University of Manitoba Mathletics
Miscellaneous problems with solutions, 24 November 2009
Many Putnam exams have at least one or two fairly straightforward questions. Here are
a few that might bring together some of the concepts looked at this year. Almos
University of Manitoba, Mathletics 2009
Session 8: Geometry
1
Facts and denitions
Things to remember:
Proper drawing helps.
Vector geometry and dot product.
Similar triangles.
Complex numbers for planar problems.
Calculus applications of integration.
University of Manitoba, Mathletics 2009
Session 5: Inequalities
1
Facts and denitions
AM-GM inequality: For a1 , a2 , . . . , an 0,
a1 + a2 + + an
(a1 a2 an )1/n ,
n
with equality i all ai s are equal.
Cauchys inequality: For reals a1 , . . . , an , b1 ,
MATH 3740 Methods of Advanced Calculus Winter 2008 Midterm 5 Solutions
Remarks: This examination is 3 hours. It consists of 10 questions, for a total of 140 marks. Notes, texts, calculators and cellphones are not allowed. Please write your answers in the
MATH 3740 Methods of Advanced Calculus Fall 2007 Midterm 1
Solutions
Remarks: The marks are low in general. They range from 12 to 38.5 (the maximum possible mark was 40), with an average of 21.91 and standard deviation of 6.6. Only two copies have more th
Analysis
University of Manitoba, Mathletics 2009
1
Some Facts and Examples
Need to know denitions and properties of continuous/dierentiable functions, sequences,
innite series, etc. (everything that is studied in our Calculus sequence) - those concepts
a