Caltech
Ma 17: How to Solve it
Fall 2013
Homework 9
Solutions
Problem 1. (Putnam A1/1951) Let A be a real 4 4 skew-symmetric matrix (i.e., At = A). Prove
that det A 0.
Proof. We claim that all eigenvalues of A are pure imaginary. Indeed, let be an eigenva
Caltech
Ma 17: How to Solve it
Fall 2013
Homework 7
Solutions
Problem 1. (Putnam B2/2000) Prove that
gcd(m, n) n
n
m
is an integer for all integers 1 m n.
Proof. Let d = gcd(m, n) we claim that x, y Z such that xm + yn = d. Indeed, write m = dm
and n = dn
Caltech
Ma 17: How to Solve it
Fall 2013
Homework 8
Solutions
Problem 1. (Putnam A1/2001) 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 .
Proof. Note that x (y x) = (y x) y )