Pucc-H:dlk@pur-ee.UUCP (Hao-Nhien Vu) (12/09/83)
Here is the promised part B. Have more fun.
Hao-Nhien Vu (pur-ee!Pucc-H:dlk, or pur-ee!vu)
Ps. I don't have the answer to all of these. Only those I, and/or other people
I know, got.
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B-1:
Let v a vertex (corner) of a cube C with edges of length 4. Let S
be the largest sphere that can be inscribed in C. Let R be the region
consisting of all points p between S and C such that p is closer to v than
any other vertex of the cube. Find the volume of R.
B-2:
For positive integers n, let C(n) be the number of representations
of n as a sum of nonincreasing powers of 2, where no power is used more than
three times. For example, C(8) = 5 since the representations for 8 are:
8, 4+4, 4+2+2, 4+2+1+1, 2+2+2+1+1
Prove or disprove that there is a polynomial P(x) such that C(n) = [P(n)]
for all positive integer n ; here [u] denotes the greatest integer less than
or equal to u.
B-3:
Assume that the differential equation
y''' + p(x)y'' + q(x)y' + r(x)y = 0
has solutions y1(x), y2(x), and y3(x) on the whole real line such that
y1(x) **2 + y2(x) **2 + y3(x) **2 = 1
for all real x. Let
f(x) = (y1'(x))**2 + (y2'(x))**2 + (y3'(x))**2 .
Find constants A and B such that f(x) is a solution to the differential
equation
y' + A p(x)y = B r(x)
B-4:
Let f(n) = n + [sqrt(n)] where [x] is the largest integer less than or
equal to x. Prove that, for every positive integer m, the sequence
m, f(m), f(f(m)), f(f(f(m))), ...
contains at least one square of an integer.
B-5:
Let || u || denote the distance from the real number u to the nearest
integer. (For example, || 2.8 || = 0.2 = || 3.2 || ) For positive integers
n, let
n
a(n) = (1/n) Int ||n/x|| dx
1
Determine lim{n-->oo} a(n) . You may assume the identity
2 2 4 4 6 6 8 8
- - - - - - - - ..... = pi / 2
1 3 3 5 5 7 7 9
B-6:
Let k be a positive integer, let m = 2**k + 1 , and let r != 1 be
a complex root of z**m - 1 = 0. Prove that there exist polynomials P(z)
and Q(z) with integer coefficients such that
( P(r) )**2 + ( Q(r) )**2 = -1.