Concrete mathematics : a foundation for computer science

98 INTEGER FUNCTIONS
31 Prove or disprove: 1x1 + \yJ + Lx + y] 6 12x1 + [ZyIJ .
32 Let (Ix(( = min(x - 1x1, [xl -x) denote the distance from x to the nearest
integer. What is the value of
x 2kllx/2kJJ2 ?
k
(Note that this sum can be doubly infinite. For example, when x = l/3
the terms are nonaero as k + -oo and also as k + +oo.)
Exam problems
33 A circle, 2n - 1 units in diameter, has been drawn symmetrically on a
2n x 2n chessboard, illustrated here for n = 3:
a How many cells of the board contain a segment of the circle?
b Find a function f(n, k) such that exactly xc:: f(n, k) cells of the
board lie entirely within the circle.
34 Let f(n) = Et=, [lgkl.
Find a closed form for f(n) , when n 3 1.
L Provethatf(n)=n-l+f([n/2~)+f(~n/Z])foralln~l.
35 Simplify the formula \(n + 1 )‘n! e] mod n. Simplify it, but
36 Assuming that n is a nonnegative integer, find a closed form for the sum
don’t change the
value,
x
l