Caltech
Ma 17: How to Solve it
Fall 2013
Homework 1
Polynomials
Problem 1. Let n, k be positive integers and suppose that the polynomial x2k xk + 1 divides
x2n + xn + 1. Prove that x2k + xk + 1 divides x2n + xn + 1.
Problem 2. Find all polynomials P (x) =
Caltech
Ma 17: How to Solve it
Fall 2013
Homework 3
Solutions
Problem 1. (IMC A1/2010) Let 0 < a < b, prove that
b
2
2
2
(x2 + 1)ex dx ea eb .
a
b2
(x
a
Proof. Since x2 + 1 2x, we get
2
+ 1)ex dx
b
a
2
2
2
2xex dx = ea eb .
Problem 2. Let f : [0, 1] R be
Caltech
Ma 17: How to Solve it
Fall 2013
Homework 2
Solutions
Problem 1. (A1/2008) Let f : R2 R be a function such that
f (x, y ) + f (y, z ) + f (z, x) = 0, x, y, z R.
Prove that there exists a function g : R R such that
f (x, y ) = g (x) g (y ), x, y R.
Caltech
Ma 17: How to Solve it
Fall 2013
Homework 1
Solutions
Problem 1. Let n, k be positive integers and suppose that the polynomial x2k xk + 1 divides
x2n + xn + 1. Prove that x2k + xk + 1 divides x2n + xn + 1.
(IMC B1, 2008)
Proof. Let f (x) = x2k xk
MA 17 HOW TO SOLVE IT
LECTURE 9
TONY WING HONG WONG
1. Probability
1.1. Theory.
Probability problems are usually quite deep. Apart from probability theory, like independence, expectation etc, you may also need combinatorics, analysis and many other tools.
MA 17 HOW TO SOLVE IT
LECTURE 8
TONY WING HONG WONG
1. Geometry
1.1. Theory.
Most geometry problems are of high-school level, so if you are good at geometry in IMO
level, then you probably want to try on these problems in the Putnam competition.
Pure Geo
MA 17 HOW TO SOLVE IT
LECTURE 7
TONY WING HONG WONG
1. Number Theory
1.1. Theory.
Most number theory problems are related to even-odd parity, divisibility, primes, perfect
squares etc. The variety of problems is great, so there is no xed pattern in solvin
MA 17 HOW TO SOLVE IT
LECTURE 6
TONY WING HONG WONG
1. Linear Algebra
1.1. Theory.
Jordan normal form/Jordan canonical form
For all square matrix A, there exists a complex matrix P such that P 1 AP = J is a
Jordan normal form. The Jordan normal form enco
MA 17 HOW TO SOLVE IT
LECTURE 5
TONY WING HONG WONG
1. Analysis
(2011 B3) (*) Let f and g be (real-valued) functions dened on an open interval containing
0, with g nonzero and continuous at 0. If f g and f /g are dierentiable at 0, must f be
dierentiable
Caltech
Ma 17: How to Solve it
Fall 2013
Homework 4
Solutions
Problem 1. (Putnam B1, 2010) Is there an innite sequence of real numbers a1 , a2 , a3 , . such that
am + am + am + . . . = m
1
2
3
for every positive integer m?
Solution. If there were such a s
Caltech
Ma 17: How to Solve it
Fall 2013
Homework 5
Solutions
Problem 1. (Putnam A1/2009) Let f be a real-valued function on the plane such that for every
square ABCD in the plane f (A) + f (B ) + f (C ) + f (D) = 0. Does it follows that f (P ) = 0 for al
Caltech
Ma 17: How to Solve it
Homework 8
Fall 2013
Liubomir Chiriac
Problem 1. Consider a set with a binary operation on (i.e., for each x, y , x y ).
If (x y ) x = y , x, y , prove that x (y x) = y , x, y .
Problem 2. Let S be a set of real numbers whic
Caltech
Ma 17: How to Solve it
Homework 7: Number Theory
Fall 2013
Liubomir Chiriac
Problem 1. Prove that
gcd(m, n) n
n
m
is an integer for all integers 1 m n.
Problem 2. Let n be a positive integer such that n + 1 is divisible by 24. Prove that the sum o
Caltech
Ma 17: How to Solve it
Homework 6: Combinatorics
Fall 2013
Liubomir Chiriac
Problem 1. Consider S = cfw_1, 2, . . . , 2n and let M S be a subset of cardinality |M | = n + 1.
Prove that
(a) there exist two elements a, b M such that a and b are rela
Caltech
Ma 17: How to Solve it
Homework 5: Geometry
Fall 2013
Liubomir Chiriac
Problem 1. Let f be a real-valued function on the plane such that for every square ABCD in the
plane
f (A) + f (B ) + f (C ) + f (D) = 0.
Does it follows that f (P ) = 0 for al
Caltech
Ma 17: How to Solve it
Homework 4: Sequences
Fall 2013
Liubomir Chiriac
Problem 1. Is there an innite sequence of real numbers a1 , a2 , a3 , . such that
am + am + am + . . . = m
1
2
3
for every positive integer m?
Problem 2. Consider the power se
Caltech
Ma 17: How to Solve it
Fall 2013
Homework 3: Integrals
due on 10/22/2013 in class.
Problem 1. Let 0 < a < b, prove that
b
2
2
2
(x2 + 1)ex dx ea eb .
a
Problem 2. Let f : [0, 1] R be a continuous function such that for any x, y [0, 1]
xf (y ) + yf
Caltech
Ma 17: How to Solve it
Fall 2013
Homework 2
Functional Equations
Problem 1. Let f : R2 R be a function such that
f (x, y ) + f (y, z ) + f (z, x) = 0, x, y, z R.
Prove that there exists a function g : R R such that
f (x, y ) = g (x) g (y ), x, y R
Caltech
Ma 17: How to Solve it
Fall 2013
Homework 6
Solutions
Problem 1. Consider S = cfw_1, 2, . . . , 2n and let M S be a subset of cardinality |M | = n + 1. Prove that
(a) there exist two elements a, b M such that a and b are relatively prime.
(b) (B2/
MA 17 HOW TO SOLVE IT
LECTURE 4
TONY WING HONG WONG
1. Calculus
(2009 A6) Let f : [0, 1]2 R be a continuous function on the closed unit square such that f
x
1
1
and f exist and are continuous on the interior (0, 1)2 . Let a = 0 f (0, y )dy , b = 0 f (1, y
MA 17 HOW TO SOLVE IT
LECTURE 3
TONY WING HONG WONG
1. Polynomials
(1952 A1) The polynomial p(x) has all integral coecients. The leading coecient, the constant term, and p(1) are all odd. Show that p(x) has no rational roots.
(2011 B2) (*) Let S be the se
MA 17 HOW TO SOLVE IT
LECTURE 2
TONY WING HONG WONG
1. Amendment in Course Policies
I am very sorry to say that there is a need for a change to the non-compulsory homework
policy in this class. I hope this does not upset anyone, since suitable amount of h
MA017 2010 HOMEWORK 3
(1) Start with two piles of chips of sizes x, y . Two players alternately take any number of chips from one pile or the same number of chips from each pile, and the winner is the one who takes the last chip. Assuming both players pla
MA017 2010 HOMEWORK 2
(1) Write n ones on the board, and repeat the process erase any two numbers a, b and write a+b until 4 one number x remains. Show that x 1/n. Solution. The sum 1/a n, taken over the numbers a on the board, is non-increasing: for any
MA017 2010 HOMEWORK 1
(1) Let a1 , . . . , an be real numbers, and dene f (x) = j =1 aj sin(jx). Suppose |f (x)| | sin x| for all n real x. Prove, by induction, that | j =1 jaj | 1. (Considering f (0) gives a slick proof, but I am asking you to use induct
MA017 2010 HOMEWORK 5
Due 2010-11-02 Tue.
n +1 (1) One can prove n + k+1 = n+1 by pure thought as follows. The (k + 1)-element subsets of k k cfw_1, . . . , n +1 are of two mutually exclusive types: those that contain n +1, and those that do not; the form
MA017 2010 HOMEWORK 4
Due 2010-10-26 Tue. The Putnam problems on this set are 2, 6, 7, 8. (1) Show that n n+1 (1)k n 1 1 k = . (k + 1)2 n+1 k
k=0 k=1
(2) Prove that all terms of the sequence a1 = a2 = a3 = 1, are integers. (3) Evaluate (for |x| < 1) x2 .
MA017 2010 HOMEWORK 3
Due 2009-10-19 Tue. (1) Start with two piles of chips of sizes x, y . Two players alternately take any number of chips from one pile or the same number of chips from each pile, and the winner is the one who takes the last chip. Assum
MA017 2010 HOMEWORK 2
Due 2010-10-12 Tue. (1) Write n ones on the board, and repeat the process erase any two numbers a, b and write a+b until 4 one number x remains. Show that x 1/n. (2) Given nitely many points in the plane, any three of which form a tr
MA017 2010 HOMEWORK 1
Due 2010-10-05 Tue. n (1) Let a1 , . . . , an be real numbers, and dene f (x) = j =1 aj sin(jx). Suppose |f (x)| | sin x| for all n real x. Prove, by induction, that | j =1 jaj | 1. (Considering f (0) gives a slick proof, but I am as