Questions to the 65rd annual Putnam exam:
Source: http://www.math.niu.edu/~rusin/problems-math/
A-1.
Basketball star Shanille O'Keal's team statistician keeps track of the
number, S(N), of successful free throws she has made in her first N
attempts of the season. Early in the season, S(N) was less than 80%
of N, but by the end of the season, S(N) was more than 80% of N.
Was there necessarily a moment in between when S(N) was exactly 80% of N ?
A-2.
For i = 1,2 let T_i be a triangle with side lengths a_i, b_i, c_i,
and area A_i. Suppose that a_1 <= a_2, b_1 <= b_2, c_1 <= c_2,
and that T_2 is an acute triangle. Does it follow that A_1 <= A_2 ?
A-3.
Define a sequence { u_n } by u_0 = u_1 = u_2 = 1, and thereafter by the
condition that
( u_n u_{n+1} )
det( ) = n!
( u_{n+2} u_{n+3} )
for all n >= 0. Show that u_n is an integer for all n.
(By convention, 0! = 1.)
A-4.
Show that for any positive integer n there is an integer N such that
the product x_1 x_2 ... x_n can be expressed identically in the form
x_1 x_2 ... x_n =
\sum_{i=1}^N c_i ( a_{i1} x_1 + a_{i2} x_2 + ... + a_{in} x_n )^n
where the c_i are rational numbers and each a_{ij} is one of the
numbers, -1, 0, 1 .
A-5.
An m x n checkerboard is colored randomly: each square is independently
assigned red or black with probability 1/2 . We say that two squares,
p and q, are in the same connected monochromatic component if there is
a sequence of squares, all of the same color, starting at p and ending
at q, in which successive squares in the sequence share a common side.
Show that the expected number of connected monochromatic regions is
greater than m n / 8 .
A-6.
Suppose that f(x,y) is a continuous real-valued function on the unit
square 0 <= x <= 1, 0 <= y <= 1. Show that
\int_0^1 ( \int_0^1 f(x,y) dx )^2 dy +
\int_0^1 ( \int_0^1 f(x,y) dy )^2 dx
is less than or equal to
( \int_0^1 \int_0^1 f(x,y) dx dy )^2 +
\int_0^1 \int_0^1 ( f(x,y) )^2 dx dy .
B-1.
Let P(x) = c_n x^n + c_{n-1} x^{n-1} + ... + c_0 be a polynomial with
integer coefficients. Suppose that r is a rational number such that
P(r) = 0. Show that the n numbers
c_n r , c_n r^2 + c_{n-1}, c_n r^3 + c_{n-1} r^2 + c_{n-2} r, ...,
c_n r^n + c_{n-1} r^{n-1} + ... + c_1 r
are integers.
B-2.
Let m and n be positive integers. Show that
(m+n)! m! n!
----------- < ----- -----
(m+n)^{m+n} m^m n^n
B-3.
Determine all real numbers a > 0 for which there exists a nonnegative
continuous function f(x) defined on [0,a] with the property that the
region
R = { (x,y) ; 0 <= x <= a, 0 <= y <= f(x) }
has perimeter k units and area k square units for some real number k.
B-4.
Let n be a positive integer, n >= 2, and put theta = 2 pi / n.
Define points P_k = (k,0) in the xy-plane, for k = 1, 2, ..., n.
Let R_k be the map that rotates the plane counterclockwise by the
angle theta about the point P_k. Let R denote the map obtained
by applying, in order, R_1, then R_2, ..., then R_n.
For an arbitrary point (x,y), find, and then simplify, the coordinates
of R(x,y).
B-5.
Evaluate
infty 1 + x^{n+1} {x^n}
lim \prod (--------------)
x->1^- n=0 1 + x^n
[That's \lim_{x\to 1^{-}} \prod_{n=0}^{\infty}
\left( {{1+x^{n+1}}\over{1+x^n}} \right)^{x^n} in TeX -- djr]
B-6.
Let A be a non-empty set of positive integers, and let N(x) denote
the number of elements of A not exceeding x. Let B denote the set
of positive integers b that can be written in the form b = a - a' with
a \in A and a' \in A. Let b_1 < b_2 < ... be the members of B,
listed in increasing order. Show that if the sequence b_{i+1} - b_i is
unbounded, then lim_{x \to\infty} N(x)/x = 0.
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