Problems for Putnam Training

Transcription

1 Problems for Putnam Training 1 Number theory Problem 1.1. Prove that for each positive integer n, the number is not prime n n + 10 n 1 Problem 1.2. Show that for any positive integer n, there exists a positive multiple of n that contains only the digits 7 and 0. Problem 1.3. Let p be a prime number and p > 3. Let k = [2p/3], where [ ] denotes the integer part of. Prove that the sum of binomial coefficients is divisible by p 2. Recall that C 1 p + C 2 p + + C k p C k n = n! (n k)! k!. Problem 1.4. Consider the following infinite number triangle: Each number is the sum of three numbers of the previous row: the number immediately above it, the number immediately to the right and the number immediately to the left. If no number appear in one or more of these positions, the number zero is used. Prove that every row, beginning with the third row, contains at least one even number. 2 Matrices Problem 2.1. Let G be a finite set of real n n matrices {M i }, 1 i r, which form a group under matrix multiplication. Suppose that r i=1 tr(m i) = 0, where tr(a) denotes the trace of the matrix A. Prove that r i=1 M i is the n n zero matrix. 1

2 Problem 2.2. Let H be an n n matrix all of whose entries are ±1 and whose rows are mutually orthogonal. Suppose H has an a b submatrix whose entries are all 1. Show that ab n. Problem 2.3. Find a 2 2 matrix A such that ( sin(a) = 0 1 ) or prove that such A does not exist. Recall that sin(x) has the following Taylor series (with infinite radius of convergence) sin(x) = n=0 ( 1) n (2n + 1)! x2n+1. Hint for Problem 2.3. A function f(x) has a Taylor series f(x) = a n x n n=0 with the infinite radius of convergence. Express f(a) for ( ) x y A = 0 x in terms of f(x) and f (x). 3 Combinatorics Problem 3.1. For a partition π of {1, 2, 3, 4, 5, 6, 7, 8, 9}, let π(x) be the number of elements in the part containing x. Prove that for any two partitions π and π, there are two distinct numbers x and y in {1, 2, 3, 4, 5, 6, 7, 8, 9} such that π(x) = π(y) and π (x) = π (y). [A partition of a set S is a collection of disjoint subsets (parts) whose union is S.] Problem 3.2. In a community every two acquaintances have no common acquaintances. Furthermore, every two people who do not know each other have exactly two common acquaintances. Show that everybody in this community has the same number of acquaintances. Problem 3.3. Determine, with proof, the number of ordered triples (A 1, A 2, A 3 ) of sets which have the property that (i) A 1 A 2 A 3 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, and (ii) A 1 A 2 A 3 =. Express your answer in the form 2 a 3 b 5 c 7 d, where a, b, c, d are nonnegative integers. 2

3 4 Polynomials Problem 4.1. Let k be the smallest positive integer for which there exist distinct integers m 1, m 2, m 3, m 4, m 5 such that the polynomial p(x) = (x m 1 )(x m 2 )(x m 3 )(x m 4 )(x m 5 ) has exactly k nonzero coefficients. Find, with proof, a set of integers m 1, m 2, m 3, m 4, m 5 for which this minimum k is achieved. Problem 4.2. Prove that if the integers a 1, a 2,..., a n are all distinct then the polynomial (x a 1 ) 2 (x a 2 ) 2... (x a n ) cannot be written as a product of two other polynomials with integral coefficients. 5 Calculus Problem 5.1. Let f be a real function with a continuous third derivative such that f(x), f (x), f (x), f (x) are positive for all x. Suppose that f (x) f(x) for all x. Show that f (x) < 2f(x) for all x. Problem 5.2. Show that for every positive integer n the polynomial has at most one real root. P n (x) = 1 + x + x2 2! + x3 3! + + xn n! Problem 5.3. Given a point (a, b) with 0 < b < a, determine the minimum perimeter of a triangle with one vertex at (a, b), one on the x-axis, and one on the line y = x. You may assume that a triangle of minimum perimeter exists. 6 Inequalities Problem 6.1. Find the smallest positive integer n such that for every integer m with 0 < m < 1993, there exists an integer k for which m 1993 < k n < m Problem 6.2. Let a and b be positive numbers. Find the largest number c, in terms of a and b, such that a x b 1 x sinh ux u(1 x) a + bsinh sinh u sinh u for all u with 0 < u c and for all x, 0 < x < 1. (Note: sinh u = (e u e u )/2.) 3

4 Problem 6.3. Let m be a positive integer and let G be a regular (2m + 1)-gon inscribed in the unit circle. Show that there is a positive constant A, independent of m, with the following property. For any points p inside G there are two distinct vertices v 1 and v 2 of G such that p v 1 p v 2 < 1 m A m 3. Here s t denotes the distance between the points s and t. 7 Unsorted Problem 7.1. On a large, flat field, n people (n > 1) are positioned so that for each person the distances to all the other people are different. Each person holds a water pistol and at a given signal fires and hits the person who is closest. When n is odd, show that there is at least one person left dry. Problem 7.2. Five points are placed on a sphere. Prove that 4 of them lie on a closed hemisphere. Problem 7.3. We have two absolutely identical chicken eggs and a building with 100 floors. We start counting the floors from the ground floor. If we drop any of these eggs from the n-th floor of this building then the egg will break but if we drop it from the previous (n 1)-th floor then the egg will not break. The number n may equal 1. Describe an algorithm which uses these two eggs to find the number n and which uses the smallest number of steps in its worst case scenario. 8 Easy problems. Unsorted Yes! Yes! It does make sense to solve easy problems to learn typical tricks like mathematical induction, pigeonhole principle, arithmetic mod n, and others... Problem 8.1. Find the remainder from division by 7 of the number Problem 8.2. Let p be a prime number and n be a natural number which is not divisible by p. Prove that there exists a natural number m such that n m = 1 mod p. If you do not know what is mod then the latter is equivalent to the remainder of the division n m by p is 1. Problem 8.3. Prove that is divisible by

5 Problem 8.4. Find the least number A such that for any two squares of combined area 1, there exists a rectangle of area A such that the two squares can be packed in the rectangle (without interior overlap). You may assume that the sides of the squares are parallel to the sides of the rectangle. Problem 8.5. Prove the following identity by induction: ( ) 2 n(n + 1) n 3 =. 2 Problem 8.6. Prove that one cannot draw a continuous line which intersects each of the 16 segments on Figure 1 exactly once. How to draw Figure 1 on the torus so that one can draw a continuous line with the above desired property? [Torus is the surface of bagel] Figure 1: Is it possible to draw a continuous line which intersects each of the 16 segments exactly once? 5