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Ross Program Application Problem
Ross Program Application Problems
(1) The letters a_{1}, a_{2}, a_{3},
a_{4}, a_{5}, a_{6}, a_{7}represent seven
positive whole numbers.
The letters b_{1}, b_{2}, b_{3}, b_{4}, b_{5},
b_{6}, b_{7} represent the same numbers but in a different
order.
Will the product
(a_{1} – b_{1})(a_{2} – b_{2})(a_{3} – b_{3})(a_{4}
– b_{4})(a_{5} – b_{5})(a_{6} – b_{6})(a_{7}
– b_{7})
always be an even number? Explain your conclusion.
(2) Call a number “nice” if it can be expressed as a sum
of two or more consecutive
positive integers . For example, 5 and 6 are nice numbers because 5 = 2+3 and
6 = 1+2+3.
(a) Which numbers from 1 to 50 are not nice ? What’s the pattern for sizes beyond
50?
(b) Explain why the pattern you observed holds true generally.
(c) In how many different ways can 1000 be expressed as a sum of consecutive
integers? Is there a simple method to find the number of ways a given number n
can be
expressed that way? Explain.
For instance, 15 can be expressed in three ways: 15 = 1+2+3+4+5 = 4+5+6 = 7+8.
(3) A set of numbers has “the triplesum property” (or
TSP) if there exist three numbers
in the set whose sum is also in the set. [Repetitions are allowed.]
For example, the set U = {2, 3, 7} has TSP since 2 + 2 + 3 = 7, while V = {2, 3,
10} fails to have TSP.
(a) Suppose the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} is separated into
two parts,
forming two subsets A and B.
Prove: Either A or B must have the triplesum property.
[To begin the proof, suppose that statement is false and there are sets A and B
as above, each without TSP.
If 1 lies in A then 3 = 1 + 1 + 1 must be in B. Complete the proof that this
situation is impossible.]
(b) Is a similar result true when the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is
separated into two parts?
(4) Suppose 100 dots are arranged in a square 10 × 10
array, and each dot is colored red
or blue.
(a) Prove that this array must contain a “monochromatic” rectangle. That is, no
matter
how the red and blue colors are assigned, there must be either a set of four red
dots that
form a rectangle or else a set of four blue dots that form a rectangle.
Note: Don’t consider the colors of the dots inside that rectangle. Just the four
corner points.
Use only those rectangles having horizontal and vertical sides.
(b) Does this result remain true for smaller rectangular arrays of dots?
To begin, find a 4 × 5 array that admits no monochromatic rectangle.
Must a monochromatic rectangle exist in a 5 × 5 array? In a 4 × 6 array?
(5) A collection of n numbers a_{1}, a_{2},
. . . , a_{n}, has the following properties:
The sum of those n numbers is 500.
The sum of the smallest three of those numbers is 48.
The sum of the largest two of those numbers is 35.
(Note: Repetitions are allowed. Several of the a_{j} might be equal to
one another.)
(a) What are the possible values of n ? Explain your reasoning.
(b) Which of those values n occur if we require all the numbers a_{j} to
be integers?
(6) Suppose p and q are odd integers. Show that the
quadratic equation
x^{2} + px + q = 0
has no rational roots .
(Recall: a rational number is a fraction m/n where m and n are integers.
This number is a “ root ” of our equation if: (m/n) ^{2} + p(m/n) + q =
0.)
(7) A point in the plane is a lattice point if it has
integer coordinates : A = (m, n) is a
lattice point provided both m and n are integers. Given a point P = (r, s) in
the plane,
consider all the distances from P to lattice points.
Find a point P with the property that all those distances are different.
Justify your answer. That is, prove that the point you chose has the
stated property.
For instance, the point P = (0, π) won’t work because P is equally distant from
(1, 0) and (1, 0).
(8) What numbers can be expressed as an alternatingsum of
an increasing sequence of
powers of 2 ?
To form such a sum, choose a subset of the sequence 1, 2, 4, 8, 16, 32, 64, . .
. (these
are the powers of 2). List the numbers in that subset in increasing order (no
repetitions
allowed), and combine them with alternating plus and minus signs. For example,
(a) Is every positive integer expressible in this fashion? If so, give a
convincing proof.
(b) There can be more than one expression of this type for a given number. For
instance 5 = 1 – 4 + 8 and 5 = –1 + 2 – 4 + 8. Given a number n, how many
different
ways are there to write n in this way? Explain why your answer is correct.
(9) Which of the problems here did you enjoy the most? Why?
We hope you enjoyed working on these problems!
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