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Brazil Undergrad MO
2008 Brazil Undergrad MO
2008 Brazil Undergrad MO
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Brazil Undergrad MO
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Real sequence
Prove that there are real numbers
a
1
,
a
2
,
.
.
a_1, a_2, ..
a
1
,
a
2
,
..
such that: i) For all real numbers x, the serie f(x) \equal{} \sum_{n \equal{} 1}^\infty a_nx^n converge; ii) f is a bijection of R to R; iii) f'(x) >0; iv) f(Q) = A, where Q is the set of rational numbers and A is the set of algebraic numbers.