Download A Crash Course in Problem Solving

Lecture 2 (Basic Techniques)
Some Basic Techniques
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Drawing a Picture
Reformulate the Problem
Use Symmetry, Create Symmetry
Symmetry in Calculus
Problem 1:
Mentally (or graphically) calculate (if exists):
x
1
2
lim  cos t dt
x  x
0
Idea:
Need to understand the meaning of the double
angle formula:
1 1
2
cos t 
 cos( 2t )
2 2
Note: L’Hopital’s Rule does not apply here. Why?
Symmetry in Geometry
Problem 2:
Find the length of the shortest path along the
outer surface of a cube between two opposite
corners.
Idea:
Draw a flattened picture of the cube.
Problem 3:
Find the length of the shortest path from the point
(3,5) to the point (8,2) that touches both the xaxis and the y-axis.
Idea:
Use symmetry about the x- and the y- axes.
Symmetry in Combinatorics
(The Art of Counting)
Problem 4
How many subsets of the set X={1,2,3,…,109}
have the property that the sum of the elements
of the subset is greater than 2997?
Idea:
Consider the map sending each subset S  X to
c
its complement S = X  S.
Symmetry in Algebra
Problem 5
Show that:
(a + b)(b + c)(c + a)  8abc,
for all positive numbers a, b, and c, with
equality iff a = b = c.
Idea:
Use the Arithmetic-Geometric mean inequality.
The Arithmetic/Geometric Mean Inequality:
Show that for x, y > 0,
x y
xy 
2
Generalize the corresponding inequality for n
positive numbers.
Problem 6:
Let ai, bi > 0, for i = 1, 2,…, n. Show that:
 a1 a2
an  b1 b2
bn 
            n 2
bn  a1 a2
an 
 b1 b2
Idea:
Use the Cauchy-Schwarz Inequality.
The Cauchy-Schwarz Inequality
x1 x2  y1 y2  z1 z2 
2
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
 x12  y12  z12 x22  y22  z 22
In other words: xy  |x||y|.
Generalize the corresponding inequality in the
nth dimensional space.
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Thank You for Coming
Wafik Lotfallah