Unformatted text preview: x 3 + 3 xy + y 3 = 1 contains only one set of three distinct points A , B and C which are the vertices of an equilateral triangle, and ﬁnd its area. [Putnam Exam, 2006, B1] Problem 5. Deﬁne a function f on the real numbers by f ( x ) = ( if x is irrational, 1 /q if x = p/q with p ∈ Z ,q ∈ N , gcd( p,q ) = 1 . Determine the set of points on which f is continuous. Problem 6. Deﬁne a sequence { u n } ∞ n =0 by u = u 1 = u 2 = 1, and thereafter by the condition that det ± u n u n +1 u n +2 u n +3 ² = n ! for all n ≥ 0. Show that u n is an integer for all n . (By convention, 0! = 1.) [Putnam Exam, 2004, A3]...
View
Full Document
 Fall '10
 RyanDaileda
 Real Numbers

Click to edit the document details