Transcript Junior/Senior Math Bowl

38th Annual
Lee Webb Math Field Day
Varsity Math Bowl
Before We Begin:
• Please turn off all cell phones while
Math Bowl is in progress.
• The students participating in Rounds 1
& 2 will act as checkers for one another,
as will the students participating in
Rounds 3 & 4.
• There is to be no talking among the
students on stage once the round has
begun.
Answers that are turned in by the checkers
are examined at the scorekeepers’ table.
An answer that is incorrect or in
unacceptable form will be subject to a
penalty. Points will be deducted from the
team score according to how many points
would have been received if the answer
were correct (5 points will be deducted for
an incorrect first place answer, 3 for
second, etc.).
• Correct solutions not placed in the given
answer space are not correct answers!
• Rationalize all denominators.
• Reduce all fractions. Do not leave
fractions as complex fractions.
• FOA stands for “form of answer”. This will
appear at the bottom of some questions.
Your answer should be written in this form.
2009
Math Bowl
Varsity
Round 1
Practice Problem – 10 seconds
What is the area of a
circle of radius  ?
Problem 1.1 – 30 seconds
Find the ordered triple that
satisfies the system
x  y  2z  4

x  y  2z  0
x  y
0

FOA: (a,b,c)
Problem 1.2 – 30 seconds
Several cannon balls are
stacked in six layers, so that
there is a 6x6 square on
the bottom, with a 5x5 layer
above that, etc. How many
cannon balls are there?
Problem 1.3 – 30 seconds
Let f  x   14  3x and
g  x   3x  1 .
Evaluate  f g  3 .
2
Problem 1.4 – 30 seconds
Determine
Arc cos(1/ e)  Arc sin(1/ e)
.
Answer in radians.
Problem 1.5 – 75 seconds
Square ABCD has area 16. E
and F are on sides BC and
CD such that AE and AF
trisect the corner at A.
What is the area of
quadrilateral AECF?
FOA:
ab c
Problem 1.6 – 15 seconds
Write
sec x
csc x
as a
simple trigonometric
function.
Problem 1.7 – 60 seconds
The x-y, y-z, and z-x planes
cut the sphere
x  y  z  36
2
2
2
into 8 parts. What is the
volume of one of these
parts?
Problem 1.8 – 45 seconds
A CD player changes the speed
of the disc in order to read the
encoded bits at the same rate.
If the disc spins at 250 rpm for
a track that is 60 mm from the
center, how many rpm are
required for another track that
is 20 mm from the center?
Problem 1.9 – 45 seconds
Find the real part of
 3  2i 
3
Problem 1.10 – 45 seconds
Consider the sequence of
digits
1234567891011121314...
What is the
th
100
digit?
Problem 1.11 – 30 seconds
Solve for y:
log5 y  log5  y  4  1
Problem 1.12 – 30 seconds
What is the principal
value of
i
i
Round 2
Problem 2.1 – 15 seconds
Simplify
3ln1/ x 
e
Problem 2.2 – 30 seconds
An angle is reported to
be
23 30 '36".
In decimal notation, this
is how many degrees?
Problem 2.3 – 30 seconds
Let
Find
g  x   2x  3
.
g  a  b  g a 
b
.
Problem 2.4 – 30 seconds
Find the exact value
of
log 3 9
.
Problem 2.5 – 15 seconds
Find an
expression
for sec
x

2
Problem 2.6 – 30 seconds
For the following
parabola, how far is the
focus from the vertex?
y x
2
Problem 2.7 – 60 seconds
Solve for k:
k
n

5040

n 5
Problem 2.8 – 15 seconds
Fill in the blank:
The orthocenter of a
triangle is the
intersection of its
___________
Problem 2.9 – 60 seconds
Jane and Carlos and their guests
had pie for dessert. They used a
special pie-cutter that cuts central
angles of any integer degree.
Everyone got exactly one piece of
pie of exactly the same size. How
many possibilities are there for
the number of guests (do not
count the 0 guest case)?
Problem 2.10 – 75 seconds
Joey clothes-pinned a card on the
front wheel of his bicycle. The
card clicks every time a spoke
strikes it. The wheel is 24” in
diameter and has 32 spokes. If
Joey rides 11 ft per second, how
many clicks are there per second?
Round off to the nearest integer.
Problem 2.11 – 30 seconds
Simplify:


log(n)  log   m 

n 3
 m 3 
314
314
Problem 2.12 – 45 seconds
Let f ( x)  x  [ x] . Put the
following in increasing order
a) f (.2)
c) f (3 / 2)
FOA: a,b,c,d
(e.g)
b) f (1)
d) f ( )
Round 3
Practice Problem – 30 seconds
Simplify
1
log 2 16  log 2 4  log 2
32
Problem 3.1 – 45 seconds
The area of an equilateral
triangle varies directly with
the square of the length of a
side. Find the constant of
proportionality.
Problem 3.2 – 30 seconds
Find the value of x  (0,  )
such that the expression
2
(cos x  sin x)
is minimal.
Problem 3.3 – 60 seconds
Calculate
2
10
 n 

 2 
n2  n  1 
FOA: fraction in lowest terms
Problem 3.4 – 60 seconds
A polyhedron has 24
vertices. Two regular
hexagons and one square
meet at each vertex. In all
there are 8 hexagons.
How many squares are
there?
Problem 3.5 – 30 seconds
In the polyhedron of the
previous problem, there are
24 vertices, 8 hexagonal
faces, and 6 square faces.
How many edges does the
polyhedron have?
Problem 3.6 – 30 seconds
Solve for x:
1
 3 2
 x 2

10 x 
 10 3 




Problem 3.7 – 60 seconds
How many points
with integer
coordinates satisfy
x  y  25
2
2
Problem 3.8 – 30 seconds
The sum of the infinite
series
1 1 1 1
1    
4 9 16 25
is equal to f ( ) for what
polynomial f ( x) ?
Problem 3.9 – 60 seconds
Zacky’s Pizzeria offers a choice
of 3 different sizes, 2 different
kinds of crusts, and 10
different kinds of toppings.
How many different pizzas can
be ordered (with at least one
topping)?
Problem 3.10 – 30 seconds
A rhombus has side length
10 and area 50. What is the
measure, in radians, of its
smallest angle?
Problem 3.11 – 60 seconds
The light in a lighthouse makes
10 revolutions per minute.
How fast does the light flash
by on the side of a boat that is
600 feet directly offshore?

Answer in feet per second in
terms of
Problem 3.12 – 60 seconds
Suppose T1, T2, T3, … is an
infinite sequence of similar
triangles. The perimeter of each
triangle is 80% as much as the
previous triangle. If the area of
the first triangle is 63, find the
sum of the areas of all the
triangles.
Round 4
Problem 4.1 – 60 seconds
Find the first five digits after the
decimal point of the following
rational number:
1
7
1
1
15 
1
Problem 4.2 – 45 seconds
A gum manufacturer randomly puts
a coupon in 1 of every 4
packages. What is the
probability of getting at least one
coupon if 4 packages are
purchased?
Problem 4.3 – 60 seconds
A triangle has
vertices at (3,4),
(6,9), and (11,2).
What is its area?
Problem 4.4 – 45 seconds
A rectangle of length 36
and height 6 is centered
at the origin. What is the
equation of the circle that
goes through all the
vertices of the rectangle?
Problem 4.5 – 30 seconds
If you draw two cards
randomly from a standard
deck, what is the probability
that you get two of a kind (2
kings or 2 sevens, etc)?
Problem 4.6 – 15 seconds
Which letter of the
Greek alphabet is
?

FOA: 1st , 2nd, or 3rd etc.?
Problem 4.7 – 45 seconds
Evaluate:


0
dx
2
x 1
Problem 4.8 – 45 seconds
Let  be a complex
number such that
2
    1  0.
Find   
Problem 4.9 – 60 seconds
It takes 7 days for 5 chickens
to lay 2 dozen eggs. How
many days will it take 21
chickens to lay 30 dozen
eggs?
Problem 4.10 – 30 seconds
Randy and forty-four other people
are situated in a circle. Randy
passes a soccer ball to the twelfth
person on his right. This is
repeated until the ball comes
back to Randy. How many
people do not touch the ball?
Problem 4.11 – 60 seconds
22 / 7 is the best rational
approximation to  that has
denominator less than 10. It is
accurate to 2 places. There is
another approximation with
denominator 113 that is accurate to 6
places. Find its numerator.
Problem 4.12 – 60 seconds
Let f ( x) be the number of points in
the 1st quadrant with integer
coordinates whose distance
back to the origin is less than x .
Determine
f ( x)
lim x  2
x