Internet Amateur Mathematics Society Newsletter 4 IAMS@quack.kfu.com >From The Editor Hello everyone. The fourth newsletter is here (finally). Hope you enjoy it. I am still looking for articles or interviews etc. All helps are very welcome. The second contest is in session, please participate. IAMS Contest 2 Problem 1 benjie@wales%hip-hop.suvl.ca.us Problem: If b>1 and x>0 and (2x)^{log_b 2} - (3x)^{log_b 3}=0, what is the value of x? Solution: For this solution, we write log for log_b. The equation is equivalent to (2x)^{log 2}=(3x)^{log 3} {2^{log 2}}/{3^{log 3}} = {x^{log 3}}/{x^{log 2}} = x^{log 3 - log 2} Taking the logarithm of each side, we get (log 2)^2 - (log 3)^2 = (log 3 - log 2)log x -(log 2 + log 3) = log x -log 6 = log 1/6 = log x x = 1/6 Here is a list of people who submitted the correct answer: mokie@cco.caltech.edu ariels@cs.huji.ac.il arodgers@dcs.qmw.ac.uk martino@rob.csata.it amits@cory.Berkeley.EDU Congratulations! The problem 2 of contest 2 is Let a, b be two solutions of the quadratic equation in x: x^2 - 2px + p^2 - 2p - 1 = 0 Find a real value p such that 0.5 {(a-b)^2-2}/{(a+b)^2+2} is an integer. It is due next monday. Integration Fever: About two weeks ago, IAMS caught an Integration Fever, here are couple of integration problems submitted. From: noring@netcom.com (Jon Noring) Evaluate the integral \int_{0}^{1} x^x dx (Hint: Use power series) From: David G Radcliffe (radcliff@csd4.csd.uwm.edu) The integral you mentioned ( \int_0^{\pi} \sqrt{1 + \sin x} dx ) looks familiar (and you are right, the value is 4). When I was a freshman, I over slept and missed a calculus exam. The makeup exam was quite difficult, and I think that this integral was on the test. I was not able to solve it at the time. Another integral I remember being unable to solve was \int \sec^3 x dx. Here are two tough integrals which have clever solutions. Can you evaluate them? \int_0^{\pi/2} {dx}/{1 + \sqrt{\tan x}} \int_0^1 \sqrt{1 - x^3} - \sqrt[3]{1 - x^2} dx (Hints: \int \sec^3 x dx could be integrated by parts; the second integral could be integrated using trigonometry substitution) Paradoxes I recently posted a request of paradoxes, the replies where wonderful (thanks guys), and I thought I would write a little summary here. >From t-davidw@microsoft.com Zeno's paradox: "suppose I'm walking from point A to point B. To get to point B, I haveta pass a halfway point which is halfway between the two points. But to get to the halfway point, I haveta make it half way to the halfway point, so I gotta get to a quarter way point. And so on, ad infinitum. Thus I can never get anywhere." >From erickw@sfu.ca Name Unknown: "What is "the smallest positive integer that cannot be uniquely described in fifteen or less English words"?" Banach-Tarski paradox: (a consequence of the Axiom of Choice) "a unit sphere can be partitioned into a finite number of sets, which can then be "reassembled" into two disjoint unit spheres. One might not really consider this a paradox, since these are not really pieces but sets, which don't necessarily have a volume (measure)." From: Ariel Scolnicov Russle Paradox's variations: Liar's paradox: This statement is false. The barber-shop paradox: "the barber shaves all men who don't shave themselves; who shaves the barber?" The library catalogue: "a library contains two [bound] catalogues - one of books which refer to themselves, one of books which don't; which catalogue do you put the second catalogue in?" >From benjie@wales.suvl.ca.us Achilles' Paradox: No matter how fast you are, you will never catch me because during the time you spent catching me, I moved further. Black Hole Paradox: The centrifugal force near a black hole is inward instead of outward. Further Readings: GODEL, ESCHER, BACH: AN ETERNAL GOLDEN BRAID. Douglas R. Hofstadter, 1990, Vintage. DOES GOD PLAY DICE? THE MATHEMATICS OF CHAOS. Ian Stewart, Basil Blackwell, 1990. COMPUTER INVESTIGATIONS IN THE SEMANTICS OF PARADOX: CHAOTIC LIARS, FRACTALS AND STRANGE ATTRACTORS. Gary Mar and Patrick Grim in Philosophy and Computing. A PARTLY TRUE STORY. Ian Stewart, Scientific American, Feb., 1993. BLACK HOLES AND THE CENTRIFUGAL FORCE PARADOX. Marek Artur Abramowicz, Scientific American, March, 1993. Last Words Nothing special to say, and as usual, Suggestions are always welcome! Happy mathing, Benjie