Internet Amateur Mathematics Society Newsletter #1 3/13/1993 >From The Editor Hello everyone. This is the first IAMS newsletter, hope you will like it. I am thinking of having a newsletter once every two weeks or so. I will generate one in ascii, one in LaTeX. We will need a ftp site to store them, any suggestions? BTW, I am not an editor for any publication, as you can see ! Contests California Mathematics League Contests Rules: Each contest is 30 minutes. Calculators are allowed. Answers must be exact or have 4(or more) significant digits, correctly rounded. Contest Number 1 1-1. Jack wrote 100 positive numbers between 0 and 3, but he did not write the number 1. Jill wrote the reciprocal of every one of Jack's number. How many of these 200 umbers were between 0 and 1 ? 1-2. What is the only integer n for which sqrt-(n), sqrt-(n+1) and sqrt-(n+2) are the lengths of the sides of a right triangle? 1-3. How many equilateral triangles with side-length 1 are required to completely cover the interior of an equilateral triangle with side length 10? 1-4. What is the only real number x which satisfies sqrt-(1992) = 1992 * sqrt-(x) ? 1-5. The World Series of baseball, a famous sporting event, is played between two teams. As soon as either team wins 4 games, that team is declared World Champions (no game can end in a tie). If a World Series is played between two teams of equal ability (so that each team's probability of winning any game is 1/2), what is the probability that World Champions are declared after only 4 games? 1-6. In a country whose currency consists of $6 bills and $11 bills, the price of every item sold is a whole number of dollars that can be paid (exactly) using only these two types of bills. What is the largest whole number of dollars which could not be the price of an item sold in this country? Contest Number 2 2-1. The number 9 can be written as the sum of 9 consecutive integers. What is the product of these integers? 2-2. The sum of the two (linear) factors of x^2 - 26x + 144 is subtracted from the sum of the two (linear) factors of x^2 - 25x + 144. What is the value of the result? 2-3. If x = -644, determine, in simplest form, the value of | | | x | - x | - x |. 2-4. What are all values of x which satisfy (2x-1)/(x+1) <= 1? 2-5. What is the distance from the origin to the graph of y = sqrt-(6x-x^2 -9) ? 2-6. Two congruent semicircles lie on the diameter of a third semicircle, each tangent to the other two. A small circle is tangent to all three semicircles. If the area of the shaded region is 45pi, what is the length of a radius of the small circle? Graph was given. The shaded area is the area of the big semicircle - the area of the small circle - the total area of the two small semicircle. Contest Number 3 3-1. What is the only integer n for which sqrt-(n^2 +1) is also an integer? 3-2. What are all values of x which satisfy x(x^2 - 1993) = x(x^2 - x) ? 3-3. The lengths of the diaonals of a parallelogram are 10 and 24. If the length of one side of the parallelogram is 13, what is the perimeter of the parallelogram? 3-4. The entries in a 7x7 magic square array of numbers are the integers 1 through 49 inclusive. The sum of the entries in every row, column and both major diagonals is the same, and is called the magic sum. What is the value of this magic sum? 3-5. There are exactly 25 prime bumbers less than 100. For how many other positive integers n (other than 100) are there exactly 25 prime numbers less than n? 3-6. Among all ordered pairs of real numbers (x,y) which satisfy x^4 + y^4 = x^2 + y^2, what is the largest value of x? Contest Number 4 4-1 If 2^(x-3)=1 and 5^(y+2)=1, what is the value of (2^x)(5^y)? 4-2 The product of 1993 consecutive integers is 0. What is the greatest possible value for the largest one of these integers? 4-3 If N is a positive integer, and if N^50 is a 16-digit number, what is the value of N? 4-4 Find the area of the region bounded by the graphs of the equations x=-2, y=0, x=3, and y=|x+2| + |x-3|. 4-5 What is the perimeter of a right triangle with hypotenuse 65 which can be circumscribed about a circle with radius 12? 4-6 Let [x] denote the greatest integer <= x. For example, [2.345]=2. Let denote the fractional part of x, so that [x] + = x. For example, <2.345>=0.345, and [2.345]+<2.345>=2.345. Also, three positive numbers a, b, and c form a geometric progression if a/b = b/c. What is the only positive real number x for which , [x] and x form a geometric progression? Contest 5 5-1 If A=1990x1991x1992x1993x1994x1995x1996 and B=1993^7, which is larger, A or B? 5-2 If x, y, and z are unequal positive integers, what is the smallest possible value of a for which x^1+y^2+z^3=a? 5-3 Solve for x: log_{2}(x+1)=2log_{2}(x-1} 5-4 There are three ways that the product of the first 23 positive integers, known as 23!, can be wroitten as the product of n consecutive positive integers, where n>1. One of the values of n is 23. What are the other two values of n? 5-5 What are all real values of x which satisfy (sqrt-(21+x) + sqrt-(21-x)) --------------------------- = 21/x ? (sqrt-(21+x) - sqrt-(21-x)) 5-6 The bisector of any angle of a triangle divides the side opposite that angle intro segments proportional to the other two sides. In the diagram, the bisector of an angle of the triangle shown divides the side opposite that angle into segments with lengths 2 and 5. If the perimeter of this triangle is an integer, what is the largest possible value of the perimeter? Diagram: a triangle that has a angle bisector from top divide the base line into segments of 2 and 5. Mu Alpha Theta Math Contests Rules: Time Limit 40 Minutes. No Calculators. Contest #1 1. November 18, 1992 falls on Wednesday. In what year after 1992 will November 18 first fall on Wednesday? 2. A november number is defined as a positive integer greater than 2 such that the sum of its positive factors is one less than twice the number. For example, 2 is the smallest november number since its positive factors are 1 and 2; 2+1=2(2)-1. Find the fourth smallest november number. 3. James Eyeyou promised the contents of his piggy bank to the math coach who came closest to guessing the number coins in the bank. The guesses by 6 coaches were: 61, 43, 52, 41, 29, 58 Of these guesses, one was off by 4, another by 5, and others by 14, 18, 11, and 6. How many coins were in James' piggy bank? 4. Find all real numbers such that: log_{10} (3x^2 + 5x - 1)=0 5. Find, in simplified radical form, the largest area of a rectangle meeting all the following: a. the diagonal is the square root of an integer. In condition c, this integer is called the radicand. b. the diagonal is less than 10. c. the radicand is a multiple of 8, but not a multiple of 16. d. at least one of the sides of the rectangle is an integer. 6. When Altoonahedge calculated her pre-calculus average, she wrote down what she thought were her six scores: 91, 79, 92, 84, 96, 98. According to Altoonahedge, her average was 90. However, Altoonahedge's actual average was a whole number between 80 and 90 because she had inadvertently interchanged the digits in one of the six numbers. What was her actual average? 7. Two right triangles each have areas of 210. If each of lengths of the six sides of the two triangles is a whole number, give the lengths of the two hypotenuses. 8. Who were those two gorgeous girls at last summber's Mu Alpha Theta Convention in Princeton? Here are some opinions: Adam: Gitchie, gitchie, goo. There were Allyson and Mary. David: Their poetry was incredible. I'll never forget Allyson and Sirinya. Marty: I worked calculator problems with them everyday. I say they were Sirinya and Taska. Jim: I know beauty when I see it, and I tell you they were Taska and Madonna. Well, what can you expect from such forgetful boys? Marty was right with one name and wrong with the other. So were two of the other three boys. Alas, the fourth boy was completely wrong. If all the girls mentioned are gorgeous, which one can we deduce could not possibly have been at the convention? 9. A puppy notices 12 letters on the floor that are arranged to spell: mu alpha theta The puppy playfully knocks the letters hither and you across the floor. Being an intelligent puppy (s)he then arranges the 12 letters at random in order from left to right. Find the probability that the 3rd through the 7th letters taken in order spell: alpha. 10. Given triangle ABS with AC=9, BC=7, and AB=8. Find, in simplified radical form, the exact value of the sine of one-half of angle A. 11. Find the sum of 1(2)+2(3)+3(4)+4(5)+5(6)+...+49(50). 12. In the addition alphametic shown below, each unique letter is representing a unique digit. As you answer give the digit represented by the letter R. B E A V E R + T I G E R -------------- R A B B I T First International Olympiad, 1959 1959/1. Prove that the fraction 21n + 4 ------- 14n + 3 is irreducible for every natural number n. 1959/2. For what real values of x is sqrt( x + sqrt(2x - 1) ) + sqrt( x - sqrt(2x - 1) ) = A , given (a) A = sqrt(2), (b) A = 1, (c) A = 3, where only non-negative real numbers are admitted for square roots? 1959/3. Let a,b,c be real numbers. Consider the quadratic equation in cos x: 2 a cos x + b cos x + c = 0 . Using the numbers a,b,c, form a quadratic equation in cos 2x whose roots are the same as those of the original equation. Compare the equations in cos x and cos 2x for a = 4, b = 2, c = -1. 1959/4. Construct a right triangle with given hypoteneuse c such that the median drawn to the hypoteneuse is the geometric mean of the two legs of the triangle. 1959/5. An arbitrary point M is selected in the interior of the segment AB. The squares AMCD and MBEF are constructed on the same side of AB, with the segments AM and MB as their respective bases. The circles circumscribed about these squares, with centers P and Q, intersect at M and also at another point N. Let N' denote the point of intersection of the straight lines AF and BC. (a) Prove that the points N and N' coincide. (b) Prove that the straight lines MN pass through a fixed point S independent of the choice of M. (c) Find the locus of the midpoints of the segments PQ as M varies between A and B. 1959/6. Two planes, P and Q, intersect along the line p. The point A is given is the plane P, and the point C in the plane Q; neither of these points lies on the straight line p. Construct an isoceles trapezoid ABCD (with AB parallel to CD) in which a circle can be inscribed, and with vertices B and D lying in the planes P and Q respectively. Posts From: "David A. Wagner" Subject: Extension of Pythagorean formula Got a question/puzzle for you all, maybe it'll be interesting... For a while I was trying to figure out something interesting about an "extension" of the Pythagorean formula a^2 + b^2 = c^2. An article in rec.puzzles posted over the summer involved finding a solution in the integers to a similar equation, a^2 + b^2 + c^2 = d^2. Computer search found an answer for some small numbers (I don't remember what they were but I could easily duplicate the search if you wish :-) Anyways, I started wondering about a more general form, finding n integers where the sum of the squares of the numbers is a square number itself. With n=2 you can actually generate all solutions; put a=m^2-n^2, b=2mn, c=m^2+n^2. Then if you let m, n run over all positive integers, you'll enumerate all solutions to the a^2 + b^2 = c^2 - and furthermore I think that any solution can be expressed in the form above for some m, n. Is there any generalization to n=3 or higher numbers? I never really got anywhere here *sigh* I do know (by computer search again :) that there are solutions for n=4; that's about all the further I got on the question. So, is this a standard/simple problem? Any ideas? David Wagner dawagner@princeton.edu From: algebra@leland.stanford.edu To: iams%wales@hip-hop.suvl.ca.us > > Got a question/puzzle for you all, maybe it'll be interesting... > > For a while I was trying to figure out something interesting about > an "extension" of the Pythagorean formula a^2 + b^2 = c^2. An article in > rec.puzzles posted over the summer involved finding a solution in the integers > to a similar equation, a^2 + b^2 + c^2 = d^2. Computer search found an > answer for some small numbers (I don't remember what they were but I could > easily duplicate the search if you wish :-) > Are they (3,4,12,13)? > Is there any generalization to n=3 or higher numbers? I never > really got anywhere here *sigh* I do know (by computer search again :) > that there are solutions for n=4; that's about all the further I got on > the question. Surely you don't need a comp search for n=4. Just add 4 1's together. 8-) > > So, is this a standard/simple problem? Any ideas? I don't have much of an idea for a general formula but I could suggest a recursive method for finding a solution for any n based on the Pythagorean formula for n=2. As (3,4,12,13) suggests, we could take the familiar triples (3,4,5) and (5,12,13) and add them. Cancellation of 5^2 results in (3,4,12,13). If you want a soln for n = 4 just find a triple beginning with 13^2 (13,84,85) and do the cancellation thingy again. It is a simple matter to show that the square on the right hand side of the eqaution always satisfies a Pythagorean triple. So you can keep adding triples till you get a solution for any n you like. From: "David A. Wagner" Subject: Another one... I got a lot of neat responses to the last question I asked. Thanks, guys! I think I'll be reading for a while in the library here about number theory before I totally understand some of the proofs for that question - but it's good stuff! [Now if I can just convince one of my professors to teach an introductory number theory seminar next semester... *grin*] A few days ago I started thinking about a problem that I think is real neat - I just thought I'd share what seems to be a neat brain teaser... Probably it's some classic problem in statistics or computer science but I'm enjoying trying to work it out on my own :-) Last year I took an introductory algorithms course and one of the homeworks was to find a good way to generate m distinct sorted random integers in the range 1 to n, inclusive. (What I mean by random is this: pick a series of numbers from 1 to n, with each integer being chosen with probability 1/n. Keep accumulating integers - ignore repeats - until you have m distinct numbers.) That algorithm can be quite slow if m is close to n. Consider what happens when you're picking the last few numbers: you've already selected almost all of the integers in 1..n, so to get just one new number you have to make a lot of calls to the random number generator. But there's a trick that will reduce this problem: if m > n/2, then find (n-m) numbers in the range 1..n, and then return the complement of that set. [Then it should take time proportional to n + m.] There are lots of ways to do this. One way is to shuffle all integers 1..n (with each permutation chosen with equal probability), keep only the first m numbers in the array, and sort them. [This takes m log m time.] If you're willing to do some statistics, you can probably do this with time linear in m. For example, one idea I'm tossing around is to find the probability distribution for the median of m numbers in the range 1..n. Then my routine can pick the median by generating a random number with this distribution. Suppose it chooses some value r to be the median: then I recursively pick (m/2-1) numbers in the range 1 to (r-1), and pick (m/2) numbers in the range (r+1) to n. [Actually you have to be a little cleaner than that and deal with the case where m isn't odd, etc.] If I've done the calculations correctly, the probability that the median of 2k+1 integers in the range 1..n is r has the form Binomial[r-1, k] Binomial[r+1, k] / Binomial[n, 2k+1]. Unfortunately I don't know a fast way to generate a random number with this distribution given only a random number generator which returns numbers from the uniform distribution. If I could find a good way to do this, it would yield a algorithm that takes time linear in m. Since that's a lower bound for the time any algorithm to do this problem can take (I think, under an appropriate model), this is pretty interesting. I've thought of a few ways similar to the last method. Say we want to pick numbers from the range 1..100. One algorithm picks j, the number of integers that were chosen and were in the range 1..50. Then you can recursively pick j numbers in the range 1..50 and (m-j) numbers in the range 51..100. A calculation indicates that the probability of there being j out of m integers in the range 1..r (out of the total big range 1..n) is as follows: Binomial[r, j] Binomial[n-r, m-j] / Binomial[n, m]. Again, I'd like to find some function f which would transform a uniform distribution into one of the above form - but haven't managed to yet. :-) You could also consider the distribution of the intervals between consecutive integers. I'm not real sure about this particular calculation, but I seem to get that the probability of finding the largest of m integers in the range 1..n to be i is Binomial[i-1, m-1] / Binomial[n, m]. Then after you pick the largest of the m integers, I believe you can pick the next biggest one by recursively selecting (m-1) numbers in the range 1 to (i-1), inclusive. I'm not real sure about this one, though, cause it seems like there could be some tricky conditional probability stuff going on. Well that about exhausts all that I've come up with in the past few days. Phew, this turned out to be a longer letter than I expected! Here's hoping you found this question half as exciting as I did! Now I can't wait to hear everyone else's suggestions and criticisms pour in... David Wagner dawagner@princeton.edu From: David G Radcliffe Subject: Re: Extension of Pythagorean formula > > Got a question/puzzle for you all, maybe it'll be interesting... > > For a while I was trying to figure out something interesting about > an "extension" of the Pythagorean formula a^2 + b^2 = c^2. An article in > rec.puzzles posted over the summer involved finding a solution in the integers > to a similar equation, a^2 + b^2 + c^2 = d^2. Computer search found an > answer for some small numbers (I don't remember what they were but I could > easily duplicate the search if you wish :-) Hint: Prove that if d can be written as a sum of three squares, then d^2 can also be written as a sum of three squares. Have fun, Dave Radcliffe radcliff@csd4.csd.uwm.edu From: David G Radcliffe Subject: a^2 + b^2 + c^2 = d^2 I have a procedure which will produce all solutions to a^2 + b^2 + c^2 = d^2 in positive integers. Let a and b be any two positive integers, not both odd. Find all factorizations pq = a^2 + b^2 with p > q and p + q even. Let c = (p - q) / 2, and d = (p + q) / 2. You can check that a^2 + b^2 + c^2 = d^2. In fact, all solutions can be found by this procedure. Dave Subject: Re: Extension of Pythagorean formula From: johnh@instruction.CS.ORST.EDU a1^2+a2^2+a3^2+...+an^2=b^2 is a standard distance formula which usually is referred to as the pythagorean formula, even though the ancient for was limited to only the first two terms. Using three terms is equivalent to finding a distance in three-space(the points on a sphere). As to the problem of finding all the points(integer valued points that is)I would probably write a program which will find all the circle in the top half of the sphere(or hyper sphere) that have integer valued heights, and then finf all the points on the cirlces that correspond to integer coordinates. Just my two credits. John E. Holeman >From ttgrq@info.win.tue.nl Fri Mar 5 05:43:59 1993 Subject: TAPE-TRAY-TALK Nowadays, most people own videorecorders. So obviously, most people own VIDEO TAPES. When you buy the thing, you'll probably purchase about 10 tapes, to start with. Now for some reason, this number increases constantly (whereas the number of times you actually watch the damn things is constantly decreasing). Anyway after a year or so, you'll own about a 100 of them on 90 of which you dont know what's on them or can't decyphre the remarks ("lst 20 m. film, doc. strt 1265 nice!...") scribbled on the back. Now this problem is not about remarks but about how to STORE the cassettes. Most people have some kind of (rectangular) drawer to put them in. Not so this friend of mine.... When I got there, their was a huge pile of tapes stacked randomly on top of a CIRCULAR SERVING TRAY. I had some time to spare, so I tried TO FIT AS MANY TAPES ON THE SERVING TRAY AS POSSIBLE, in one layer, written remarks facing up. Which wasn't easy but very enjoyable. Enough crap, here's the formal problem: "Given a circular tray of diameter 100, and an unlimited number of upright video tapes (rectangles of 1 x 10), WHAT'S YOUR BEST APPROXIMATION THE MAX NUMBER OF TAPES THAT FIT ON THE TRAY, staying within the borderline of the circle, and how are they arranged ?" It's a 2D-problem, so just ONE LAYER. have fun Andreas "Prove their exist no uninteresting numbers" Gammel, ttgrq@info.win.tue.nl From: David G Radcliffe Subject: A few questions... Here is my attempt at writing questions for a high school math contest. None of these problems require calculus or advanced mathematics. Please let me know what you think. 1. How many was can you write 675 as the sum of three or more consecutive odd integers? 2. Find all integers n for which n^2 + 3n + 9 is a square. 3. Determine the number of integer solutions of 1/x + 1/y = 1/12. 4. Find the perimeter of a right triangle, given the hypoteneuse and the radius of an inscribed circle. 5. Let a, b, and c be integers. Show that if an^2 + bn + c is divisible by seven for three consecutive integers, then it is divisible by seven for all integers. 6. Suppose ABC is a triangle with median AD. Prove that AD < (AB + AC) / 2. From: ttgrq@info.win.tue.nl (Andreas Gammel) Subject: BOGGLE In the game of BOGGLE, you use a throwing dice, i.e. a cube, whose six faces do NOT show dots, but the letters B, O, G, G, L and E. The object of the game is to form the word "BOGGLE" by throwing the dice over and over again. The order is irrelevant, and letters you already have are simply skipped. So the following throwing result is valid: G,L,L,O,O,G,L,E,L,B(the game ends here cause you gathered all necesarry letters. The length of this trial was 10. It can of course be longer or shorter! Now here's the puzzle: 1. "What's the expected number of throws E(BOGGLE), you need to form the word BOGGLE, where the order of the letters is irrelevant" 2. "What if the order IS relevant?" in which the expectation E is defined as: E(BOGGLE) = SUM ( L * P(getting BOGGLE in L throws) ) (over all possible values of L) where P = propability Have fun Andreas ttgrq@info.win.tue.nl From: vhansen@ipf.bau-verm.uni-karlsruhe.de (Wolfgang von Hansen) Subject: question on exp(ix) Hi everybody, ix I finally accepted, that e = cosx + isinx I'd like to know how this equeation can be prooved geometrically. Wolfgang vhansen@ipfs.bau-verm.uni-karlsruhe.de From: ttgrq@info.win.tue.nl (Andreas Gammel) Subject: wanted: MISSING LINK PUZZLES Now and then you see these puzzles where a logical story is told, of which some vital clues are omitted, which makes them sound very strange and illogical. The object of the game is to explain the logic of the story, i.e. find the missing clues. The players may ask questions, to which only yes/no is answered. An example: A man comes to a hotel, and says: "Damn, now I'm broke!" Now you start asking questions, like Is it a big hotel (no), is it smaller than an apple(yes), is the man a bussiness man (no) etc.etc.. By picking smart questions you can finally find out that, in this case, the man was playing MONOPOLY and got to a field with a hotel on it. So the problem is always to find the MISSING LINK. DOES ANYONE HAVE ANY MORE OF THESE PUZZLES ? HOW ARE THEY CALLED ? ARE THERE FTP SITES WITH THESE PUZZLES ? Here are the few I've heard (and have the solution to). In all cases the question is "EXPLAIN WHAT HAPPENED, AND WHY". I. A man comes to a hotel and says :"Damn, now I'm broke!". II. Ceasar and Cleopatra are lying on a bed. They are dead. The window is open. III. Two people sit in their cars on a street, both of them dead. IV. A man walks into a salloon and asks for a glass of water. The bartender takes out his gun and points it at the man, to which the man says: "Thank you", after which he leaves the salloon. V. A man lives on the 12th floor of a building. Every morning he takes the elevator from the 12th to 1st floor and goes to work. Every night he takes the elevator from the 1st to the 10th (!) floor and then takes the stairs to the 12th floor where he lives. EXCEPT when it rains, then he goes all the way to the 12th floor at night. VI. A man sits at home, when the bell rings. It's the postman who brings him a package. The man asks the postman to wait, opens the package, checks the contents, nods approvingly, closes the package, writes a new address on top, and returns it to the waiting postman who takes it with him. VII. A man lies dead in a field. Next to him are a couple of matches. VIII. A person lies dead in the street. A car stands nearby. IX. ....... For questions, suggestions, solutions, and other interesting stuff, but especially new MISSING LINKS, please respond to: ttgrq@info.win.tue.nl or Rec.puzzles or IAMS@quack.kfu.com (the Internet Amateur Mathematics Society, mailing list, applications to IAMS-request@quack.kfu.com) Bye . /| / | / | -----/---+ ndreas /________________________________________________________ ttgrq@info.win.tue.nl Last Words I want to include an interview next time in the newsletter, any volunteers? The interview will be on your experience with mathematics.