Varsity Math Bowl
March 11 2006
Sample Question: Find the x-intercept of the line y  3x  12 .
1. Solve the following equation for x: y 
3x  2
.
4x  5
2. Values of the functions f and g are given in the following tables:
n
f(n)
1
5
2
6
3
3
4
1
5
2
6
4
7
7
8
9
9
10
10
8
n
1
g(n) 10
2
3
3
4
4
5
5
6
6
8
7
2
8
1
9
7
10
9
Find f ( g ( f ( g ( f ( g (4)))))) .
3. Let f ( x)  5 x  7 and g ( x)  x  3 . Find | f ( g (4))  g ( f (4)) | .
2
4. Let f ( x)  ( x  1)( x  2)( x  3)( x  4)( x  5) . Find
f (6)  1 .
5. Ravi has loaded 12 songs on his MP3 player- 4 rock, 4 soul, and 4 jazz. He sets
the player to play each song once in random order. What is the probability that
the 4 rock songs will get played first? Answer as a fraction in lowest terms.
2
6. Consider the region between the parabola y  x and the line y  10 . How
many points are in this region whose coordinates are both positive integers.
7. The speed of light is 3  10 meters per second. If a certain type of lightwave
8
7
was length 4  10 meters, how many of this type of wave can pass a fixed point
in one second?
8. A three-inch diameter gyroscope spins 200 rotations per minute. How fast is a
point on the rim moving? Answer in inches per second, rounded off to the first
decimal place.
9. Given the system of equations
x  y  10
y  z  12
x  z  14
find 3x  3 y  3z .
10. Suppose that ( x  y)  16 and ( x  y )  36 . How many possible values of y
are there?
11. Let f ( x)  1/ 2  cos x . Find the smallest root of f that is greater than 10.
Your answer should be in terms of  and written as a fraction in lowest terms.
2
2
12. Suppose the value of a tractor depreciates 10% each year. After four years, the
value is closest to what whole number percentage of the original value?
Round II
 3 4
 4 6
A


 3 5.
 2 3


1
1
2
2. Given that x  2  7 and assuming that x  0 , find x  .
x
x
1. Find the lower left entry of the matrix A such that 
3.
Suppose two nonnegative numbers sum to 160. If the first number is increased
by 25% and the second is decreased by 25%, what is the largest their new sum
can be?
4. In a round-robin tournament, ten players each play each other and there are no ties.
Nine of the players each won 4 games. How many games did the last player win?
5. A beaker contains 200 milliliters of 40% potassium chloride. How much water
must be added to dilute it to become a 5% solution?
6. In the interval 0    2 , how many solutions are there to the equation
(2sin(3 )  1)2  0 ?
7. In the diagram, the segment AB of length 6 is tangent to the circle at B. Point A
is 4 units from the nearest point of the circle, E. What is the length of the
diameter of the circle?
A
6
B
4
E
C
D
8. For any positive integer n, let f (n)  n / d , where d is the largest proper divisor
of n. For example f (8)  2, f (35)  5, f (23)  23 . Find the maximum of
{ f (1000), f (10000), f (77), f (11), f (55), f (121), f (144), f (169)}.
Answer in evaluated form – not in terms of the function.
1
9. Suppose f ( x)  2 x  3 . Find f (17) .
10. Let f ( x)  3x  1 and h( x)  ( f ( x))  f ( x ) . Find h(3) .
11. Consider the three equations:
2
2
cos3x  4cos3 x  3cos x
cos3x  4cos x  3cos3 x
cos3x  3cos3 x  2cos x
Which of the following statements is true:
A. None of these equations is an identity
B. Only the first is an identity
C. Only the second is an identity
D. Only the third is an identity
E. Only the first and second are identities
F. Only the second and third are identities
G. Only the first and third are identities
H. All of them are identities
12. A library issues cards with 4 digit numbers. None of the them begin or end with 0.
Further more none of the numbers have the middle two digits both 0. How many
different numbers can there be? Answer must be written out – not in the form of
a product.
Round III
1. For sets A and B, define A-B as the set of elements that are in A but not in B.
Also let A B  ( A  B)  ( B  A) . Suppose F  {3,6,9,12} and
E  {2,4,6,8,10,12} . What is the sum of the elements of E F ?
2. Suppose you start at 0 on a number line and move according to the following rule:
you repeatedly flip a (fair) coin and for each head you move to the right half a
unit and for each tail you move to the left half a unit. What is the probability that
you will eventually move to a position to the right of 200? Answer as a
percentage.
3. In the given diagram, AE=2, BE=4, DE=5. What is the length CE?
B
A
E
D
C
4. Let f ( x)  1sin x  2cos x  3tan x  4sec x  5cot x  6csc x . Find, in simplest
radical form, f ( / 6) .
5. Values of the functions f are given in the following table:
n
1
2
f(n)
5
6
Let f1  f , f 2  f
3
3
4
1
f , f3  f
5
2
f
6
4
7
8
7
9
f etc. What is f 2006 (6) ?
9
10
10
8
6. Let l be the line 5 x  12 y  3  0 . What is the distance from l to the point
(5,2)?
x2 y 2
7. The ellipse

 1 contains the point (4,3). What is the slope of the
32 18
tangent line to the ellipse at this point?
1/ 4
8. Evaluate
 sin
1
x  cos 1 xdx
0
9. How many positive factors does 1000 have?
10. A sine wave has amplitude 4. If the smallest number in the range is 3, what is the
largest number in the range?
a
11. Find the largest value of a such that 30! is divisible by 3 .
12. Suppose three standard dice are rolled. What is the probability that their product
is a prime number? Answer as a fraction in lowest terms.
Round IV
1. Suppose a tennis ball is dropped off a 125 meter building. Assuming the
acceleration of gravity is 10 meters per second per second and neglecting air
resistance, how fast, in meters per second, will the ball be going when it hits the
ground?
2. Suppose the line corresponding to y  x is rotated 15 degrees counterclockwise
around the origin. What is the equation of the new line?
3. A rectangle R has vertices at A=(0,0) B=(4,0) C=(4,3) D=(0,3). Suppose R
“rolls” along the x-axis until R is has its original orientation How long is the path
traced out by point A? I.e. R rotates clockwise around B, until C hits the axis,
then R rotates around C until D hits the axis, then rotates around D until A hits the
axis, and finally rotates around A until B hits the axis.
4. A circle of radius 1 is rotated around a line 2 units from the center of the circle.
What is the volume of the resulting torus? Answer in terms of  .
5. For any real numbers a and b, what is the maximum possible value of

b
a
9  x 2dx ?
6. At noon, John rides his bike north at 4mph and Sara walks west at 3 mph.
Assuming their speeds remain constant, how fast is the distance between them
increasing at 2:00?
2
7. Let f ( )  (cos  sin  ) . Find f ( /12) . Answer as a fraction.
8. Evaluate lim x0
sin 3 x
. Answer as a fraction.
2x
9. Let f ( x)  x  3x  4 and g  f
3
1
. Find g '(8) . Answer as a fraction.
10. Three dominoes – each of length 4 - are stacked up, each one overhanging the one
it is on top of. What is the furthest the top domino could stick out over the bottom
domino without tipping over? That is, in the picture below, find the maximum
possible value for x.
x
11. Suppose f (0)  f (1)  1 and f (n) 
n 1
 f (i) . Find
f (7) .
i 0
12. Two numbers x and y differ by 99. The two numbers are both squares and there
are no squares in between them. We also know that x<y. What is y?