 (matriceal equation)

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12/14/2019
Solve in M2(R) \mathcal{M}_2(\mathbb{R}) the equation X3+X+2I=0. X^3+X+2I=0.
Florian Dumitrel
Matriceslinear algebraequationsCayley-Hamilton
Endomorphism with exactly one fixed point

Source:

12/14/2019
Let be a finite group G G having an endomorphism η \eta that has exactly one fixed point.
a) Demonstrate that the function f:G⟶G f:G\longrightarrow G defined as f(x)=x−1⋅η(x) f(x)=x^{-1}\cdot\eta (x) is bijective. b) Show that G G is commutative if the composition of the function f f from a) with itself is the identity function.
functiongroup theoryendomorphismsmorphism