IMO Shortlist 1996 problem N3
Dodao/la:
arhiva2. travnja 2012. A finite sequence of integers
![a_0, a_1, \ldots, a_n](/media/m/4/5/a/45a875189a0ae4e5fbc7fd19c6ea5da3.png)
is called quadratic if for each
![i](/media/m/3/2/d/32d270270062c6863fe475c6a99da9fc.png)
in the set
![\{1,2 \ldots, n\}](/media/m/e/2/9/e2912619e481d147c60603264a61af13.png)
we have the equality
a.) Prove that any two integers
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
and
![c,](/media/m/a/f/e/afe009c3cb1977d857fb63152c9b04bd.png)
there exists a natural number
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
and a quadratic sequence with
![a_0 = b](/media/m/4/b/9/4b989a88de20c94c22b50b20e7d365fc.png)
and
b.) Find the smallest natural number
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
for which there exists a quadratic sequence with
![a_0 = 0](/media/m/0/4/7/0471eca23d871c11edd98d44cf6ecffb.png)
and
%V0
A finite sequence of integers $a_0, a_1, \ldots, a_n$ is called quadratic if for each $i$ in the set $\{1,2 \ldots, n\}$ we have the equality $|a_i - a_{i-1}| = i^2.$
a.) Prove that any two integers $b$ and $c,$ there exists a natural number $n$ and a quadratic sequence with $a_0 = b$ and $a_n = c.$
b.) Find the smallest natural number $n$ for which there exists a quadratic sequence with $a_0 = 0$ and $a_n = 1996.$
Izvor: Međunarodna matematička olimpijada, shortlist 1996