1

Part of 1998 Hungary-Israel Binational

Problems(2)

solve 5^x - 3^y = 16

Source: 9th-th Hungary-Israel Binational Mathematical Competition 1998

Find all positive integers

x

y

5^{x}-3^{y}= 16

modular arithmeticnumber theory proposednumber theory

Source: 9th-th Hungary-Israel Binational Mathematical Competition 1998

A player is playing the following game. In each turn he ﬂips a coin and guesses the outcome. If his guess is correct, he gains

1

point; otherwise he loses all his points. Initially the player has no points, and plays the game until he has

2

points. (a) Find the probability

p_{n}

that the game ends after exactly

n

ﬂips. (b) What is the expected number of ﬂips needed to ﬁnish the game?

probabilityexpected valuecombinatorics unsolvedcombinatorics