2

Part of 2008 Hungary-Israel Binational

Problems(2)

Hungary-Israel Binational 2008\2

Source:

For every natural number

t

f(t)

is the probability that if a fair coin is tossed

t

times, the number of times we get heads is 2008 more than the number of tails. What is the value of

t

f(t)

attains its maximum? (if there is more than one, describe all of them)

probabilitycombinatorics proposedcombinatoricscoins

Hungary-Israel Binational 2008\5

Source: recursive sequence, prove that the terms are integers

a_n

is defined as follows: a_0\equal{}1, a_1\equal{}1, a_{n\plus{}1}\equal{}\frac{1\plus{}a_{n}^2}{a_{n\minus{}1}}. Prove that all the terms of the sequence are integers.

inductionalgebra proposedalgebra