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Putnam 2008 QUESTIONS
Posted:
Dec 7, 2008 3:30 AM
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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 n-gon in counterclockwise
order.
Prove that at least one of f(x) and g(x) has degree greater than or
equal to n-1 .
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 ln|G|
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
4-dimensional 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^2-1) are distinct modulo
p^2.
Show tht h(0), h(1), .., h(p^3-1) 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 _k-limited_ if | sigma(i) - i | <= k for all i.
Prove that the number of k-limited permutations of {1, 2, ..., n} is
odd
if and only if n = 0 or 1 (mod 2k+1).
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