Putnam Practice Problems #3
October 20, 2015
(in no particular order)
(1) Anu plays a solitaire game with the following rules: the integers from 1 to 2015 are printed
on a piece of paper, and a coin is placed on one of the numbers. On each turn, Anu can
move the coin from the number it’s on to any number that can be obtained by multiplying
by a prime or dividing by a prime. Anu’s goal is to move the coin from one number to
another in the fewest possible moves. Find the smallest number M such that Anu can
move the coin between any two numbers on the paper in at most M moves.
(2) Find all quadruples (w, x, y, z) of real numbers such that
1 1
1
1
+ + = .
w + x + y = z and
w x y
z
(3) Suppose you have 2n points in the plane, no three of which are collinear, and that n of the
points are colored pink and the other n are colored violet. Show that n line segments can
be drawn so that each line segment has one pink endpoint and one violet endpoint, and no
two line segments intersect (including at their endpoints).
(4) Let
a1,1 a1,2 a1,3 . . .
a2,1 a2,2 a2,3 . . .
a3,1 a3,2 a3,3 . . .
..
..
..
...
.
.
.
be a doubly infinite array of positive integers, and suppose each positive integer appears
exactly eight times in the array. Prove that am,n > mn for some pair of positive integers
(m, n).
(5) Calculate
Z
0
1
X
2014
n
(−1)
√
n
1−
x2015−n
dx.
n=1
(6) Let n be a positive integer, and let e = 2.71828 . . . denote the base of the natural logarithm.
Prove that when (n + 1)!/e is rounded off to the nearest integer, the result is always a
multiple of n.
(7) Let f (x) and g(x) be two differentiable functions such that f (0) = g(0) = 0 and f (1) =
g(1) = 1. Suppose that f 0 (x) > 0 and g 0 (x) > 0 for every x ∈ (0, 1). Prove that there exist
x1 , x2 ∈ (0, 1) such that f (x1 ) = g(x2 ) and f 0 (x1 ) = g 0 (x2 ).
(continued on next page)
(8) For any positive integer k, let Mk be the k × k matrix whose entries are 10, 102 , . . . ,
2
10k , placed in order as you spiral inwards in a clockwise direction starting from the upper
left-hand entry. For example,

 1
10 102 103
M3 = 108 109 104  .
107 106 105
Let dk be the last (rightmost) nonzero digit of | det Mk |. For example,
det M3 = −9,899,990,100,000,000,000,
and so d3 = 1. Find, with proof, d2015 .