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	<title>Mathematical Food for Thought</title>
	<link>http://wangsblog.com/jeffrey</link>
	<description>Serves a Daily Special and an All-You-Can-Eat Course in Problem Solving. Courtesy of me, Jeffrey Wang.</description>
	<lastBuildDate>Sun, 08 Jul 2007 00:22:04 +0000</lastBuildDate>
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	<item>
		<title>What&#8217;s Your Function? Topic: Algebra. Level: AMC/AIME.</title>
		<description><![CDATA[Problem: Given two positive reals  and , show that there is a continuous function  that satisfies .
Solution: There are several special cases that are interesting to look at before we make a guess as to what type of function  will be. First we consider the case . This immediately gives
.
It should not [...]]]></description>
		<link>http://wangsblog.com/jeffrey/?p=333</link>
			</item>
	<item>
		<title>Addition At Its Finest. Topic: Calculus/S&amp;S.</title>
		<description><![CDATA[Problem: Evaluate  where  is a real number with .
Solution: Looking at that all too common denominator, we do a partial fraction decomposition in hopes of telescoping series. The summation becomes
.
Common Taylor series knowledge tells us that
,
which convenient fits the first part of the summation. As for the second part, we get

from the same [...]]]></description>
		<link>http://wangsblog.com/jeffrey/?p=332</link>
			</item>
	<item>
		<title>The Smaller The Better. Topic: Calculus.</title>
		<description><![CDATA[Problem: Given a complicated function , find an approximate local minimum.
Solution: The adjective complicated is only placed so that we assume there is no easy way to solve  to immediately give the solution. We seek an algorithm that will lead us to a local minimum (hopefully a global minimum as well).
We start at an [...]]]></description>
		<link>http://wangsblog.com/jeffrey/?p=331</link>
			</item>
	<item>
		<title>Colorful! Topic: Calculus.</title>
		<description><![CDATA[Theorem: (Green&#8217;s Theorem) Let  be a simply connected plane region whose boundary is a simple, closed, piecewise smooth curve  oriented counterclockwise. If  and  are continuous and have continuous first partial derivatives on some open set containing , then
.
&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8211;
Problem: Evaluate , where  is the boundary of the region enclosed by  [...]]]></description>
		<link>http://wangsblog.com/jeffrey/?p=330</link>
			</item>
	<item>
		<title>More Integrals&#8230; *whine*. Topic: Calculus.</title>
		<description><![CDATA[Definition: (Jacobian) If  is the transformation from the -plane to the -plane defined by the equations  and , then the Jacobian of  is denoted by  or by  and is defined by
,
i.e. the determinant of the matrix of the partial derivatives (also known as the Jacobian matrix). Naturally, this can be [...]]]></description>
		<link>http://wangsblog.com/jeffrey/?p=329</link>
			</item>
	<item>
		<title>One By One, We&#8217;re Making It Fun. Topic: Calculus/S&amp;S.</title>
		<description><![CDATA[Theorem: (Stolz-Cesaro) Let  and  be sequences of real numbers such that  is positive, strictly increasing, and unbounded. If the limit

exists, then the following limit also exists and we have
.
&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8211;
Theorem: (Summation by Parts) If  and  are two sequences, then
.
&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8211;
Problem: Let  be a sequence of real numbers such that  converges. [...]]]></description>
		<link>http://wangsblog.com/jeffrey/?p=328</link>
			</item>
	<item>
		<title>Bigger Means Better. Topic: Algebra/Inequalities/Sets. Level: Olympiad.</title>
		<description><![CDATA[Definition: A set  is said to be convex if, for any  and , we have .
&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8211;
Definition: Let  be a real-valued function defined on a convex set . We say that  is convex if, for all  and , we have
.
&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8211;
Theorem: (Jensen&#8217;s Inequality) Let  be a real-valued, convex function defined on [...]]]></description>
		<link>http://wangsblog.com/jeffrey/?p=327</link>
			</item>
	<item>
		<title>A Square of Divisors. Topic: Number Theory. Level: AIME/Olympiad.</title>
		<description><![CDATA[Problem: (1999 Canada &#8211; #3) Determine all positive integers  with the property that . Here  denotes the number of positive divisors of .
Solution: So, testing some small numbers yields  as a solution. We claim that this is the only such solution.
Clearly, since  is a square, we can use a variant of [...]]]></description>
		<link>http://wangsblog.com/jeffrey/?p=326</link>
			</item>
	<item>
		<title>Ready For AP Calculus? Topic: Calculus/S&amp;S. Level: AIME/Olympiad.</title>
		<description><![CDATA[Problem: Evaluate .
Solution: Let . Then , a well-known Taylor series. So we want to integrate this:

by parts using  and . Substituting  in the last integral, we have
.
So . Thus

for some constant . Using our knowledge that , , and  by L&#8217;Hopital twice, we see that

Then setting  we obtain  and [...]]]></description>
		<link>http://wangsblog.com/jeffrey/?p=325</link>
			</item>
	<item>
		<title>Just A &#8220;Natural&#8221; Thing. Topic: Algebra. Level: AIME/Olympiad.</title>
		<description><![CDATA[Problem: Factor .
Solution: Consider the polynomial . We have
,
so we want to factor the second term when . Call it  so that . Consider the relation
.
Since  is a root of the LHS, we factor it out of the RHS as well to get
.
Dividing through by  and rearranging, we obtain the nice expression
.
Letting [...]]]></description>
		<link>http://wangsblog.com/jeffrey/?p=324</link>
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