Solutions to the Problems on Derivatives and Integrals
1: Let 0 < k < 1, and let f (x) be dierentiable with f (x) k for all x R. Show that f (x) has a xed point.
Solution: If f (0) = 0 then 0 is a xed point of f . Suppose that f (0) = b > 0. For any x > 0

Combinatorics
1: Find the number of words of length n on the alphabet cfw_0, 1 with exactly m blocks of the
form 01.
2: Find the number of words of length n on the alphabet cfw_0, 1, 2, 3 with an even number of
zeros.
3: Find the number of words of length

Solutions to the Problems on Sequences, Series and Products
6
. Determine whether cfw_an converges, and if so then nd the limit.
an + 1
6an + 6
6
6
36
=
Solution: Note that an+2 =
=6
= 6
. If the sequence of odd terms cfw_a2k+1
an+1 + 1
an + 7
an + 7
an

Solutions to the Number Theory Problems
1: Show that (2 +
3)n is odd for every positive integer n.
Solution: Notice that (2+ 3)n +(2 3)n =
i
2ni 3 +
i
(1)i n 2ni 3 = 2
i
i=0
i=0
which is an even number, and we have 0 < (2 3)n < 1, so (2 + 3)n is odd.
n
n

Solutions to Some of the Problems on Polynomials
1: Let p(x) be a polynomial over Z with the property that for some integer k , p(k ), p(k + 1) and p(k + 2) are
all multiples of 3. Show that p(n) is a multiple of 3 for every integer n.
Solution: Recall th

Solutions to the Combinatorics Problems
1: Find the number of words of length n on the alphabet cfw_0, 1 with exactly m blocks of the form 01.
Solution: There are n 1 locations between the digits in such a word. Let us call a location at which the
digits

Solutions to the Problems Using Invariants or Monovariants
1: Show that if 25 people play in a ping pong tournament then, at the end of the tournament,
the number of people who played an even number of games is odd.
Solution: Each game involves two people

The Faculty of Mathematics at the University of Waterloo
in association with
The Centre for Education in Mathematics and Computing
presents
The Tenth Annual
Small c Competition
for First and Second Year Students
Saturday 25 September 2010
Time: 1 hour
Cal

The Faculty of Mathematics at the University of Waterloo
in association with
The Centre for Education in Mathematics and Computing
presents
The Ninth Annual
Small c Competition
for First and Second Year Students
Friday 02 October 2009
Time: 1 hour
Calcula

The Faculty of Mathematics at the University of Waterloo
in association with
The Centre for Education in Mathematics and Computing
presents
The Eighth Annual
Small c Competition
for First and Second Year Students
Friday 33 September 2008
Time: 1 hour
Calc

The Faculty of Mathematics at the University of Waterloo
in association with
The Centre for Education in Mathematics and Computing
presents
The Seventh Annual
Small c Competition
for First and Second Year Students
Saturday 22 September 2007
Time: 1 hour
C

The Faculty of Mathematics at the University of Waterloo
in association with
The Centre for Education in Mathematics and Computing
and
The Canadian Mathematics Competition
presents
The Sixth Annual
Small c Competition
for First and Second Year Students
Sa

SPECIAL K
Saturday November 5, 2011
10:00 am - 1:00 pm
1: Find the number of sequences a1 , a2 , , a6 with each ai cfw_1, 2, 3, 4 such that
a1 < a2 , a2 > a3 , a3 < a4 , a4 > a5 , a5 < a6 and a6 > a1 .
2: Find the number of solutions to the congruence x2

Solutions to the Special K Problems, 2010
1: Find the minimum possible discriminant = b2 4ac of a quadratic f (x) = ax2 + bx + c which satises the
requirement that f f f (0) = f (0).
Solution: Let f (x) = ax2 + bx + c. Then f (0) = c so we have
f f f (0)

SPECIAL K
Saturday November 6, 2010
10:00 am - 1:00 pm
1: Find the minimum possible discriminant = b2 4ac of a quadratic f (x) = ax2 + bx + c
which satises the requirement that f f f (0) = f (0).
2: Show that for every integer a, there exist innitely many

Solutions to the Special K Problems, 2009
1: Determine the number of ways the digits 1, 2, 3, , 8 can be arranged to form an 8-digit number which is
divisible by 11.
Solution: In order for an arrangement to give a multiple of 11, the alternating sum of th

SPECIAL K
Saturday November 7, 2009
9:00 am - 12:00 noon
1: Determine the number of ways the digits 1, 2, 3, , 8 can be arranged to form an 8-digit
number which is divisible by 11.
2: Find the largest integer n such that x8 x2 is a multiple of n for every

Solutions to the Special K Problems, 2008
1: Find the value of min max (x2 + xy ) and the value of max min (x2 + xy ).
|y |1 |x|1
|x|1 |y |1
y2
2
y2
4.
Solution: For xed y , let f (x) = x2 + xy = x +
The graph of f (x) is a parabola which is concave
up. F

SPECIAL K
Saturday November 1, 2008
9:00 am - 12:00 noon
1: Find the value of min max (x2 + xy ) and the value of max min (x2 + xy ).
|y |1 |x|1
|x|1 |y |1
2: Let f (x) = x3 5x + 1 and let g (x) =
graph of y = f (g (x).
x1
. Find the number of x-intercept

SPECIAL K
Saturday 27 October 2007
9 a.m. to 12 noon
1. The notation a89b means the four-digit (base 10) integer whose thousands digit is a, whose
hundreds digit is 8, whose tens digit is 9, and whose units digit is b.
Determine all pairs of non-zero digi

SPECIAL K
Saturday 04 November 2006
9 a.m. to 12 noon
1. A party of 100 mathies went to the circus. The total charge for admission was $95. For faculty,
the charge was $10. For grad students, the charge was $2.50. For undergrads, the charge was
$0.50. Det

SPECIAL K
Saturday 29 October 2005
9 a.m. to 12 noon
1. Suppose that A and B are points on the parabola y = x2 , with the tangent line to the parabola
which is parallel to AB . Show that the midpoint of AB , the point at which is tangent to
the parabola,

SPECIAL K
Saturday 06 November 2004
9 a.m. to 12 noon
1. Determine all possible pairs (x, y ) which satisfy
x + y + x + y = 56
x y + x y = 30
2. A square is drawn inside a rectangle of length a and width b, with one vertex of the square on the
11
1
diagon

SPECIAL K
Saturday November 1, 2003
9:00 am - 12:00 noon
1: How many times between midday and midnight is the hour hand of a clock at right angles
to the minute hand?
2: At the vertices of a cube are written eight distinct positive integers. On each of th

SPECIAL K
Saturday November 2, 2002
9:00 am - 12:00 noon
1: Xavier and Yolanda play a game on a board which consists of a narrow strip which is one
square wide and n squares long. They take turns at placing counters, which are one square
wide and two squa

SPECIAL K
Saturday October 27, 2001
9:00 am - 12:00 noon
1
= 2001, where the summation taken is over all non-empty subsets
i1 i2 ik
cfw_i1 , i2 , , ik of the set cfw_1, 2, , 2001.
1: Prove that
2: A diameter AB of a circle intersects a chord CD at the po

SPECIAL K
Saturday October 28, 2000
9:00 am - 12:00 noon
1: Let x > 1 be a real number, and n > 1 be an integer. Prove that
x1
n
.
x < 1+
n
2: Find the smallest (by area) right-angled triangle with integral sides in which a square with
integral sides can

Solutions to the Bernoulli Trials Problems for 2011
1: There exists a positive integer n such that for every integer a with 1,000 a 1,000,000, a is prime if and
only if gcd(a, n) = 1.
Solution: This is TRUE. Indeed we can take n to be the product of all p