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Part of 2015 Romania National Olympiad

Problems(4)

artihmetic progression of square roots

Source: Romanian National Olympiad 2015, grade IX, p.1

Show that among the square roots of the first

2015

natural numbers, we cannot choose an arithmetic sequence composed of

45

arithmetic sequencealgebranumber theory

complex equation involving cyclic sum

Source: Romania National Olympiad 2015, grade x, p.1

Find all triplets

(a,b,c)

of nonzero complex numbers having the same absolute value and which verify the equality:

\frac{a}{b} +\frac{b}{c}+\frac{c}{a} =-1

complex numbersabsolute valuealgebra

A characterization of the zero function

Source: Romanian National Olympiad, grade xi, p.1

Find all differentiable functions

f:\mathbb{R}\longrightarrow\mathbb{R}

that verify the conditions: \text{(i)} \forall x\in\mathbb{Z} f'(x) =0
\text{(ii)} \forall x\in\mathbb{R} f'(x)=0\implies f(x)=0

functionreal analysis

A characterization of Boolean rings

Source: Romania National Olympiad 2015, grade xii, p.1

Let be a ring that has the property that all its elements are the product of two idempotent elements of it. Show that: a)

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is the only unit of this ring. b) this ring is Boolean.

superior algebraRing Theoryidempotency