Math 289 โ Winter 2006 โ Problem Set 3
Due February 1, 2006
1.(10) Can a number nconsisting of 600 sixes and some zeros be a square?
2.(10) A class with nstudents meets for the first time. Everyone shakes hands with
everyone else. Prove that at every moment during the greeting ceremony there are always
two students who have shaken the same number of hands up to that time.
3.(10) Let a > 0. Define a0=โaand an+1 =โa+an. Find lim
nโโ
an.
4.(10) Twenty pairwise distinct positive integers are all smaller than 70. Prove that among
their pairwise differences there are four equal numbers.
5.(15) In the sequence 1,9,3,5,8,5,1,9,3,8,1, . . ., every digit from the fifth on is the sum
of the preceding four digits modulo 10 (i.e. 1 + 9 + 3 + 5 = 18 โ8, 9 + 3 + 5 + 8 = 25 โ5,
etc.). Does one of the following words ever occur in the sequence
(a) 1234 (b) 3289 (c) 1935 (d) 2193 ?
6.(15) The length of each side of a convex quadrilateral ABCD is less than 24. Let Pbe
any point inside ABCD. Prove that there exists a vertex, say A, such that |PA|<17.
7.(15) There are nidentical cars on a circular track. among all of them, they have just
enough gas for one car to complete a lap. Show that there is a car which can complete a
lap by collecting gas from the other cars on its way around.
[Hint: You may want to use Mathematical Induction.]
8.(20) We define an infinite binary sequence as follows: Start with 0 and repeatedly replace
each 0 by 001 and each 1 by 0.
(a) Is the sequence periodic?
(b) What is the 1,000th digit of the sequence?
(c) What is the place number of the 10,000th one in the sequence?
(d) Try to find a formula for the positions of the ones (3,6,10,13, . . .) and a formula for
the positions of the zeros.
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