1

Part of 1999 Hungary-Israel Binational

Problems(2)

Hungary-Israel Binational 1999\1

Source: Recursively defined polynomials

f(x)

is a given polynomial whose degree at least 2. Define the following polynomial-sequence: g_1(x)\equal{}f(x), g_{n\plus{}1}(x)\equal{}f(g_n(x)), for all

n \in N

r_n

be the average of

g_n(x)

's roots. If r_{19}\equal{}99, find

r_{99}

algebrapolynomialalgebra proposed

Hungary-Israel Binational 1999\4

Source: Prove that a sequence containts only integers

c

is a positive integer. Consider the following recursive sequence: a_1\equal{}c, a_{n\plus{}1}\equal{}ca_{n}\plus{}\sqrt{(c^2\minus{}1)(a_n^2\minus{}1)}, for all

n \in N

. Prove that all the terms of the sequence are positive integers.

inequalitiesalgebrapolynomialalgebra proposed