Concrete mathematics : a foundation for computer science

You know you’re
in college when the
book doesn’t tell
you how to pronounce
‘Dirichlet’.
Exercises
Warmups
3 EXERCISES 95
1 When we analyzed the Josephus problem in Chapter 1, we represented
an arbitrary positive integer n in the form n = 2m + 1, where 0 < 1 < 2”.
Give explicit formulas for 1 and m as functions of n, using floor and/or
ceiling brackets.
2 What is a formula for the nearest integer to a given real number x? In case
of ties, when x is exactly halfway between two integers, give an expression
that rounds (a) up-that is, to [xl; (b) down-that is, to Lx].
3 Evaluate 1 \m&]n/a] , w hen
m and n are positive integers and a is an
irrational number greater than n.
4 The text describes problems at levels 1 through 5. What is a level 0
problem? (This, by the way, is not a level 0 problem.)
5 Find a necessary and sufficient condition that LnxJ = n[xJ , when n is a
positive integer. (Your condition should involve {x}.)
6 Can something interesting be said about Lf(x)J when f(x) is a continuous,
monotonically decreasing function that takes integer values only when
x is an integer?
‘7 Solve the recurrence
X, = n, for 0 6 n < m;
x, = x,-,+1, for n 3 m.
8 Prove the Dirichlet box principle: If n objects are put into m boxes,
some box must contain 3 [n/ml objects, and some box must contain
6 lnhl.
9 Egyptian mathematicians in 1800 B.C. represented rational numbers between
0 and 1 as sums of unit fractions 1 /xl + . . . + 1 /xk, where the x’s
were distinct positive integers. For example, they wrote $ + &, instead
of 5. Prove that it is always possible to do this in a systematic way: If
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