IMO Shortlist 1983 problem 19
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arhiva2. travnja 2012. Let
![(F_n)_{n\geq 1}](/media/m/f/b/7/fb77e3a318ded9a0c9eb19b549cd5da9.png)
be the Fibonacci sequence
![F_1 = F_2 = 1, F_{n+2} = F_{n+1} + F_n (n \geq 1),](/media/m/2/d/0/2d092f3b18e18461b4f32c1c090113b3.png)
and
![P(x)](/media/m/c/d/7/cd7664875343d44cd5f96a566b582b0e.png)
the polynomial of degree
![990](/media/m/a/f/8/af8bfc964d04ac8bfafee7a8d0974f90.png)
satisfying
![P(k) = F_k, \qquad \text{ for } k = 992, . . . , 1982.](/media/m/d/5/2/d52bb934a13052984f540d41ef199476.png)
Prove that
%V0
Let $(F_n)_{n\geq 1}$ be the Fibonacci sequence $F_1 = F_2 = 1, F_{n+2} = F_{n+1} + F_n (n \geq 1),$ and $P(x)$ the polynomial of degree $990$ satisfying
$$P(k) = F_k, \qquad \text{ for } k = 992, . . . , 1982.$$
Prove that $P(1983) = F_{1983} - 1.$
Izvor: Međunarodna matematička olimpijada, shortlist 1983