# EASY PUTNAM PROBLEMS

Remark. The problems in the Putnam Competition are usually
very hard, but practically

every session contains at least one problem very easy to solve—it still may need
some sort

of ingenious idea, but the solution is very simple. This is a list of “easy”
problems that have

appeared in the Putnam Competition in past years—Miguel A. Lerma

**2009-A1.** Let f be a real-valued function on the plane such that for every
square ABCD in the

plane, f(A)+ f(B)+ f(C)+ f(D) = 0. Does it follow that f(P) = 0 for all points P

in the plane?

**2009-B1. **Show that every positive rational number can be written as a
quotient of products of

factorials of (not necessarily distinct) primes. For example,

**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.

**
2008-A2. **Alan and Barbara play a game in which they take turns filling
entries of an initially

empty 2008 × 2008 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?

**2008-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.)

**Find all values of α for which the curves and**

2007-A1.

2007-A1.

are tangent to each other.

**Let f be a polynomial with positive integer coefficients. Prove that if n is a positive**

2007-B1.

2007-B1.

integer, then f(n) divides f(f(n) + 1) if and only if n = 1. [Note: one must assume

f is nonconstant.]

**2006-A1.**Find the volume of the region of points (x, y, z) such that

**2006-B2.** Prove that, for every set
of n real numbers, there exists a nonempty

subset S of X and an integer m such that

**2005-A1.** Show that every positive integer is a sum of one or more numbers
of the form 2^{r}3^{s},

where r and s are nonnegative integers and no summand divides another. (For
example,

23 = 9 + 8 + 6.)

**2005-B1. **Find a nonzero polynomial P(x, y) such that
for all real numbers a.

(Note: is the greatest integer less than or
equal to v.)

**2004-A1. **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?

**2004-B2. **Let m and n be positive integers. Show that

**2003-A1.** Let n be a fixed positive integer. How many ways are there to
write n as a sum of

positive integers, , with k an arbitrary
positive integer and

? For example, with n = 4 there are four
ways: 4, 2+2,

1+1+2, 1+1+1+1.

**2002-A1.** Let k be a fixed positive integer. The n-th derivative of
has the form P

where is a polynomial. Find
.

**2002-A2. **Given any five points on a sphere, show that some four of them
must lie on a closed

hemisphere.

**2001-A1. **Consider a set S and a binary operation *, i.e., for each a, b ∈
S, a * b ∈ S. Assume

(a *b) * a = b for all a, b ∈ S. Prove that a * (b * a) = b for all a, b ∈ S.

**2000-A2**. Prove that there exist infinitely many integers n such that n,
n+1, n+2 are each the

sum of the squares of two integers. [Example: .]

**1999-A1.** Find polynomials f(x), g(x), and h(x), if they exist, such that
for all x,

**1998-A1. **A right circular cone has base of radius 1
and height 3. A cube is inscribed in the

cone so that one face of the cube is contained in the base of the cone. What is
the

side-length of the cube?

**1997-A5.** Let denote the number of
ordered n-tuples of positive integers
such

that . Determine whether
is even or odd.

**1988-B1.** A composite (positive integer) is a product ab with a and b not
necessarily distinct

integers in {2, 3, 4, . . . }. Show that every composite is expressible as
xy+xz+yz+1,

with x, y, and z positive integers.