Internet Amateur Mathematics Society Newsletter 2 IAMS@quack.kfu.com >From The Editor Hello everyone. This is the second IAMS newsletter, hope you will like it. We are still in need of a ftp site to store the newsletters, any suggestions? I am preparing for two AP tests (May 6th and 12th), and so I will be busy for a couple of weeks. I will try to post some more contests, but don't count on it. UCSB College of Creative Studies Mathematics Competition, Final Round benjie@wales\%hip-hop.suvl.ca.us You have 50 minutes, and no calculators. Enjoy. 1. Find 3 pairs, (m,n), of positive integers such that m^2 - 7n^2 = 1 2. A regular tetrahedron contains a sphere of radius 1. The length of a side of the tetrahedron must then be at least how long? 3. Find all real values of k for which the equation x^3 - kx - 1 = 0 has two roots equal. 4. In a parallelogram with acute angle 30 degree, the diagonals are in the ratio of sqrt{2+sqrt{3}} to sqrt{2-sqrt{3}}. Find the ratio of the sides. 5. How many positive integers are there smaller than 210 and having no common factor with 210 other than 1? David Radcliffe posted his solutions, and I still have them. If you want a copy, email me. 1992 Putnam Exam From: David A. Wagner (dawagner@phoenix.Princeton.EDU) Problem A1 Prove that f(n) = 1-n is the only integer-valued function defined on the integers that satisfies the following conditions. (i) f(f(n)) = n, for all integers n; (ii) f(f(n + 2) + 2) = n for all integers n; (iii) f(0) = 1. Problem A2 Define \alpha to be the coefficient of x^{1992} in the power series expansion about x = 0 of (1 + x)^{\alpha}. Evaluate \int_{0}^{1} \alpha (-y-1)(\frac{1}{y+1}+ \frac{1}{y+2} + \frac{1}{y+3} + ... + \frac{1}{y+1992}) dy Problem A3 For a given positive integer m, find all triples (n,x,y) of positive integers, with n relatively prime to m, which satisfy (x^2 + y^2)^m = (xy)^n. Problem A4 Let f be an infinitely differentiable real-valued function defined on the real numbers. If f(\frac{1}{n}) = \frac{n^{2}}{n^2+1}, n = 1,2,3,..., compute the values of the derivatives f^{k}(0), k = 1,2,3,... . Problem A5 For each positive integer n, let a_{n} = {0 if the number of 1's in the binary representation of n is even, 1 if the number of 1's in the binary representation of n is odd.} Show that there do not exist positive integers k and m such that a_{k+j} = a_{k+m+j} = a_{k+2m+j}, for 0 <= j <= m - 1. Problem A6 Four points are chosen at random on the surface of a sphere. What is the probability that the center of the sphere lies inside the tetrahedron whose vertices are at the four points? (It is understood that each point is independently chosen relative to a uniform distribution on the sphere.) Problem B1 Let S be a set of n distinct real numbers. Let A_S be the set of numbers that occur as averages of two distinct elements of S. For a given n >= 2, what is the smallest possible number of distinct elements in A_S? Problem B2 For nonnegative integers n and k, define Q(n,k) to be the coefficient of x^k in the expansion of (1+x+x^2+x^3)^n. Prove that Q(n,k) = \sum_{j=0}^n {n \choose j} {n \choose k - 2j} where {a \choose b} is the standard binomial coefficient. (Reminder: For integers a and b with a >= 0, {a \choose b} = a!/b!(a-b)! for 0 <= b <= a, and {a \choose b} = 0 otherwise.) Problem B3 For any pair (x,y) of real numbers, a sequence a_n(x,y)_{n>=0} is defined as follows: a_0(x,y) = x, a_{n+1}(x,y) = \frac{a_n(x,y)^2 + y^2}{2} for all n >= 0. Find the area of the region {(x,y)|(a_n(x,y))_{n>=0}\ converges}. Problem B4 Let p(x) be a nonzero polynomial of degree less than 1992 having no nonconstant factor in common with x^3 - x. Let \frac{d^{1992}}{dx^{1992}}\times \frac{p(x)}{x^3-x}=\frac{f(x)}{g(x)} for polynomials f(x) and g(x). Find the smallest possible degree of f(x). Problem B5 Let D_n denote the value of the (n-1) by (n-1) determinant 3 1 1 1 ... 1 1 4 1 1 ... 1 1 1 5 1 ... 1 1 1 1 6 ... 1 . . . . ... . 1 1 1 1 ... n+1 Is the set \frac{D_n}{n!} for n>=2 bounded? Problem B6 Let M be a set of real n by n matrices such that (i) I \in M, where I is the n by n identity matrix; (ii) if A \in M and B \in M, then either AB \in M or -AB \in M, but not both; (iii) if A \in M and B \in M, then either AB = BA or AB = -BA; (iv) if A \in M and A \neq I, there is at least one B \in M such that AB = -BA. Prove that M contains at most n^2 matrices. Selected Posts: From: Harry.Cohen@who.cc.trincoll.edu Subject: a little number theory Hi Guys, how 'bout a number theory problem. For an integer N, let M be formed by reversing N's digits (if N=3296, then M=6923) Prove that N-M is divisible by 9. From: Ray J Cornwall Subject: Re: a little number theory A number is divisible iff the sum of its digits are divisible. Since a mod 9 - b mod 9 = (a-b) mod 9, and N and M will have the same sum of digits, then N-M mod 9 will equal 0, and N-M is divisible by 9. Course, I'm doing this in my head, so I may have made a serious error... Ray From: David G Radcliffe} Subject: Tests for divisibility A number is divisible by 3 if and only if the sum of its digits is divisible by 3. A number is divisible by 9 if and only if the sum of its digits is divisible by 9. A number is divisible by 2^k if and only if the last k digits together form a number divisible by 2^k. For example, 19392 is divisible by 8, because 392 is divisible by 8. A number is divisible by 5^k if and only if the last k digits together form a number divisible by 5^k. A number is divisible by 11 if and only if the quantity obtained by alternately adding and subtracting its digits is divisible by 11. Example: 5162949 is divisible by 11, because 5-1+6-2+9-4+9 = 22. To test whether a number is divisible by 7, 11, or 13, divide the digits into groups of three, starting from the right. Alternately add and subtract the groups. If the total is divisible by 7, 11, or 13, then so is the original number (and coversely). Example: Let N = 48,286,615. 48 - 286 + 615 = 377. 377 is divisible by 13, so N is divisible by 13. N is not divisible by 7 or 11. David Radcliffe radcliff@csd4.csd.uwm.edu From: Bryan D Levenson Subject: Tests for divisibility To determine whether a number is divisible by (10^k)+1, divide the number up into groups of k, beginning at the right. Alternately add and subtract the groups. Iff this sum is divisible by (10^k)+1, then the original number is. EXAMPLE: Is 75,548 divisible by 101=(10^2)+1? 7-55+48=0, which is divisible by 101, therefore 75,548 is divisible by 101 (75,548=101\times748) A number is divisible by c=a\timesb, where a and b are relatively prime, if it is divisible by a and b. EXAMPLE: Is 30 divisible by 12=2\times 6=3\times 4? Using 2 and 6 which are NOT relatively prime, we see that 30 is divisible by 2 and by 6, but not by 12. Using 3 and 4 which are relatively prime, we see that 30 is divisible by 3 and by 4 and hence it is divisible by 12. From: David G Radcliffe Subject: Yet another divisibility test Some last words on divisibility: The rules I gave for divisibility by 7 or 13 are very efficient if one is using a calculator, but they are somewhat cumbersome for manual computation. Below is an alternative method. Let n be a given positive integer, and let d_{k}d_{k-1}...d_{0} be its decimal representation. Let a = d_0 - d_3 + d_6 - d_9 +... and b = d_1 - d_4 + d_7 - d_10 +... and c = d_2 - d_5 + d_8 - d_11 +.... Then (1) n=a+3b+2c (mod 7), and (2) n=a-3b-4c (mod 13). Example: n = 31,415,926,535,897,932,384,622 a = 2 - 4 + 2 - 7 + 5 - 6 + 5 - 1 = -4 b = 2 - 8 + 3 - 9 + 3 - 2 + 1 - 3 = -13 c = 6 - 3 + 9 - 8 + 5 - 9 + 4 = 4 a + 3b + 2c = -35 = 0 (mod 7) a - 3b - 4c = 19 = 6 (mod 13) Conclusion: n is divisible by 7, but leaves a remainder of 6 when divided by 13. Can you figure out why this works? Try to invent a similar test for divisibility by 37. Last Words I want to include an interview next time in the newsletter, any volunteers? The interview will be on your experience with mathematics. Suggestions are always welcome!