Math 289 – Fall 2006 – Problem Set 8
Due November 2, 2006
1.(10) Let Bnis an n×nmatrix with entries 1,2, . . . , n2, defined as
B2=1 2
3 4, B3=
123
456
789
, B4=
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
, . . .
We choose nentries of Bnsuch that exactly one entry is chosen in each column
and each row and sum them up. What are all the possible values of this sum?
2.(10) Let mand nare positive integers and let a0, a1, . . . , am−1and b0, b1, . . . , bn−1
are real non-negative numbers such that
(a0+a1x+· · ·+am−1xm−1+xm)·(b0+b1x+· · ·+bn−1xn−1+xn) = 1+x+x2+· · ·+xm+n.
Prove that each aiand each bjequals either 0 or 1.
3.(15) Let Cbe a unit cube with a uniform density. One cuts an eight of a unit
sphere with a center in one of the vertices of the cube from C. Where is the center
of mass of the remaining part of C?
4.(15) Suppose P∞
n=1 anconverges. Do the following re-ordered sums have to con-
verge as well?
(a) a1+a2+ (a4+a3)+(a8+a7+a6+a5)+(a16 +a15 +· · · +a9) + a32 +. . .
(b) a1+a2+ (a3+a4) + (a5+a7+a6+a8) + (a9+a11 +a13 +a15 +a10 +a12 +
a14 +a16) + a17 +a19 +. . . .
