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Topic:
Putnam 2008 QUESTIONS
Replies:
2
Last Post:
Dec 8, 2008 5:55 AM




Putnam 2008 QUESTIONS
Posted:
Dec 7, 2008 3:30 AM


Here are the questions on the 2008 Putnam exam held today (Dec 6 2008)
A1. Let f : R^2 > R be a function such that f(x,y)+f(y,z)+f(z,x)=0 for all real numbers x,y, and z. Prove that there exists a function g : R > R such that f(x,y) = g(x)g(y) for all real numbers x and y.
A2. Alan and Barbara play a game in which they take turns filling entries of an initially empty 2008x2008 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 if the determinant of the resulting matrix is nonzero; Barbara wins if it is zero. Which player has a winning strategy?
A3. Start with a finite sequence a_1, a_2, ..., a_n of positive integers. If possible, choose two indices j<k such that a_j does not divide a_k, replace a_j and a_k respectively by gcd(a_j, a_k) and lcm(a_j, a_k) respectively. Prove that if this process is repeated, it must eventually stop and the final sequence does not depend on the choices made. (Note: gcd means greatest common divisor and lcm means least common multiple.)
A4. Define f : R > R by f(x) = { x , if x <= e { x f(ln x) , if x > e.
Does sum_{n=1}^{oo} 1/f(n) converge?
A5. Let n >=3 be an integer. Let f(x) and g(x) be polynomials with real coefficients such that the points (f(1),g(1)), (f(2),g(2)), ..., (f (n),g(n)) in R^2 are the vertices of a regular ngon in counterclockwise order. Prove that at least one of f(x) and g(x) has degree greater than or equal to n1 .
A6. Prove that there exists a constant c > 0 such that in every nontrivial finite group G there exists a sequence of length at most c lnG with the property that each element of G equals the product of some subsequence. (The elements of G in the sequence are not required to be distinct. A _subsequence_ of a sequence is obtained by selecting some of the terms, not necessarily consecutive, without reordering them; for example 4, 4, 2 is a subsequence of 2, 4, 6, 4, 2, but 2, 2, 4 is not.)
B1. What is the maximum number of rational points that can lie on a circle in R^2 whose center is not a rational point? (A _rational point_ is a point both of whose coordinates are rational numbers.)
B2. Let F_0(x) = ln x . For n >=0 and x > 0, let F_{n+1} = \int_0^x F_n(t) dt. Evaluate lim_{n>oo} { n! F_n(1) } / { ln n } .
B3. What is the largest possible radius of a circle contained in a 4dimensional hypercube of side length 1 ?
B4. Let p be a prime number. Let h(x) be a polynomial with integer coefficients such that h(0), h(1), ..., h(p^21) are distinct modulo p^2. Show tht h(0), h(1), .., h(p^31) are distinct modulo p^3 .
B5. Find all continuously differentiable functions f: R > R such that for every rational number q, the number f(q) is rational and has the same denominator as q. (The denominator of a rational number q is the unique positive integer b such that q = a/b for some integer a with gcd (a,b)=1.) (Note: gcd means greatest common divisor.)
B6. Let n and k be positive integers. Say that a permutation sigma of {1, 2, ..., n} is _klimited_ if  sigma(i)  i  <= k for all i. Prove that the number of klimited permutations of {1, 2, ..., n} is odd if and only if n = 0 or 1 (mod 2k+1).



