2. Lecture 2 (Basic Techniques)

3. Some Basic Techniques Drawing a Picture Reformulate the Problem
Use Symmetry, Create Symmetry

4. Symmetry in Calculus

5. Problem 1:
Mentally (or graphically) calculate (if exists):

6. Idea: Need to understand the meaning of the double angle formula:
Note: L’Hopital’s Rule does not apply here. Why?

7. Symmetry in Geometry

8. Problem 2:
Find the length of the shortest path along the outer surface of a cube between two opposite corners.

9. Idea:
Draw a flattened picture of the cube.

10. Problem 3:
Find the length of the shortest path from the point (3,5) to the point (8,2) that touches both the x-axis and the y-axis.

11. Idea:
Use symmetry about the x- and the y- axes.

12. Symmetry in Combinatorics
(The Art of Counting)

13. 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?

14. Idea:
Consider the map sending each subset S  X to its complement Sc = X  S.

15. Symmetry in Algebra

16. 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.

17. Idea:
Use the Arithmetic-Geometric mean inequality.

18. The Arithmetic/Geometric Mean Inequality: Show that for x, y > 0,
Generalize the corresponding inequality for n positive numbers.

19. Problem 6: Let ai, bi > 0, for i = 1, 2,…, n. Show that:

20. Idea:
Use the Cauchy-Schwarz Inequality.

21. The Cauchy-Schwarz Inequality
In other words: xy  |x||y|.
Generalize the corresponding inequality in the nth dimensional space.

22. Thank You for Coming
Wafik Lotfallah