2

Part of 1995 Vietnam National Olympiad

Problems(2)

Sequence of numbers

Source: Vietnam NMO 1995, Problem 2

The sequence (a_n) is defined as follows: a_0\equal{}1, a_1\equal{}3 For

n\ge 2

, a_{n\plus{}2}\equal{}a_{n\plus{}1}\plus{}9a_n if n is even, a_{n\plus{}2}\equal{}9a_{n\plus{}1}\plus{}5a_n if n is odd. Prove that 1) (a_{1995})^2\plus{}(a_{1996})^2\plus{}...\plus{}(a_{2000})^2 is divisible by 20 2) a_{2n\plus{}1} is not a perfect square for every natural numbers

n

number theory proposednumber theory

greater than 1995

Source: Vietnam NMO 1995, Problem 5

Find all poltnomials

P(x)

with real coefficients satisfying: For all

a>1995

, the number of real roots of P(x)\equal{}a (incuding multiplicity of each root) is greater than 1995, and every roots are greater than 1995.

algebrapolynomialalgebra proposed