HOMEWORK
MATH 409
(typically due Fri):
Solutions available here
Please spend at least 1 hour EVERY
night reading the material/looking at the proofs/making sure you understand
the details.
NOTE: IT IS NOT ALWAYS THE CASE THAT
PROBLEMS ARE WELL-STATED -- SOMETIMES YOU NEED TO EMAIL ME AND SAY YOU THINK
IT IS TOO VAGUE!
- Week 11 and on: Prepare for Putnam!
- Week 10: Nov 7 - Nov 11
- Homework for Wednesday: Think about A1 problems, try to be ready to
present something.
- Homework \#8: Due Wednesday November 9: \#1: There are four people who want to
cross a bridge. They all begin on the same side. You have 17 minutes to get
all of them across to the other side. It is night. There is one flashlight.
A maximum of two people can cross at one time. Any party who crosses, either
one or two people, must have the flashlight with them. The flashlight must
be walked back and forth, it cannot be thrown, etc. Each walks at a
different speed. A pair must walk together at the rate of the slower. Their
times are 1, 2, 5 and 10 minutes to cross. How do you get them across in 17
minutes? \#2: Generalizing Conway's Checker Problem: Consider the
three-dimensional version, where we again have an integer lattice and now
have checkers at all points with coordinates (x,y,z) with z non-positive. We
jump as before, but can now do north/south, east/west, and up/down. Prove
there is a positive number N such that we cannot get a checker to a point (x,y,N)
in finitely many moves. (Note: you can probably see how to generalize to
arbitrarily many dimensions, the bound N will grow the dimensions, but the
bound and the truth could be far apart). \#3: 1989A2 Evaluate \(\int_0^a
\int_0^b \exp\left(\max\{b^2 x^2, a^2 y^2\}\right) dy dx\) where \(a, b\)
are positive.
- Homework: do two Project Euler problems you have not done yet and
be prepared to present.
- Homework: do two Project Euler problems you have not done yet and
be prepared to present. It will not be before Monday November 14th.
- Week 9: Oct 31 - Nov 4
- Homework for Monday, October 31: Think about \(\int_0^\infty \sin(x)/x dx\).
- Homework \#8: Due Wednesday November 9: \#1: There are four people who want to
cross a bridge. They all begin on the same side. You have 17 minutes to get
all of them across to the other side. It is night. There is one flashlight.
A maximum of two people can cross at one time. Any party who crosses, either
one or two people, must have the flashlight with them. The flashlight must
be walked back and forth, it cannot be thrown, etc. Each walks at a
different speed. A pair must walk together at the rate of the slower. Their
times are 1, 2, 5 and 10 minutes to cross. How do you get them across in 17
minutes? \#2: Generalizing Conway's Checker Problem: Consider the
three-dimensional version, where we again have an integer lattice and now
have checkers at all points with coordinates (x,y,z) with z non-positive. We
jump as before, but can now do north/south, east/west, and up/down. Prove
there is a positive number N such that we cannot get a checker to a point (x,y,N)
in finitely many moves. (Note: you can probably see how to generalize to
arbitrarily many dimensions, the bound N will grow the dimensions, but the
bound and the truth could be far apart). \#3: 1989A2 Evaluate \(\int_0^a
\int_0^b \exp\left(\max\{b^2 x^2, a^2 y^2\}\right) dy dx\) where \(a, b\)
are positive.
- Homework: do two Project Euler problems you have not done yet and
be prepared to present.
- Week 8: Oct 24 - 28
- Homework for Monday, October 31: Think about \(\int_0^\infty \sin(x)/x dx\).
- Homework \#8: Due Wednesday November 9: \#1: There are four people who want to
cross a bridge. They all begin on the same side. You have 17 minutes to get
all of them across to the other side. It is night. There is one flashlight.
A maximum of two people can cross at one time. Any party who crosses, either
one or two people, must have the flashlight with them. The flashlight must
be walked back and forth, it cannot be thrown, etc. Each walks at a
different speed. A pair must walk together at the rate of the slower. Their
times are 1, 2, 5 and 10 minutes to cross. How do you get them across in 17
minutes? \#2: Generalizing Conway's Checker Problem: Consider the
three-dimensional version, where we again have an integer lattice and now
have checkers at all points with coordinates (x,y,z) with z non-positive. We
jump as before, but can now do north/south, east/west, and up/down. Prove
there is a positive number N such that we cannot get a checker to a point (x,y,N)
in finitely many moves. (Note: you can probably see how to generalize to
arbitrarily many dimensions, the bound N will grow the dimensions, but the
bound and the truth could be far apart). \#3: 1989A2 Evaluate \(\int_0^a
\int_0^b \exp\left(\max\{b^2 x^2, a^2 y^2\}\right) dy dx\) where \(a, b\)
are positive.
- Homework: do two Project Euler problems you have not done yet and
be prepared to present.
- Week 7: October 17 - 21, 2022
-
Readings on monovariants:
- For Wednesday: think about which is larger:
\(e^\pi\) or \(\pi^e\). You are NOT allowed to use a computer to
calculate anything; try to prove elementarily which wins.
- For Friday: Prepare two project Euler problems so you can present one at
the board. Email me what you want to do.
Week 6: October 10 - 14
- Mon: No Class; Wed: Fri:
- Readings on monovariants:
- Read some of the Geometry handouts:
- Homework \#6: Due Wednesday, October 19: \#1: Let \(a_1, a_2, \dots, a_n\) be
positive integers. Show a subset sums to a multiple of \(n\). \#2: Given any
\(n\), show there is a number \(x_n\) whose digits are only 0's and 7's such
that \(n\) divides \(x_n\). \#3: Consider the previous problem. Find such a
number for \(n=2017\); what is the smallest such number? \#4: Show that if
\(n\) divides a Fibonacci number that it divides infinitely many Fibonacci
numbers. \#5: For all positive real numbers \(a, b, c\) show that \(a^a b^b
c^c >= a^b b^c c^a\).
Week 5: Oct 3-7
(possibly no class Friday, if class it will be remote)
- Read some of the Geometry handouts:
- Homework \#5: Due Wednesday Oct 12: (0) Show that
no matter what 5 points are chosen on the surface of a unit sphere, there is
at least one closed hemisphere containing at least 4 of the points. (1) Prove the law of cosines: if a, b
and c are the sides of a triangle and theta is the angle between a and b,
then c^2 = a^2 + b^2 - 2 a b cos(theta). (2-21) Complete the first 20
Project Euler Problems, and include in your HW a screenshot showing that you
have completed all of these. Note this problem is worth 200 points (20
questions), and is thus giving you credit for all the work you have been
doing. We will spend a future class discussing the coding and these problems, so let
me know in advance ones you find particularly interesting.
- Homework (optional): Geometry problems typically invoke extreme
reactions: some love, and some hate. If you like geometry problems look at
the resources above, and choose 1-2 problems to do and submit. You may use
these are HW exemptions for problems in future weeks (i.e., if you get full
credit on either of these, you can skip a future problem and receive full
credit).
Week 4: Sept 26-30
- Project Euler: Make sure you have done the first 15 problems by Friday.
-
Read: Recurrence relations:
-
Homework \#4: Due Wednesday October 5: \#1: Prove for \(a_i >
0\) that \((1+a_1) \cdots (1+a_n) \ge 2^n \sqrt{a_1 \cdots a_n}\). \#2:
Prove for \(a, b > 0\) that \(a/b + b/a \ge 2\), both by using an inequality
approach \emph{and} without using an inequality! \#3: Solve the double
recurrence \(f_n = f_{n-1} + 3 g_{n-1}, g_n = -3 f_{n-1} + 9 g_{n-1}\). \#4:
Define a set to be selfish if it contains its cardinality (i.e., its number
of elements) as an element; thus \(\{1,3,5\}\) is selfish, while
\(\{1,2,3,5\}\) is not. Find, with proof, the number of subsets of
\(\{1,2,...,n\}\) that are minimal selfish sets (that is, selfish sets none
of whose subsets are selfish; thus \(\{1,3,5\}\) is not minimal selfish as
\(\{1\}\) is a subset). This is a Putnam problem..... Also, make sure you
have done the first 15 Project Euler Problems.
Week 3:
Sept 19 - 23, 2022
Week 2:
Sept 12-16, 2018
- Videos Mon:
https://youtu.be/tax8BkrPB6A Wed:
https://youtu.be/gUvX8r-nnbc Fri:
-
Some suggested
coding problems you should try (click
here for Mathematica notebook, and
click here for a pdf).
- Readings: Read up on induction. There are
lots of good webpages with induction problems:
-
Homework \#2: Due Friday September 16: (1)
Go to Project Euler (https://projecteuler.net/)
and create an account for yourself, and solve the first problem. You do
not need to submit this, just email me when done. (2) Read up on
induction and do the following: If n lines are drawn in a plane, and no two
lines are parallel, how many regions do they separate the plane into?
(3) Prove that \((1 − 1/4 )(1 − 1/9) \cdots (1 − 1/n^2) = (n+1) / 2n\).
- Week 1:
September 9, 2017
HOMEWORK 2018 Version
- Week 10 and on: Prepare for Putnam, work on chapters
- Week 9: Oct 29 - Nov 2
- Homework for Monday, November 5: Think about \(\int_0^\infty \sin(x)/x dx\).
- Homework \#9: Due Friday, November 2: \#1 (worth 20 points, counts
as two problems): Generalizing Conway's Checker Problem: Consider the
three-dimensional version, where we again have an integer lattice and now
have checkers at all points with coordinates (x,y,z) with z non-positive. We
jump as before, but can now do north/south, east/west, and up/down. Prove
there is a positive number N such that we cannot get a checker to a point (x,y,N)
in finitely many moves. (Note: you can probably see how to generalize to
arbitrarily many dimensions, the bound N will grow the dimensions, but the
bound and the truth could be far apart). \#2 (worth 20 points, counts as two
problems): do a Project Euler problem you have not done yet and
be prepared to present.
- Week 8: Oct 22 - 26 (No Class Friday)
- Homework for Monday, October 29: Think about \(\int_0^\infty \sin(x)/x dx\).
- Homework \#8: Due Friday October 26: \#1: There are four people who want to
cross a bridge. They all begin on the same side. You have 17 minutes to get
all of them across to the other side. It is night. There is one flashlight.
A maximum of two people can cross at one time. Any party who crosses, either
one or two people, must have the flashlight with them. The flashlight must
be walked back and forth, it cannot be thrown, etc. Each walks at a
different speed. A pair must walk together at the rate of the slower. Their
times are 1, 2, 5 and 10 minutes to cross. How do you get them across in 17
minutes? \#2: Generalizing Conway's Checker Problem: Consider the
three-dimensional version, where we again have an integer lattice and now
have checkers at all points with coordinates (x,y,z) with z non-positive. We
jump as before, but can now do north/south, east/west, and up/down. Prove
there is a positive number N such that we cannot get a checker to a point (x,y,N)
in finitely many moves. (Note: you can probably see how to generalize to
arbitrarily many dimensions, the bound N will grow the dimensions, but the
bound and the truth could be far apart). \#3: 1989A2 Evaluate \(\int_0^a
\int_0^b \exp\left(\max\{b^2 x^2, a^2 y^2\}\right) dy dx\) where \(a, b\)
are positive.
- Homework: do two Project Euler problems you have not done yet and
be prepared to present.
- Week 7: October 15 - 19, 2018
-
Readings on monovariants:
- For Wednesday: think about which is larger:
\(e^\pi\) or \(\pi^e\). You are NOT allowed to use a computer to
calculate anything; try to prove elementarily which wins.
- Homework \#6: Due Friday, October 19: \#1: Let \(a_1, a_2, \dots, a_n\) be
positive integers. Show a subset sums to a multiple of \(n\). \#2: Given any
\(n\), show there is a number \(x_n\) whose digits are only 0's and 7's such
that \(n\) divides \(x_n\). \#3: Consider the previous problem. Find such a
number for \(n=2017\); what is the smallest such number? \#4: Show that if
\(n\) divides a Fibonacci number that it divides infinitely many Fibonacci
numbers. \#5: For all positive real numbers \(a, b, c\) show that \(a^a b^b
c^c >= a^b b^c c^a\).
- Homework \#7: Due October 26, 2018: \#1, \#2, \#3, \#4 (counts as four
problems): Show that any decomposition of \(N\) as a sum of Fibonacci
numbers cannot have fewer summands than the Zeckendorf decomposition. Is
there a monovariant that can help?
Week 6: October 8 - 12 (no class Monday).
- Mon: No Class; Wed:
https://youtu.be/hDini6X5UY4 Fri:
https://youtu.be/vi1utgfeQHU
- Readings on monovariants:
- Read some of the Geometry handouts:
- Homework \#6: Due Friday, October 19: \#1: Let \(a_1, a_2, \dots, a_n\) be
positive integers. Show a subset sums to a multiple of \(n\). \#2: Given any
\(n\), show there is a number \(x_n\) whose digits are only 0's and 7's such
that \(n\) divides \(x_n\). \#3: Consider the previous problem. Find such a
number for \(n=2017\); what is the smallest such number? \#4: Show that if
\(n\) divides a Fibonacci number that it divides infinitely many Fibonacci
numbers. \#5: For all positive real numbers \(a, b, c\) show that \(a^a b^b
c^c >= a^b b^c c^a\).
Week 5: Oct 1-5
(no class Friday)
- Read some of the Geometry handouts:
- Homework \#5: Due Friday Oct 5: (0) Show that
no matter what 5 points are chosen on the surface of a unit sphere, there is
at least one closed hemisphere containing at least 4 of the points. (1) Prove the law of cosines: if a, b
and c are the sides of a triangle and theta is the angle between a and b,
then c^2 = a^2 + b^2 - 2 a b cos(theta). (2-21) Complete the first 20
Project Euler Problems, and include in your HW a screenshot showing that you
have completed all of these. Note this problem is worth 200 points (20
questions), and is thus giving you credit for all the work you have been
doing. We will spend a future class discussing the coding and these problems, so let
me know in advance ones you find particularly interesting.
- Homework (optional): Geometry problems typically invoke extreme
reactions: some love, and some hate. If you like geometry problems look at
the resources above, and choose 1-2 problems to do and submit. You may use
these are HW exemptions for problems in future weeks (i.e., if you get full
credit on either of these, you can skip a future problem and receive full
credit).
Week 4: Sept 24-28
- Project Euler: Make sure you have done the first 15 problems by Friday.
-
Read: Recurrence relations:
-
Homework \#4: Due Friday, Sept 28, 2018: \#1: Prove for \(a_i >
0\) that \((1+a_1) \cdots (1+a_n) \ge 2^n \sqrt{a_1 \cdots a_n}\). \#2:
Prove for \(a, b > 0\) that \(a/b + b/a \ge 2\), both by using an inequality
approach \emph{and} without using an inequality! \#3: Solve the double
recurrence \(f_n = f_{n-1} + 3 g_{n-1}, g_n = -3 f_{n-1} + 9 g_{n-1}\). \#4:
Define a set to be selfish if it contains its cardinality (i.e., its number
of elements) as an element; thus \(\{1,3,5\}\) is selfish, while
\(\{1,2,3,5\}\) is not. Find, with proof, the number of subsets of
\(\{1,2,...,n\}\) that are minimal selfish sets (that is, selfish sets none
of whose subsets are selfish; thus \(\{1,3,5\}\) is not minimal selfish as
\(\{1\}\) is a subset). This is a Putnam problem..... Also, make sure you
have done the first 15 Project Euler Problems.
Week 3:
Sept 17 - 21, 2018
Week 2:
Sept 10-14, 2018
Week 1:
September 7, 2017
Kansas State Math Competition:
http://www.math.ksu.edu/main/events/parker-mathcomp/