Final
Problem!!
11/20/06
Week
#13 Mon. Nov. 20 - Fri. Dec. 1
Problem 13 (25 points) In a
certain competition, a player scores either M or N points at the end of each
round, were M and N are given positive integers. The player notices that her
cumulative score can take any positive integer value except for those in a
finite set P, where |P| =1003. If 60 is
not in P, find M and N.
Copies of this problem (and answers to the old problems) are
available from the shelves in front of the Math office, MH 308, or online at http://www.math.sjsu.edu/~jackson. Solutions to problem #13 should be submitted
to the Math office by Fri. Dec. 1 at noon.
Remember that a correct solution should also contain an explanation of
the answer. Make sure that your name,
student ID number, and the division you are participating in (undergraduate or
graduate or SJMC) are printed legibly at the top of the page. A correct solution to problem #13 and the
final results of the 2006 Problem Solving competition will be posted on Mon.
Dec. 4.
11/14/06
Week
#12 Mon. Nov. 13 - Wed. Nov. 22
Problem 12 (20 points) A rectangle
is 'weakly integral' if at least one of its dimensions is an integer.
a) Suppose a large rectangle is
tiled with five weakly integral rectangles as in the figure below. Show that the large rectangle is also weakly
integral.
h
g
f
a
y
b x e
c d
b)
Prove or disprove: If a large rectangle can be tiled with a collection of
weakly integral rectangles, the large rectangle must be weakly integral.
Copies of this problem (and answers to the old problems) are
available from the shelves in front of the Math office, MH 308, or online at http://www.math.sjsu.edu/~jackson. Solutions to problem #12 should be submitted
to the Math office by Wed. Nov. 22 at noon.
Remember that a correct solution should also contain an explanation of
the answer. Make sure that your name,
student ID number, and the division you are participating in (undergraduate or
graduate or SJMC) are printed legibly at the top of the page. A correct solution to problem #12 will be
posted on Mon. Nov. 27.
11/06/06
Week
#11 Mon. Nov. 6 - Wed. Nov. 15
Problem 11 (15 points) Show that
there do not exist positive real numbers x, y, and z, which satisfy the
following system of equations (where a ^ b represents a to the bth power).
(x + y
+ z) ^ x = 6/7
(x + y
+ z) ^ y = 7/8
(x + y
+ z) ^ z = 8/9.
10/30/06
Week
#10 Mon. Nov. 6 - Wed. Nov. 15
Problem 10 (14 points) Is it
possible to inscribe a triangle with angles of 45 degrees, 60 degrees, and 75
degrees in a unit circle (x ^2 + y ^ 2 = 1) so that each vertex of the triangle
has rational coordinates?
10/23/06
Week #9 Mon. Oct. 23 - Wed. Nov. 1
Problem 9
(13 points) For all positive integers m,n show that
[(3m)!(3n)!]/[m!n!(m+n)!(n+m)!] is always an integer.
10/17/06
Week #8 Mon. Oct. 16 - Wed. Oct. 25
Problem 8 (12
points) A certain island has x red, y blue, and z yellow chameleons, where
x,y,z are positive integers. Whenever
two chameleons that are different colors meet, they can both change their color
to the third color. For which triples
(x,y,z) of positive integers could all the chameleons ever end up being the
same color.
10/10/06
Week #7 Mon. Oct. 9 - Wed. Oct. 18
Problem 7 (11 points) Let n^4 represent the 4th power of n. Determine all non-negative integral solutions of (n1, n2, n3, ... , n14), if any, of the Diophantine equation
(n1)^4 + (n2)^4 + ... + (n14)^4 = 1,599.
10/03/06
Week
#6 Mon. Oct. 2 – Wed. Oct. 11
Problem 6 (10 points) Does there exist a continuous function
f: R ->
R such that the image of every rational number is irrational and the image of
every irrational is rational.
9/25/06
Week
#5 Mon. Sept. 26 – Wed. Oct. 4
Problem 5
(9 points) If P(x) denotes a polynomial of degree n such that P(k) = k/(k+1)
for k = 0, 1, 2, ..., n, determine P(n+1).
9/18/06
Week #4 Mon. Sept. 19 – Wed. Sept. 27
Problem 4
(8 points) In a certain town it began snowing before noon and continued snowing
at a constant rate until dark. At noon
a crew set out along a highway, clearing the snow from it as they went. They cleared two miles in the first two
hours but only one mile in the next two hours.
If the crew clears equal volumes of snow in equal times, at what time
did it begin to snow?
9/11/06
Week
#3 Tues. Sept. 12 – Wed. Sept. 20
Problem 3
(7 points) Given a set of 1004 integers between 1 and 2006 (inclusive), show
that at least one member of the set must divide another member of the set.
9/5/06
Week #2
Tues. Sept. 5 – Wed. Sept. 13
Problem 2 (6 points) There is a plane with n > 1 seats, and lined up to
board are n passengers, each of whom has an assigned seat. The first passenger,
who has amnesia and cannot remember his assigned seat, sits in a random seat.
Thereafter, as each person in line boards the plane, they look to see if their
assigned seat is available. If it is, they sit in it. If it is not, they sit in
a random open seat. What is the probability that the last person to board sits
in his/her assigned seat?
8/15/06
Week
#1 Wed. Aug. 23 – Wed. Sept. 6