MathDB

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Part of 2014 Saint Petersburg Mathematical Olympiad

Problems(2)

Building function

Source: St Petersburg Olympiad 2014, Grade 11, P1

10/24/2017
Let f(x)f(x) is such function, that f(x)=1f(x)=1 for integer xx and f(x)=0f(x)=0 for non integer xx. Build such function using only variable xx, integer numbers, and operations +,−,∗,/,[.]+,-,*,/,[.](plus, minus, multiply,divide and integer part)
number theoryfunction
Algebra with trinomial

Source: St Petersburg Olympiad 2014, Grade 10, P1

10/26/2017
f(x)f(x) is square polynomial and a≠ba \neq b such that f(a)=b,f(b)=af(a)=b,f(b)=a. Prove that there is not other pair (c,d)(c,d) that f(c)=d,f(d)=cf(c)=d,f(d)=c
algebra