IMO Shortlist 1992 problem 19
Dodao/la:
arhiva2. travnja 2012. Let
![f(x) = x^8 + 4x^6 + 2x^4 + 28x^2 + 1.](/media/m/d/4/a/d4a72d354edaac578333b9306ff0a7cb.png)
Let
![p > 3](/media/m/d/1/4/d14b51fb9611c16eac843d287f8c9fdf.png)
be a prime and suppose there exists an integer
![z](/media/m/d/2/4/d241a79f1fdd0ce9a8f3f91570ba5d62.png)
such that
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
divides
![f(z).](/media/m/e/b/1/eb1e70fbebece1fdf3394836255c976e.png)
Prove that there exist integers
![z_1, z_2, \ldots, z_8](/media/m/e/d/3/ed3081b2124d04b4a98f962a3b1a2e2b.png)
such that if
![g(x) = (x - z_1)(x - z_2) \cdot \ldots \cdot (x - z_8),](/media/m/5/f/c/5fc97357c4505eb91bd5b01477bbcc40.png)
then all coefficients of
![f(x) - g(x)](/media/m/c/1/3/c13eadaa4ff8e7fa326c4cd4babf3e77.png)
are divisible by
%V0
Let $f(x) = x^8 + 4x^6 + 2x^4 + 28x^2 + 1.$ Let $p > 3$ be a prime and suppose there exists an integer $z$ such that $p$ divides $f(z).$ Prove that there exist integers $z_1, z_2, \ldots, z_8$ such that if $$g(x) = (x - z_1)(x - z_2) \cdot \ldots \cdot (x - z_8),$$ then all coefficients of $f(x) - g(x)$ are divisible by $p.$
Izvor: Međunarodna matematička olimpijada, shortlist 1992