First Putnam Practice Meeting

Hosted by Quo-Shin Chi

First Putnam Practice Meeting Come to our first practice session, 4-6pm, Friday, September 7, in Room 241, Compton Hall, if you are interested in solving problems of the sort in the following. 1. Find the smallest positive integer j such that for every polynomial p(x) with integer coefficients and for every k, the integer d jp(x) dxj |x=k, which is the jth derivative of p(x) at k, is divisible by 2018. 2. Suppose S is a finite set of points in the plane such that the area of the triangle ∆ABC is at most 1 whenever A, B, C are in S. Show that there is a triangle of area 4 that (together with its interior) covers the set S. 3. Let f be a real-valued function on the plane such that for every square ABCD in the plane, f(A) + f(B) + f(C) + f(D) = 0. Does it follow that f(P) = 0 for all points P in the plane? 4. Alan and Barbara play a game in which they take turns filling entries of an initially empty 2018 × 2018 array. Alan plays first. At each turn, a player chooses a real number and places it in a vacant entry. The game ends when all the entries are filled. Alan wins when the determinant of the resulting matrix is nonzero; Barbara wins if it is zero. Which player has a winning strategy? 5. Show that every positive integer is a sum of one or more numbers of the form 2r3 s , where r and s are nonnegative integers and no summand divides another. (For example, 23 = 9 + 8 + 6.) 6. Prove that there are infinitely many integers n such that n, n+1, n+2 are each the sum of the squares of two integers. 7. Show that every nonzero coefficient of the Taylor series of (1 − x + x 2 )e x about x = 0 is a rational number whose numerator (in lowest terms) is either 1 or a prime number.