A Quickie
by t0rajir0u, May 2, 2007, 2:10 AM
A short fun problem.
Problem 1: (Engel, Chapter 4, #67) Prove that, among any seven real numbers
, there exist two such that
.
Solution: The middle should instantly remind you of tangent subtraction, while the RHS should remind you of
. The interpretation is almost automatic:
Let
. Then WLOG 
, an interval of length 
. Among seven such 
we can choose two that are strictly less than 
apart by Pigeonhole. Let these two be 
. Then
As desired. The right inequality turns out to be strict. QED.
One of many unexpected applications of Pigeonhole. The wording of the problem made the pigeons more or less obvious, but the holes much less so. Nearly all of the difficulty of these types of problems is in discerning the identity of the pigeons and the holes.
Practice Problem 1: (Engel, Chapter 4, #61) There are
points inside a circle of radius 
. Prove that there exists a ring with inner radius 
and outer radius 
covering ten of these points.
Practice Problem 2: (Engel, Chapter 4, #76) Any of the
points 
in space has a smaller distance from point 
than from all the other points 
. Prove that 
.
(Can you generalize to lower and/or higher dimensions?)
Problem 1: (Engel, Chapter 4, #67) Prove that, among any seven real numbers
, there exist two such that
.Solution: The middle should instantly remind you of tangent subtraction, while the RHS should remind you of
. The interpretation is almost automatic:Let
. Then WLOG 
, an interval of length 
. Among seven such 
we can choose two that are strictly less than 
apart by Pigeonhole. Let these two be 
. Then
As desired. The right inequality turns out to be strict. QED.
One of many unexpected applications of Pigeonhole. The wording of the problem made the pigeons more or less obvious, but the holes much less so. Nearly all of the difficulty of these types of problems is in discerning the identity of the pigeons and the holes.
Practice Problem 1: (Engel, Chapter 4, #61) There are
points inside a circle of radius 
. Prove that there exists a ring with inner radius 
and outer radius 
covering ten of these points.Practice Problem 2: (Engel, Chapter 4, #76) Any of the
points 
in space has a smaller distance from point 
than from all the other points 
. Prove that 
.(Can you generalize to lower and/or higher dimensions?)
,
must be at least some number of radians, and then calculating the area of a circle on a sphere... that doesn't sound particularly easy. I bet there's a better way.
and then apply Pigeonhole so one angle is 
and get a contradiction. In 3-D, I can't see a good way to apply a similar argument.
April 2009
March 2009