Problem 1. Suppose f is a real function with domain R which has the intermediate value property: If f (a) < c < f (b), then f (x) = c for some a < x < b.
Suppose also for every rational r, that the set of all x with f (x) = r is closed.
Prove that f is co
Name '- " . '
. Date '-
MathCounts Competitions
' (2008) ,._' l. The ndings from a recent analysis of rental housing costs 1. 3
are given in the table. How much more expensive is the
cheapest monthly rental cost for a 3 Bedroom in the District
of Columbia
1. In this Number Wall, you add the
numbers next to each other and write
the sum in the block directly above the
two numbers. Which number will be in
the block labeled n?
1. _
46
15
n 4 8 7
2. How many integers between 500 and 1000 contain both the
dig
MATHCOUNTS
2004
Chapter Competition
Sprint Round
Problems 130
Name
School
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO.
This round of the competition consists of 30 problems. You will
have 40 minutes to complete the problems. You are not
allowed to use
MATHCOUNTS
2013
Chapter Competition
Target Round
Problems 1 and 2
Name
School
DO NOT BEGIN UNTIL YOU ARE INSTRUCTED
TO DO SO.
This section of the competition consists of eight problems, which will
be presented in pairs. Work on one pair of problems will b
MATHCOUNTS
2013
Chapter Competition
Sprint Round
Problems 130
HONOR PLEDGE
I pledge to uphold the highest principles of honesty and integrity as a Mathlete. I will neither
give nor accept unauthorized assistance of any kind. I will not copy anothers work
MATHCOUNTS CHAPTER 2011
TARGET ROUND
3 vertices of square ABCD are located at A(5, -1), B(7, 1) and D(3,1). Find the coordinates of
point C.
What is the units digit of 3^2011?
The sale price that Mr. Adams paid for a 10-ft by 12-ft piece of carpet was the
a
3
= and b = 10, what is the valueof a?
b
5
1. _
If
triangles
2. _
Square ABCD, shown here, has diagonals AC and BD that intersect at E. How
many triangles of any size are in the figure?
A
D
E
B
C
3. _
When the integers 1 to 100 inclusive are written,
MATHCOUNTS
2007
Chapter Competition
Target Round
Problems 1 and 2
Name
School
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO.
This section of the competition consists of eight problems,
which will be presented in pairs. Work on one pair of
problems will b
2
4
1. What common fraction is exactly half-way between and ?
3
5
1. _
2. The stem and leaf plot represents the heights, in inches, of the
players on the Spring Vale Middle School girls basketball
team. What is the mean height of the players on the tea
2008
Chapter Competition 1. What is the average student headcount for the spring terms of the 1- Students
0203, 03-04 and 0405 academic years? Express your answer
to the nearest whole number.
Total Student Headcount (2082-2003 to 2005-2006)
Fall Ter
2014
Chapter Competition
Target Round
Problems 1 & 2
Name
School
DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO
DO SO.
This section of the competition consists of eight problems, which will be
presented in pairs. Work on one pair of problems will be completed a
2005
Chapter Competition
Sprint Round
Problems 130
Name
School
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO.
This round of the competition consists of 30 problems.
You will have 40 minutes to complete the problems. You
are not allowed to use calculators
MATHCOUNTS
2004
Chapter Competition
Target Round
Problems 1 and 2
Name
School
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO.
This round of the competition consists of eight problems, which
will be presented in pairs. Work on one pair of problems will
be
1. The sum of two positive integers is 9. What is the least
possible sum of their reciprocals? Express your answer as a
decimal to the nearest hundredth.
1. _
2. Two integers are relatively prime if they have no common
factors other than 1 or 1. What is t
QUESTIONS FOR QUIZ 1
D. PAGONAKIS
The aim of this paper is to pose two challenging exam problems on the material
we have covered in class. The solutions are not included in order to challenge the
readers and give them a chance to test their knowledge on i
PROBLEM SET #10 , 18.100C
DIMITRIOS PAGONAKIS
Problem 1. Prove that if f, g : [a, b] R are Riemann-Stieltjes integrable (for
some ), then so is the function max(f, g).
Solution. Note that we can write the function max(f, g) = |f +g| + |f g| (lecture
2
2
n
PROBLEM SET #8 , 18.100C
DIMITRIOS PAGONAKIS
Problem 1. Let (fn ) and (gn ) be two sequences of functions [a, b] (R), each of
which converges uniformly,
lim fn = f,
lim gn = g.
n
n
Suppose that f and g are bounded. Show that then, (fn gn ) also converges
2014
Chapter Competition
Sprint Round
Problems 130
HONOR PLEDGE
I pledge to uphold the highest principles of honesty and integrity as a Mathlete. I will neither give nor
accept unauthorized assistance of any kind. I will not copy anothers work and submit
1. Michelle bought three golf balls and one
bottle of water before starting her game.
She paid $12.00 for them. After nine
holes, she bought one additional golf
ball and four additional bottles of water.
For these she paid $9.50. How much does a
bottle o
1. In the integer 45,075,123, the 2 represents the value 20. By what
factor would the value represented by the 5 in the thousands place
have to be multiplied to equal the value represented by the 5 in the
millions place?
1. _
2. Twenty-seven increased
MATHCOUNTS
2012
I Chapter Competition I
Sprint Round
Problems 130
HONOR PLEDGE
I pledge to uphold the highest principles of honesty and integrity as a Mathlcte9.'l will neither
give nor accept unauthorized assistance of any kind. I will
MATHCOUNTS
2007
Chapter Competition
Sprint Round
Problems 130
Name
School
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO.
This section of the competition consists of 30 problems.
You will have 40 minutes to complete all the problems.
You are not allowed t
2005
Chapter Competition
Target Round
Problems 1 and 2
Name
School
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO.
This round of the competition consists of eight problems,
which will be presented in pairs. Work on one pair of
problems will be completed a
Raytheon Company 2': US. Department of Defense a
Northrop Grumman Foundation a
National Society of Professional Engineers *
Bezos Family Foundation 2': ConocoPhillips S
CNA Foundation a Texas Instruments Incorporated *
ThinkFun * 3M Foundation
ANTI-
MATHCOUNTS
2006
Chapter Competition
Target Round
Problems 1 and 2
Name
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO.
This section of the competition consists of eight problems,
which will be presented in pairs. Work on one pair of
problems will be compl