Ross Program 2021 Application Problems
ProfsHoarseWailsRoll
This document is part of the application to the Ross Mathematics Program, and
will remain posted at https://rossprogram.org/students/to-apply from January
through March.
The Admissions Committee will start reading applications in March 2021. The dead-
line for applications is April 1, but for adequate consideration of your application, it
is best to submit your solutions some weeks before the end of March.
Work independently on the problems below. We are interested in seeing how you
approach unfamiliar math problems, not whether you can nd answers by searching
through web sites or books, or by asking other people.
Submit your own work on these problems.
For each problem, explore the situation (with calculations, tables, pictures, etc.),
observe patterns, make some guesses, test the truth of those guesses, and write logical
proofs when possible. Where were you led by your experimenting?
Include your thoughts (but not your scratch-paper) even if you might not have found
a complete solution. If you’ve seen one of the problems before (e.g. in a class or
online), please include a reference along with your solution.
We are not looking for quick answers written in minimal space. Instead, we hope
to see evidence of your explorations, conjectures, and proofs written in a readable
format.
The quality of mathematical exposition, as well as the correctness and
completeness of your solutions, are factors in admission decisions.
PDF format required.
Convert your problem solutions into a PDF le. You may type the solutions using
LATEX or a word processor, and convert the output to PDF format.
Alternatively, you may scan your solutions from a handwritten paper copy, and con-
vert the output to PDF. (Use dark pencil or pen and write on only one side of
the paper.) Submitting photos of your work is not recommended since le sizes of
photos are often too large. (The Ross system cannot accept les much larger than
5 megabytes.) Rather than photographs, you might use a scan” feature on your
camera.
Note: Unlike the problems here, each Ross Program course concentrates deeply on one subject.
These problems are intended to assess your general mathematical background and interests.
1
Problem 1
Suppose A = (an) = (a1; a2; a3; : : : ) is an increasing sequence of positive integers.
A number c is called A-expressible if c is the alternating sum of a nite subsequence
of A. To form such a sum, choose a nite subset of the sequence A, list those numbers
in increasing order (no repetitions allowed), and combine them with alternating plus
and minus signs. We allow the trivial case of one-element subsequences, so that each
an is A-expressible.
Denition. Sequence A = (an) is an alt-basis” if every positive integer is uniquely
A-expressible. That is, for every integer m > 0, there is exactly one way to express
m as an alternating sum of a nite subsequence of A.
Examples. Sequence B = (2n?1) = (1; 2; 4; 8; 16; : : : ) is not an alt-basis because
some numbers are B-expressible in more than one way. For instance 3 = ?1 + 4 =
1 ? 2 + 4.
Sequence C = (3n?1) = (1; 3; 9; 27; 81; : : : ) is not an alt-basis because some numbers
(like 4 and 5) are not C-expressible.
(a) Let D = (2n ? 1) = (1; 3; 7; 15; 31; : : : ). Note that:
1 = 1, 2 = ?1 + 3, 3 = 3, 4 = ?3 + 7, 5 = 1 ? 3 + 7,
6 = ?1 + 7, 7 = 7, 8 = ?7 + 15, 9 = 1 ? 7 + 15, . . .
Prove that D is an alt-basis.
(b) Can some E = (4; 5; 7; : : : ) be an alt-basis? That is, does there exist an
alt-basis E = (en) with e1 = 4; e2 = 5, and e3 = 7? Justify your answer.
The rst few values seem to work: 1 = ?4 + 5, 2 = ?5 + 7, 3 = ?4 + 7.
(c) Can F = (1; 4; : : : ) be an alt-basis? That is, does there exist an alt-basis F = (fn)
with f1 = 1 and f2 = 4 ?
(d) Investigate some other examples. Is there some fairly simple test to determine
whether a given sequence A = (an) is an alt-basis?
2
Problem 2
A polynomial f(x) has the factor-square property (or FSP) if f(x) is a factor of f(x2).
For instance, g(x) = x ? 1 and h(x) = x have FSP, but k(x) = x + 2 does not.
Reason: x?1 is a factor of x2 ?1, and x is a factor of x2, but x+2 is not a factor of x2 +2.
Multiplying by a nonzero constant preserves” FSP, so we restrict attention to poly-
nomials that are monic (i.e., have 1 as highest-degree coecient).
What patterns do monic FSP polynomials satisfy?
To make progress on this topic, investigate the following questions and justify your
answers.
(a) Are x and x ? 1 the only monic FSP polynomials of degree 1?
(b) List all the monic FSP polynomials of degree 2.
To start, note that x2, x2 ? 1, x2 ? x, and x2 + x + 1 are on that list.
Some of them are products of FSP polynomials of smaller degree. For instance,
x2 and x2 ?x arise from degree 1 cases. However, x2 ?1 and x2 +x+1 are new,
not expressible as a product of two smaller FSP polynomials.
Which terms in your list of degree 2 examples are new?
(c) List all the new monic FSP polynomials of degree 3.
Note: Some monic FSP polynomials of degree 3 have complex coecients that are not real.
Can you make a similar list in degree 4 ?
(d) Are there monic FSP polynomials (of some degree) that have real number
coecients, but some of those coecients are not integers?
Explain your reasoning.
3
Problem 3
Rossie is a simple robot in the plane, with Start position at the origin O, facing the
positive x-axis.
An angle  is entered into Rossie’s memory. He can take only two actions:
S: Rossie steps one meter in the direction he is facing.
R: Rossie stays in place and rotates counterclockwise through angle .
Notation: A string of symbols S and R (read from left to right) represents a sequence
of Rossie’s moves. For instance, SRRSS indicates that Rossie steps one meter along
the x-axis, rotates through angle 2, and then steps two meters in that new direction.
In the questions below, we consider only those sequences of actions that include at
least one S.
(a) For which  can a sequence of actions result in Rossie’s return to Start?
(Then by repeating that sequence of actions, Rossie will retrace the same path.)
For example, with  = 2=3 = 120, the actions SRSRSR cause Rossie to trace
an equilateral triangle and return to Start.
(b) Suppose  is the angle pictured below, with cos() = ?1=3. Note that  is
approximately 109:47.
With this angle , explain why the
actions
SSSRSSRSSS
cause Rossie to return to O.
x
y

Those moves return Rossie to O but he is not at Start:
He is not facing the positive x-axis.
With that , is there some sequence of actions that returns Rossie to Start?
Justify your answer.
(c) Investigate the following question:
Which angles allow Rossie return to O? (Not necessarily facing the positive x-axis)
Provide more examples of such angles. Are there some angles  that allow Rossie
to return to O, but only after tracing some path more complicated than a triangle
or a regular polygon?
(d) Are there some angles  for which Rossie can never return to O?
Explain your reasoning.
4
Problem 4
Suppose f = (f1; f2; f3; : : : ) is a sequence of integers. For 0  k  n, dene the
f-bin” numbers [ nk
]f as follows: Dene [n0
]f = 1, and for k  1 let

n
k

f
=
fnfn?1


fn?k+1
fk fk?1


f1
:
If In = n, then [nk
]I =
?n
k


is the usual binomial coecient.
Another example: Dene L = (1; 3; 4; 7; 11; 18; : : : ) by setting L1 = 1; L2 = 3 and
Ln = Ln?1 + Ln?2 for n > 2.
Here are lists of some of the f-bin numbers [ nk
]f for the sequences f = I and f = L.
0 1 2 3 4 5 6
0 1
1 1 1
2 1 2 1
3 1 3 3 1
4 1 4 6 4 1
5 1 5 10 10 5 1
6 1 6 15 20 15 6 1
…
…
…
…
…
…
…
…
. . .
I – bin numbers
0 1 2 3 4 5 6
0 1
1 1 1
2 1 3 1
3 1 4 4 1
4 1 7 28
3 7 1
5 1 11 77
3
77
3 11 1
6 1 18 66 231
2 66 18 1
…
…
…
…
…
…
…
…
. . .
L- bin numbers
Denition. Sequence f is binomid if all the f-bin numbers [ nk
]f are integers.
Equivalently: f is binomid when, for each k  1:
Every product of k consecutive terms fnfn?1


fn?k+1 is an integer multiple of the
product of the rst k consecutive terms fkfk?1


f1.
Since every binomial coecient
?n
k


is an integer, the sequence I is binomid. The
table above shows that the sequence L is not binomid.
(a) Dene sequences Pn = 2n = (2; 4; 8; : : : ), Qn = n2 = (1; 4; 9; : : : ), and
Dn = 2n = (2; 4; 6; : : : ). In each case, nd a simple formula for [ nk
], check that it
is an integer, and conclude that P; Q and D are binomid.
(b) Is the sequence Mn = 2n ? 1 binomid? Justify your answer.
(c) Is the sequence Tn = n(n + 1) binomid?
As a rst step, verify that
 n
2

T
= Tn Tn?1
T2 T1
= n(n+1)
6
(n?1)n
2 is always an integer.
(d) Find some further examples of binomid sequences. Are there some interesting
conditions on a sequence f that imply that f is binomid?
Other notations for
?n
k


include nCk and C(n; k).
5

Ross Program 2021 Application Problems
ProfsHoarseWailsRoll
This document is part of the application to the Ross Mathematics Program, and
will remain posted at https://rossprogram.org/students/to-apply from January
through March.
The Admissions Committee will start reading applications in March 2021. The dead-
line for applications is April 1, but for adequate consideration of your application, it
is best to submit your solutions some weeks before the end of March.
Work independently on the problems below. We are interested in seeing how you
approach unfamiliar math problems, not whether you can nd answers by searching
through web sites or books, or by asking other people.
Submit your own work on these problems.
For each problem, explore the situation (with calculations, tables, pictures, etc.),
observe patterns, make some guesses, test the truth of those guesses, and write logical
proofs when possible. Where were you led by your experimenting?
Include your thoughts (but not your scratch-paper) even if you might not have found
a complete solution. If you’ve seen one of the problems before (e.g. in a class or
online), please include a reference along with your solution.
We are not looking for quick answers written in minimal space. Instead, we hope
to see evidence of your explorations, conjectures, and proofs written in a readable
format.
The quality of mathematical exposition, as well as the correctness and
completeness of your solutions, are factors in admission decisions.
PDF format required.
Convert your problem solutions into a PDF le. You may type the solutions using
LATEX or a word processor, and convert the output to PDF format.
Alternatively, you may scan your solutions from a handwritten paper copy, and con-
vert the output to PDF. (Use dark pencil or pen and write on only one side of
the paper.) Submitting photos of your work is not recommended since le sizes of
photos are often too large. (The Ross system cannot accept les much larger than
5 megabytes.) Rather than photographs, you might use a scan” feature on your
camera.
Note: Unlike the problems here, each Ross Program course concentrates deeply on one subject.
These problems are intended to assess your general mathematical background and interests.
1
Problem 1
Suppose A = (an) = (a1; a2; a3; : : : ) is an increasing sequence of positive integers.
A number c is called A-expressible if c is the alternating sum of a nite subsequence
of A. To form such a sum, choose a nite subset of the sequence A, list those numbers
in increasing order (no repetitions allowed), and combine them with alternating plus
and minus signs. We allow the trivial case of one-element subsequences, so that each
an is A-expressible.
Denition. Sequence A = (an) is an alt-basis” if every positive integer is uniquely
A-expressible. That is, for every integer m > 0, there is exactly one way to express
m as an alternating sum of a nite subsequence of A.
Examples. Sequence B = (2n?1) = (1; 2; 4; 8; 16; : : : ) is not an alt-basis because
some numbers are B-expressible in more than one way. For instance 3 = ?1 + 4 =
1 ? 2 + 4.
Sequence C = (3n?1) = (1; 3; 9; 27; 81; : : : ) is not an alt-basis because some numbers
(like 4 and 5) are not C-expressible.
(a) Let D = (2n ? 1) = (1; 3; 7; 15; 31; : : : ). Note that:
1 = 1, 2 = ?1 + 3, 3 = 3, 4 = ?3 + 7, 5 = 1 ? 3 + 7,
6 = ?1 + 7, 7 = 7, 8 = ?7 + 15, 9 = 1 ? 7 + 15, . . .
Prove that D is an alt-basis.
(b) Can some E = (4; 5; 7; : : : ) be an alt-basis? That is, does there exist an
alt-basis E = (en) with e1 = 4; e2 = 5, and e3 = 7? Justify your answer.
The rst few values seem to work: 1 = ?4 + 5, 2 = ?5 + 7, 3 = ?4 + 7.
(c) Can F = (1; 4; : : : ) be an alt-basis? That is, does there exist an alt-basis F = (fn)
with f1 = 1 and f2 = 4 ?
(d) Investigate some other examples. Is there some fairly simple test to determine
whether a given sequence A = (an) is an alt-basis?
2
Problem 2
A polynomial f(x) has the factor-square property (or FSP) if f(x) is a factor of f(x2).
For instance, g(x) = x ? 1 and h(x) = x have FSP, but k(x) = x + 2 does not.
Reason: x?1 is a factor of x2 ?1, and x is a factor of x2, but x+2 is not a factor of x2 +2.
Multiplying by a nonzero constant preserves” FSP, so we restrict attention to poly-
nomials that are monic (i.e., have 1 as highest-degree coecient).
What patterns do monic FSP polynomials satisfy?
To make progress on this topic, investigate the following questions and justify your
answers.
(a) Are x and x ? 1 the only monic FSP polynomials of degree 1?
(b) List all the monic FSP polynomials of degree 2.
To start, note that x2, x2 ? 1, x2 ? x, and x2 + x + 1 are on that list.
Some of them are products of FSP polynomials of smaller degree. For instance,
x2 and x2 ?x arise from degree 1 cases. However, x2 ?1 and x2 +x+1 are new,
not expressible as a product of two smaller FSP polynomials.
Which terms in your list of degree 2 examples are new?
(c) List all the new monic FSP polynomials of degree 3.
Note: Some monic FSP polynomials of degree 3 have complex coecients that are not real.
Can you make a similar list in degree 4 ?
(d) Are there monic FSP polynomials (of some degree) that have real number
coecients, but some of those coecients are not integers?
Explain your reasoning.
3
Problem 3
Rossie is a simple robot in the plane, with Start position at the origin O, facing the
positive x-axis.
An angle  is entered into Rossie’s memory. He can take only two actions:
S: Rossie steps one meter in the direction he is facing.
R: Rossie stays in place and rotates counterclockwise through angle .
Notation: A string of symbols S and R (read from left to right) represents a sequence
of Rossie’s moves. For instance, SRRSS indicates that Rossie steps one meter along
the x-axis, rotates through angle 2, and then steps two meters in that new direction.
In the questions below, we consider only those sequences of actions that include at
least one S.
(a) For which  can a sequence of actions result in Rossie’s return to Start?
(Then by repeating that sequence of actions, Rossie will retrace the same path.)
For example, with  = 2=3 = 120, the actions SRSRSR cause Rossie to trace
an equilateral triangle and return to Start.
(b) Suppose  is the angle pictured below, with cos() = ?1=3. Note that  is
approximately 109:47.
With this angle , explain why the
actions
SSSRSSRSSS
cause Rossie to return to O.
x
y

Those moves return Rossie to O but he is not at Start:
He is not facing the positive x-axis.
With that , is there some sequence of actions that returns Rossie to Start?
Justify your answer.
(c) Investigate the following question:
Which angles allow Rossie return to O? (Not necessarily facing the positive x-axis)
Provide more examples of such angles. Are there some angles  that allow Rossie
to return to O, but only after tracing some path more complicated than a triangle
or a regular polygon?
(d) Are there some angles  for which Rossie can never return to O?
Explain your reasoning.
4
Problem 4
Suppose f = (f1; f2; f3; : : : ) is a sequence of integers. For 0  k  n, dene the
f-bin” numbers [ nk
]f as follows: Dene [n0
]f = 1, and for k  1 let

n
k

f
=
fnfn?1


fn?k+1
fk fk?1


f1
:
If In = n, then [nk
]I =
?n
k


is the usual binomial coecient.
Another example: Dene L = (1; 3; 4; 7; 11; 18; : : : ) by setting L1 = 1; L2 = 3 and
Ln = Ln?1 + Ln?2 for n > 2.
Here are lists of some of the f-bin numbers [ nk
]f for the sequences f = I and f = L.
0 1 2 3 4 5 6
0 1
1 1 1
2 1 2 1
3 1 3 3 1
4 1 4 6 4 1
5 1 5 10 10 5 1
6 1 6 15 20 15 6 1
…
…
…
…
…
…
…
…
. . .
I – bin numbers
0 1 2 3 4 5 6
0 1
1 1 1
2 1 3 1
3 1 4 4 1
4 1 7 28
3 7 1
5 1 11 77
3
77
3 11 1
6 1 18 66 231
2 66 18 1
…
…
…
…
…
…
…
…
. . .
L- bin numbers
Denition. Sequence f is binomid if all the f-bin numbers [ nk
]f are integers.
Equivalently: f is binomid when, for each k  1:
Every product of k consecutive terms fnfn?1


fn?k+1 is an integer multiple of the
product of the rst k consecutive terms fkfk?1


f1.
Since every binomial coecient
?n
k


is an integer, the sequence I is binomid. The
table above shows that the sequence L is not binomid.
(a) Dene sequences Pn = 2n = (2; 4; 8; : : : ), Qn = n2 = (1; 4; 9; : : : ), and
Dn = 2n = (2; 4; 6; : : : ). In each case, nd a simple formula for [ nk
], check that it
is an integer, and conclude that P; Q and D are binomid.
(b) Is the sequence Mn = 2n ? 1 binomid? Justify your answer.
(c) Is the sequence Tn = n(n + 1) binomid?
As a rst step, verify that
 n
2

T
= Tn Tn?1
T2 T1
= n(n+1)
6
(n?1)n
2 is always an integer.
(d) Find some further examples of binomid sequences. Are there some interesting
conditions on a sequence f that imply that f is binomid?
Other notations for
?n
k


include nCk and C(n; k).
5