PUTNAM TRAINING PROBLEMS

PUTNAM TRAINING PROBLEMS (Last updated: December 3, 2003) Remark This is a list of Math problems for the NU Putnam team to be discussed during the training sessions Miguel A Lerma 1 Bag of candies In a box we have N +1 bags with N candies each (N 2) The first bag contains N orange flavored candies, the second one contains N 1 orange flavored candies and 1 lemon flavored one, the third one contains N 2 orange flavored and 2 lemon flavored, and so on We take one bag at random and try one of the candies, which turns out to be lemon flavored Then we take a second candy from the same bag What is the probability that the second candy is also lemon flavored? Hint: Use Bayes Theorem (or some equivalent argument) to determine the probability that the candy comes from the bag with k lemon-flavored candies, then compute the total probability that the second candy is also lemon-flavored 2 Rank of matrix with 0, ±1 elements Let n be a positive odd number and A an n by n matrix whose diagonal elements are 0, its non-diagonal elements are ±1, and the sum of the elements of each row is 0 Find the rank of A Hint: Think modulo 2 3 Fibonacci sum Let F n be the Fibonacci sequence 0, 1, 1, 2, 3, 5, 8,, defined recursively F 0 = 0, F 1 = 1, F n = F n 1 + F n 2 for n 2 Prove that Hint: Generating functions n=1 F n 2 n = 2 4 A triangle inequality Prove that in a triangle with sides a, b, c and opposite angles A, B, C (in radians) the following relation holds: a A + b B + c C a + b + c π 3 Hint: Assume a b c, A B C, and rewrite the inequality as something 0 5 Split squares Prove that there are infinitely many squares not multiple of 10 whose representation in base 10 can be split into two squares For instance 7 2 = 49 can be split 4 9, where 4 and 9 are squares (4 = 2 2, 9 = 3 2 ); 13 2 = 169 can be split 16 9, again two squares, etc (we exclude multiples of 10 in order to avoid trivial answers like the infinite sequence 49 = 4 9, 4900 = 4 900, 490000 = 4 90000, etc) 1

PUTNAM TRAINING PROBLEMS 2 Hint: Pose the problem as a system of equation and inequality: 10 n x 2 + y 2 = z 2 10 n 1 < y 2 < 10 n 6 John s PIN John s PIN for his teller machine consists of a 6-digit number abcdef He has bad memory, but does not want to write it down just in case someone finds it So he breaks the number in two 3-digit numbers abc and def, and with his pocket calculator finds the quotient abc/def, which he writes down A few days later he needs the PIN, and as expected he cannot remember it, but he remembers that he wrote down the quotient between the two halves of the number in a piece of paper: abc/def= 0195323246 So he takes his pocket calculator (with only basic arithmetic operations and the inversion 1/x key), punches the keys for a few seconds, scratches some numbers on a piece of paper for a few seconds more and gets the original 6-digit number What is that number and how did he find it so quickly? Hint: continued fractions 7 Average temperature (From Zeitz s book) A spherical, 3-dimensional planet has center at (0, 0, 0) and radius 20 At any point of the surface of this planet, the temperature is T (x, y, z) = (x+y) 2 +(y z) 2 degrees What is the average temperature of the surface of this planet? Hint: Find the average of T (x, y, z) + T (y, z, x) + T (z, x, y) 8 Plane coloring (Based on problem A4 from Putnam competition, 1988) Prove that if we paint every point of the plane in one of three colors, there will be two points one inch apart with the same color Is this result necessarily true if we replace three by nine? Hint: Argue by contradiction Find a property of colors of points 3 inches apart 9 Eight root of continued fraction (Problem B4 from Putnam competition, 1995) Evaluate 1 2207 8 1 2207 2207 Express your answer in the form (a + b c)/d, where a, b, c, d, are integers Hint: Define a 0 = 2207, a n = 2207 1/a n 1 (n > 0) Find limit of a n Use (x + 1/x) 2 = x 2 + 2 + 1/x 2 10 Function on rational numbers (Based on problem B4 from Putnam competition, 2001) Let S denote the set of rational numbers different from 1, 0, +1 Define f : S S by f(x) = x 1/x Let S n be defined recursively as S 0 = S, S n+1 = f(s n ) = image of S n by f for n 0 Prove that for every rational number r, r is not in S n if n is large enough Hint: Define the height of a rational number p/q expressed in lowest terms as the function H(p/q) = p + q

PUTNAM TRAINING PROBLEMS 3 11 Honey pots (AMM #11002) Pooh Bear has 2N + 1 honey pots No matter which one of them he sets aside, he can split the remaining 2N pots into two sets of the same total weight, each consisting of N pots Must all 2N + 1 pots weight the same? Hint: Pose it as a linear system of equations and find the rank of the coefficient matrix 12 Composite values of a polynomial Let p(x) be a non-constant polynomial such that p(n) is an integer for every positive integer n Prove that p(n) is composite for infinitely many values of n Hint: Find a value of n such that p(n) is a multiple of p(k) (different from p(k)) 13 Integer expression Let r be a number such that r + 1/r is an integer Prove that for every positive integer n, r n + 1/r n is an integer Hint: Induction 14 Regions on a sphere (Zeitz 2214) A great circle is a circle drawn on a sphere that is an equator, ie, its center is also the center of the sphere There are n great circles on a sphere, no three of which meet at any point They divide the sphere into how many regions? Hint: Experiment (with 1, 2, 3, 4, great circles) State a conjecture Prove it by induction 15 The round random table In the border of a perfectly circular piece of wood we choose n points at random to place legs and make a table What is the probability that the table will stand without falling? Hint: For each point P 1, P 2,, P n consider the event E i = 0 j i, where P i P j = arc from P i to P j measured anticlockwise P i P j < π for all 16 Integral involving (tan x) 2 (Problem A3 from Putnam competition, 1980) Evaluate π/2 0 dx 1 + (tan x) 2 Hint: Make the substitution u = π/2 x and use symmetry 17 Sum of sin k Prove that Hint: Complex numbers n sin k = sin n 2 sin 1 2 k=0 sin n+1 2 18 Rational distances on the unit circle (A version of this problem appeared in the IMO of 1975) Prove that there are infinitely many points on the unit circle x 2 +y 2 = 1 such that the distance between any two of them is a rational number Hint: If sin α, cos α, sin β and cos β are all rational, so are sin (α + β) and cos (α + β) Also: If sin α and cos α are rational numbers different from 0, 1 and 1, then α is not

PUTNAM TRAINING PROBLEMS 4 a rational multiple of π (See eg Appendix D of I Niven s Numbers: Rational and Irrational, NML vol 1, p 129) 19 Subsets of {1, 2,, 2n} (Zeitz, example 337) Prove that any (n + 1)-element subset of {1, 2,, 2n} contains two integers one of which divides the other one (Zeitz 3318) Prove that any (n + 1)-element subset of {1, 2,, 2n} contains two integers that are relatively prime Hint: Pigeonhole Principle 20 Rooks on a chessboard (Zeitz 3326) Forty-one rooks are placed on a 10 10 chessboard Prove that there must exist 5 rooks, none of which attack each other (Recall that rooks attack each piece located on its row or column) Hint: Pigeonhole Principle 21 Cosine expression (Zeitz 4220) Show that if x+1/x = 2 cos a, then for any integer n, x n + 1/x n = 2 cos na Hint: Complex numbers n 1 22 Product of sines (Zeitz 4224) Find a close-form expression for sin kπ n n 1 Hint: Write sin t = (e ti e ti )/2i Look at the polynomial (x e 2πik/n ) 23 Polynomial with a factor x 1 (USAMO 1976, Zeitz 4226) If P (x), Q(x), R(x), S(x) are polynomials such that k=1 P (x 5 ) + xq(x 5 ) + x 2 R(x 5 ) = (x 4 + x 3 + x 2 + x + 1)S(x), prove that x 1 is a factor of P (x) Hint: Roots of unity 24 Combinatorial identity Prove that for any positive integer n ( ) 2 ( ) 2 ( ) 2 ( ) 2 ( ) n n n n 2n + + + + = 0 1 2 n n where ( ) a b = a! (binomial coefficient) b!(a b)! Hint: Write (1 + x) 2n in two different ways 25 Coefficient of x k (Putnam 1992, Zeitz 4313) 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 k ( )( ) n n Q(n, k) = j k 2j Hint: Factor 1 + x + x 2 + x 3 j=0 k=1

PUTNAM TRAINING PROBLEMS 5 26 P-balanced sequences (Leningrad Mathematical Olympiad 1991, Zeitz 4316) A finite sequence a 1, a 2,, a n is called p-balanced if any sum of the form a k + a k+p + a k+2p + is the same for any k = 1, 2, 3, Prove that if a sequence with 50 members is p-balanced for p = 3, 5, 7, 11, 13, 17, then all its members are equal zero Hint: Look at the polynomial p(x) = a 1 + a 2 x + a 3 x 2 + + a 50 x 49 27 Constant in an interval Believe it or not the following function is constant in an interval [a, b] (a < b) Find that interval and the constant value of the function f(x) = x + 2 x 1 + x 2 x 1 Hint: u 2 = u 28 The largest factorial (Zeitz 5513) The notation n! (k) means take factorial of n k times For example, n! (3) means ((n!)!)! What is bigger, 1999! (2000) or 2000! (1999)? Hint: (No hint needed) 29 Polynomial with coefficients ±1 (Inspired on Putnam 1968, B6) Prove that a polynomial with only real roots and all coefficients equal to ±1 has degree at most 3 Hint: Find the sum of the squares of the roots, their product, and use the arithmeticgeometric mean inequality (AGM): the arithmetic mean of a list of numbers is that their geometric mean 30 An inequality (Zeitz 5539) If a, b, c > 0, prove that (a 2 b + b 2 c + c 2 a)(ab 2 + bc 2 + ca 2 ) 9a 2 b 2 c 2 Hint: Cauchy-Schwarz inequality: x i y i x 2 i yi 2 i i i 31 No two consecutive elements (Zeitz 646) Find the number of subsets of {1, 2,, n} that contain no two consecutive elements of {1, 2,, n} Hint: Divide the subsets into two classes, subsets containing element n and subsets not containing element n State a recurrence 32 Ice-cones (Zeitz 6315) Imagine that you are going to give n kids ice-cream cones, one cone per kid, and there are k different flavors available Assuming that no flavor gets mixed, find the number of ways we can give out the cones using all k flavors Hint: Solve the problem without the restriction using all k flavors Then subtract the distributions of ice-cream cones that miss at least one color Compute these using the Principle of Inclusion-Exclusion (PIE) 33 Permutations with property P (IMO 1989, Zeitz 6316) Let a permutation of {1, 2,, 2n} have property P if it contains at least two neighboring elements with difference n; for instance for n = 2 permutation 1324 has the property P ( 1 3 = 2), however 4321 does not Show that, for each n, there are more permutations with property P than without it

Hint: Principle of Inclusion-Exclusion PUTNAM TRAINING PROBLEMS 6 34 Eight-digit numbers in base 3 Let A be the set of all 8-digit numbers in base 3 (so they are written with the digits 0,1,2 only), including those with leading zeroes such as 00120010 Prove that given 4 elements from A, two of them must coincide in at least 2 places Hint: Pigeonhole Principle 35 Harmonic sum (Zeitz 7120) Show that 1 + 1/2 + 1/3 + + 1/n cannot be an integer if n 2 Hint: Look at the parity of numerator and denominator (in lowest terms) 36 Sums of digits (IMO 1975; Zeitz 7210) Let f(n) denote the sum of the digits of n Let N = 4444 4444 Find f(f(f(n))) without a calculator Hint: Think modulo 9 37 Primes of the form 4n + 3 Prove that there are infinitely many prime numbers of the form 4n + 3 Hint: Assume that the set of primes of the form 4n + 3 is finite Call their product P Look at P 2 2 38 Integer part of a huge fraction (Putnam, 1986, A2) What is the units (ie, rightmost) digit of? 10 20000 10 100 +3 Hint: Look at the expression I = 1020000 3 200 10 100 +3 39 Additive continuous functions Find all continuous functions f : R R such that f(x + y) = f(x) + f(y) for every x, y Hint: Start by studying the cases in which x is a natural number, an integer, a rational number 40 Two functions (Putnam 1991, B2) Suppose f and g are nonconstant, differentiable, real-valued functions on R Furthermore, suppose that for each pair of real numbers x and y f(x + y) = f(x) f(y) g(x) g(y) g(x + y) = f(x) g(y) + g(x) f(y) If f (0) = 0 prove that (f(x)) 2 + (g(x)) 2 = 1 for all x Hint: Define h(x) = f(x) + ig(x) and reinterpret the conditions as h(x + y) = h(x) h(y) Get a differential equation for h 41 Derivatives of a function (Putnam 1992, A4) Let f be an infinitely differentiable real-valued function defined on the real numbers If f( 1 ) = n2, n = 1, 2, 3, n n 2 + 1 compute the values of the derivatives f (k) (0), k = 1, 2, 3,

PUTNAM TRAINING PROBLEMS 7 Hint: Compare to the function g(x) = 1/(1 + x 2 ) Note that the difference h(x) = f(x) g(x) is not necessarily zero, but we can say something about the derivatives of h at zero (recall Rolle s theorem: h (x) has zeros between the zeros of h(x)) 42 Infinite product (Zeitz 8328) Compute lim n { n k=1 ( 1 + k ) } 1/n n Hint: Natural log of that is a Riemann sum of a certain integral