Putnam Practice Problems #2
October 6, 2015
(in no particular order)
(1) If the numbers 1 through 64 are placed on the sixty-four squares of a checkerboard, define
the gap to be the largest difference between two numbers on adjacent squares (two squares
are adjacent if they share a side or even a corner). Determine, with proof, the smallest
possible gap among all ways of placing the numbers on the squares.
(2) Let g : [0, 1] → R be a continuous function, not identically zero, such that g(0) = g(1) =
0. Prove that there exists a square in the plane with two vertices on the open line segment
between (0, 0) and (1, 0) and the other two vertices on the graph of g.
(3) Let A1 , A2 , . . . , An be points on a circle of radius 1. Prove that there exists a point P on the
circle such that P A1 + P A2 + · · · + P An > n. (Here, P Aj denotes the distance between
the points P and Aj .)
(4) Two players, Natalia and Bruce, take turns removing marbles from a jar which initially
contains 2015 marbles. On each turn, the player may remove any number of marbles that
is a power of 2 (including 20 = 1 marble); the player who takes the last marble wins. If
Natalia has the first turn, who has the winning strategy, and what is it?
(5) Define An to be the average value of the function
x ln(1 + x/n)
1+x
on the interval 0 ≤ x ≤ n. Determine, with proof, limn→∞ An .
(6) Let f (x), g(x) be real-valued functions defined on (0, 1). Suppose that for all x, y ∈ (0, 1),
2
f (x) − g(y) ≤ y + x .
x
y
Prove that lim f (x) exists.
x→0+
(7) For each positive integer n, define the polynomial
n
X
(−1)k k
Fn (x) =
x .
k!
k=0
Prove that for every integer k ≥ 3, there exist positve integers m 6= n such that Fm (k) =
Fn (k).
(continued on next page)
(8) What is the measure of the angle labeled x? (As the figure indicates, angle ABC is bisected
by line BD.)