Download KU Putnam Training Session Induction, Recursion, and Pigeonhole

KU Putnam Training Session
Induction, Recursion, and Pigeonhole Principle
November 5, 2015
Mat Johnson
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Induction and Recursion
Suppose we have infinitely many statements P (n), indexed by n = a, a + 1, a + 2, . . . for
some a ∈ Z, that we want to verify. Mathematical induction guarantees all the statements
P (n) are true provided
(1) P (a) is true. (This is the base step.)
(2) for each n = a, a + 1, a + 2, . . ., P (n) being true implies that P (n + 1) is true. (This
is the inductive step.)
Problems:
1. Show for all n = 1, 2, 3, . . . that
n
X
k=
k=1
n(n + 1)
.
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2. Show for all n = 1, 2, 3, . . . that
12 + 32 + 52 + . . . + (2n − 1)2 =
n(4n2 − 1)
.
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3. Show that n3 divides 2 (n + 1)n − n2 − 1 for every integer n ≥ 2.
4. The Fibonacci numbers are defied by the recursive sequence
Fn+2 = Fn+1 + Fn ,
n = 1, 2, 3, . . .
with F1 = F2 = 1. Show for every n = 1, 2, 3, . . . that
F12 + F22 + . . . + Fn2 = Fn Fn+1 .
5. Let f be a real-valued functions defined on all the positive integers by f (1) = 1 and
f (1) + f (2) + . . . + f (n) = n2 f (n), n ≥ 2.
Find and prove an explicit formula for f (n).
6. Let x1 , x2 , . . . , xn be real numbers satisfying 0 ≤ xi ≤ 1 for each i. Show that
(1 + x1 )(1 + x2 ) · · · (1 + xn ) ≤ 2n−1 (1 + x1 x2 · · · xn ).
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7. 100 passengers are going to board a plane with 100 seats. Each passenger has an
assigned seat, and the passengers board one at a time.
The first passenger on the plane can not read their seat assignment. They choose a
seat at random and sit down.
The rest of the passengers can read their seat assignments. If someone is already
sitting in their assigned seat, they choose a seat at random and sit down. If nobody
is sitting in their assigned seat, they sit in their assigned seat.
What is the probability that the last person on board will sit in their assigned seat?
8. (Problem B1, 2009 Putnam) Show every positive rational number can be written as
a quotient of products of factorials of (not necessarily unique) primes. For example,
10
2! · 5!
=
.
9
3! · 3! · 3!
9. (Problem A1, 2012 Putnam) Let d1 , d2 , d3 . . . d12 be real numbers in the open interval
(1, 12). Show there exists distinct indices i, j, k such that di , dj , dk are side lengths of
an acute triangle.
10. (Problem A1, 2005 Putnam) Show every positive integer is the sum of one or more
numbers of the form 2r 3s for some r, s ∈ {0, 1, 2, 3, . . .} such that no summand divides
the other.
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Pigeonhole Principle
The Pigeonhole Principle is one of the most obvious of mathematical statements, but typically one must be very clever in its application.
Pigeonhole Principle: If n + 1 objects are put into n boxes, then at least one box contains
two or more objects.
Problems:
1. Show that among any group of 13 people, at least two of them have birthdays in the
same month.
2. Consider a group of n married couples, how many of the 2n people must be selected
to ensure that you have selected at least one married couple?
3. In any group of n people, show there are at least two people having the same number
of friends within the group. Assume that friendship is a mutual relationship.
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4. Prove that if n + 1 distinct numbers are selected from the set {1, 2, . . . , 2n − 1}, then
some two of these numbers must sum to 2n. Show that it is also possible to select
n distinct numbers so that no two of them sum to 2n. Formulate and prove similar
statements for collections of numbers selected from {1, 2, . . . , 2n}.
5. Given n integers a1 , a2 , . . . , an , not necessarily distinct, there exists integers k and l
with 0 ≤ k < l ≤ n such that the sum ak+1 + ak+2 + . . . + al is a multiple of n.
6. Prove that any collection of eigeht distinct integers contains distinct integers x and y
such that x − y is a multiple of 7.
7. If eleven integers are selected from {1, 2, . . . , 20}, show the selection includes integers
a and b such that a − b = 2.
8. If 5 points are selected from the interior of a 1 ×√1 square, show there
√ exists two points
whose distance from each√other is at most 1/ 2. Show that 1/ 2 is best possible,
i.e. the claim is false if 1/ 2 is replaced by any smaller positive number.
9. Over a period of 44 days, suppose Big Jay plays chess against Little Jay at least once
per day, and a total of 70 games are played in all. Show there exists a period of
consecutive days during which exactly 17 days are played.
10. (Problem A2, Putnam 2002) Given any 5 points on a sphere, show some 4 of them
must lie on a closed hemisphere.
11. Let p be a prime number and consider the set Gp := {1, 2, 3, . . . (p − 1)} equipped with
multiplication. Show that given any x ∈ Gp , there exists a y ∈ Gp such that xy = 1
mod(p). (Here, Gp is really the set Zp := Z/pZ.)
12. (Problem A1, Putnam 2013) Recall that a regular icosahedron is a convex polyhedron
having 12 vertices and 20 faces; the faces are congruent to equilateral triangles. On
each face of a regular icosahedron is written a nonnegative integer such that the sum
of all 20 integers is 39. Show that there are 2 faces that share a vertex and have the
same integer written on them.
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