IMO Training 2010
Projective Geometry - Part 2
1. Consider the dilation carrying to the excircle opposite to A. Point E is mapped to F , which
must also be the point of tangency of the excircle to BC.
2. Let the excircle be tan
SOLUTIONS TO EXERCISES FOR
MATHEMATICS 205A Part 3
Spaces with special properties
Compact spaces I
Problems from Munkres, 26, pp. 170 172
Show that a nite union of compact subspaces of X is compact.
Suppose that Ai X i
DIRECTIONS: Solve each problem, choose the correct answer, and then use your pencil
to fill in the corresponding circle on your answer sheet.
Do not use too much time on any one problem. Solve the ones you can do quickly; then
return to t
SOLVING LINEAR CONGRUENCES
I have isolated proofs at the end. Fancy not, even for a moment, that this means the proofs are
unimportant! They are essential to understanding the algorithm. Rather, I thought it easier to
use this as a reference if you could
HOMEWORK 9 FOR 18.100B/C, FALL 2010
WAS DUE THURSDAY 18 NOVEMBER
(1) A function f : [a, b] R is said to be Lipschitz continuous (or just
Lipschitz) if there exists a constant A such that
|f (x) f (y)| A|x y| x, y [a, b].
Show that if f : [a, b] R is diere