Questions to the 64rd annual Putnam exam:
Source: http://www.math.niu.edu/~rusin/problems-math/
Questions from the 64th Putnam exam, Dec 6 2003.
A1
Let n be a fixed positive integer. How many ways are there to write n
as a sum of positive integers, n = a_1 + a_2 + ... + a_k, with k an
arbitrary positive integer and a_1 \le a_2 \le .. a_k \le a_1 + 1?
For example, with n=4, there are four ways: 4, 2+2, 1+1+2, 1+1+1+1.
A2
Let a1, a2, ..., a_n and b1, b2, ..., b_n be nonnegative real numbers.
Show that
(a1 a2 ... a_n)^{1/n} + (b1 b2 ... b_n)^{1/n} <=
( (a1+b1) (a2+b2) ... (a_n + b_n) )^{1/n}
A3
Find the minimum value of
| sin x + cos x + tan x + cot x + sec x + csc x |
for real numbers x.
A4
Suppose that a,b,c,A,B < C are real numbers, a\ne 0 and A \ne 0, such that
| a x^2 + b x + c | <= | A x^2 + B x + C |
for all real numbers x. Show that
| b^2 - 4 a c | <= | B^2 - 4 A C | .
A5
A Dyck n-path is a lattice path of n upsteps (1,1) and n downsteps (1,-1)
that starts at the origin O and never dips below the x-axis.
A return is a maximal sequence of contiguous downsteps that terminates
on the x-axis. For example, the Dyck 5-path illustrated has two returns,
of length 3 and 1 respectively.
*
/ \
* * *
/ \ / \
* * * *
/ \ / \
O *---------------*---*
Show that there is a one-to-one correspondence between the Dyck n-paths
with no return of even length and the Dyck (n-1) paths.
A6
For a set S of nonnegative integers, let r_S(n) denote the number of
ordered pairs (s_1, s_2) such that s_1 \in S, s_2 \in S, s_1 \ne s2,
and s_1 + s_2 = n. Is it possible to partition the nonnegative
integers into two sets A and B in such a way that r_A(n) = r_B(n)
for all n ?
================================
B1
Do there exist polynomials a(x), b(x), c(y), d(y) such that
1 + x y + x^2 y^2 = a(x) c(y) + b(x) d(y)
holds identically?
B2
Let n be a positive integer. Starting with the sequence
1, 1/2, 1/3, ..., 1/n form a new sequence of n-1 entries
3/4, 5/12, ..., (2n-1)/{2n(n-1)} by taking the averages of
two consecutive entries in the first sequence. Repeat the
averaging of neighbors on the second sequence to obtain a third
sequence of n-2 entries and continue the final sequence produced
consists of a single number x_n. Show that x_n < {2/n} .
B3
Show that for each positive integer n,
n ! = \Prod_{i=1}^n lcm{1, 2, ..., [n/i] } .
(Here lcm denotes the least common multiple, and [x] denotes the
greatest integer \le x .)
B4
Let f(z) = a z^4 + b z^3 + c z^2 + d z + e = a(z-r_1)(z-r_2)(z-r_3)(z-r_4)
where a,b,c,d,e are integers, a \ne 0. Show that if r_1 + r_2 is a
rational number, and if r_1 + r_2 \ne r_3 + r_4 then r_1 r_2 is a
rational number.
B5
Let A,B, and C be equidistant points on the circumference of a circle
of unit radius centered at O, and let P be any point in the circle's
interior. Let a, b, c be the distance from P to A, B, C respectively.
Show that there is a triangle with side lengths a, b, c and that the
area of this triangle depends only on the distance from P to O.
B6
Let f(x) be a continuous real-valued function defined on the interval
[0,1]. Show that
\int_0^1 \int_0^1 | f(x) + f(y) | dx dy > = \int_0^1 |f(x)| dx.
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