Pucc-H:dlk@pur-ee.UUCP (Hao-Nhien Vu) (12/06/83)
The forty-fourth annual William Loweel Putnam Mathematical Competition
took place today (Saturday December 03, 1983). For those who don't know it,
the competition is organized throught the U.S. and Canada for undergraduates.
It is sponsored by the Mathematical Association of America. Official solutions
are posted in the Math Monthly and usually appear eleven month after the
competition takes place ( ~ November or so).
Contestants have 3 hours in the morning to do set A, and 3 hours in the
afternoon to do set B. Here are the problems. Have fun. Set B will be posted
separately to shorten the article's length.
Hao-Nhien Vu (pur-ee!Pucc-H:dlk, or pur-ee!vu)
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A-1:
How many positive integers n are there such that n is an exact
divisor of at least one of the numbers
10**40, 20**30 ?
A-2:
The hands of an accurate clock have lengths 3 and 4. Find the
distance between the tips of the hands when that distance is increasing
most rapidly.
A-3:
Let p be in the set {3, 5, 7, 11, ...} of odd primes and let
F(n) = 1 + 2n + 3n**2 + 4n**3 + ... + (p-1)n**(p-2)
Prove that if a and b are distinct integers in { 0, 1, 2, ..., p-1}
then F(a) and F(b) are not congruent modulo p, that is, F(a) - F(b) is not
exactly divisible by p.
A-4:
Let k be a positive integer and let m = 6k - 1.
Let
2k-1 j+1
S(m) = sum (-1) C( 3j-1 , m)
j=1
For example, with k = 3,
S(17) = C(2,17) - C(5,17) + C(8,17) - C(11,17) + C(14,17)
m!
Prove that S(m) is never zero.[As usual, C(r,m) = ------------ .]
r! (m-r)!
{Note by poster: I may have made a syntax error. It may be either C(m,r) or
C(r,m) }
A-5:
Prove or disprove that there exists a positive real number u such
that [u**n] - n is an even integer for all positive integer n.
Here [x] denotes the greatest integer less than or equal to x.
A-6:
Let
x**4 x x-u
F(x) = -------- Int Int exp(u**3 + v**3) dv du
exp(x**3) 0 0
Find lim{x-->oo} F(x) or prove that it does not exist.
{N.B.P. Int denotes the integral sign. The top and bottom numbers denote
the upper and lower bound resp.}