Score
November 2, 2009
Round 1
Name ____________________________________School ____________________________ Team ________
No calculators allowed. All answers must be in exact simplified form.
1. A box of coins contains a total of $26.00 in nickels, dimes and quarters.
If there is the same number of nickels as dimes, but twice as many quarters
as nickels, how many dimes are in the box?
1. _______________
2. Three judges for a talent quest have to vote publicly on three performers
A, B, and C, listing their order of preference. In how many ways can the
judges vote so that two of them agree in their order of preference, while
the third differs?
2. _______________
3. Three circles of radius 1 are mutually tangent as shown. What is the area
of the gap enclosed between the three circles?
3. _______________
4. In a triangle ABC, let a, b, c be the sides opposite those angles A, B, C.
4. _______________
2
a+b
= 0, then
=
cos B + sin B
c
_________________
If cos A + sin A !
Score
November 2, 2009
Round 2
No calculators allowed. All answers must be in exact simplified form.
Name ____________________________________School ____________________________ Team ________
1. Laura jogs seven blocks the first day of her training program. She increases
her distance by two blocks each day. On the last day she jogs 25 blocks.
How many days was she in training?
1. _______________
2. Scores on a particular standardized test are approximately normally
distributed with a mean of 515 and a standard deviation of 90. To the
nearest whole percent, what percent of students score above 605?
2. _______________
3. The first two terms of a sequence are a, b. From then on, each term is equal
to the opposite of the previous term, plus the term before that. What is
the sixth term?
3. _______________
4. Compute 231base 5 . 42base 5 and give the answer in base 5 notation.
4. _______________
Score
November 2, 2009
Round 3
Name ____________________________________School ____________________________ Team ________
No calculators allowed. All answers must be in exact simplified form.
1.
In the figure shown, find x.
1. _____________
3
12
4
x
2.
It rained on exactly 11 days during Jane’s holiday trip. On each rainy day
it rained either in the morning or in the afternoon but not both. There are
exactly 13 afternoons when it did not rain and exactly 16 mornings when it
did not rain. How many days did the trip last?
2. _______________
3.
Three people take turns drawing a card from a shuffled deck, returning the
card and reshuffling after each draw. They agree that whoever draws a heart
first will pay for dinner. If in the first set of drawings no one draws a heart,
they draw again, taking turns in the same order, and they continue in this manner
until eventually someone draws a heart. What is the probability that the third
person will have to pay for dinner?
3. _______________
4.
If real numbers x, y satisfy (x + 5)2 + (y – 12)2 = 142, then
the minimum value of x2 + y2 equals?
4. _______________
Score
November 2, 2009
Round 4
Name ____________________________________School ____________________________ Team ________
Calculators allowed! All answers must be in exact simplified form or rounded to the nearest hundredth.
1.
Solve the equation for x:
1. _______________
x(x + 5) = (x ! 1)(x + 7)
2. In ∆ABC with C at the right angle, if sin(A) = 3/5, find tan(B).
2. _______________
3. If log ab + log bc + log ac = 10 for positive a, b and c,
what is the value of log abc ?
3. _______________
4. Here's a spooky problem to ponder -- if you dare! The ghoulish inhabitants
4. _______________
of a ghost town decide to build themselves a new home -- that is, a brand new
cemetery. They have scraped together $10,000 (by selling their gold fillings and
haunting their rich living relatives). They plan to use as much of the vacant fence
land north of Creepy Street as they need, but they will need to build a rectangular
around the new cemetery. On the three sides of the rectangle not facing Creepy Street
they will use materials costing $10 per foot of fence. But on the one side which borders
Creepy Street, they need more expensive materials costing $15 per foot. In order to make
their new home as large as possible in area, what should be the length of the fence
along Creepy Street? Remember, they have $10,000 to spend on the fence.
Score
November 2, 2009
Team Event
Name ____________________________________School ____________________________ Team ________
1. A five-digit number with all 5 digits different – let’s call it ABCDE – has
the following characteristics:
• The four-digit number DEAD is prime.
• The four-digit number BEAD is divisible by 9.
• The number B and the three-digit number ACE are both powers of E.
• C is less than A.
Find the five-digit number ABCDE.
1. _______________
2. Write the repeating decimal 0.3146146146146146146146146146 . . .
as a simplified fraction a/b where a and b are integers.
2. _______________
3. A bin contains 25 balls: 10 red, 8 yellow, and 7 blue. We draw three balls at
3. _______________
random from the bin, and we will say that we "win" if our three balls represent
exactly two colors. (That is, we "win" if we draw two balls of one color and another
ball of a different color.) What is the probability of winning this game?
4. A parabola opens downward and has its vertex on the line y = 2. It passes
through the points (0,1) on the y-axis and (2,0) on the x-axis. If its vertex
is the point (a,2), find the value of a.
4. _______________
5. The circle and the square at the right have the
same center and the same area. If the circle
has radius 1, what is the length of segment AB?
5. _______________
A
6. Suppose line n is tangent to the curve y = f(x) at the point
(10, 6) as shown below. Find f '(10)
B
6. _______________
Score
November 2, 2009
Round 1
Name ____________________________________School ____________________________ Team ________
No calculators allowed. All answers must be in exact simplified form.
1. A box of coins contains a total of $26.00 in nickels, dimes and quarters.
If there is the same number of nickels as dimes, but twice as many quarters
as nickels, how many dimes are in the box?
5N + 10D + 25Q = 2600
N=D
2N = Q
Substituting into the first equation,
5N + 10N + 25.2N=2600.
65N = 2600
N = 40
1. _______________
40
So, D = 40 and Q = 80.
2. Three judges for a talent quest have to vote publicly on three performers
A, B, and C, listing their order of preference. In how many ways can the
judges vote so that two of them agree in their order of preference, while
the third differs?
2. _______________
90
There are 3 pairs of judges that can agree. There are six possible orders of
preference for the two who agree and five other orders for the judge who disagrees.
3.6.5 = 90 ways.
#
3"
2
3. _______________
3. Three circles of radius 1 are mutually tangent as shown. What is the area
of the gap enclosed between the three circles?
The area of the gap is the area of the equilateral
triangle minus the area of the three sectors (which,
together, are half a circle).
1
$ "12
$
A = " 2" 3 #
= 3 # square units
2
2
2
!
!
!
a+b
4. In a triangle ABC, let a, b, c be the sides opposite those angles A, B, C.
If cos A + sin A !
2
a+b
= 0, then
=
cos B + sin B
c
Rewrite the ratios in terms of the sides:
b a
2
+ " a b =0
c c c+c
b+ a 2
" a +b = 0
c
c
a+b
2
Then, let x =
to get x " = 0 . Multiply by x to get x 2 " 2 = 0 , x ≠ 0.
c
x
x
=
±
2
Solving, this gives
. But these are lengths so x > 0.
!
!
!
!
= 2
4. _______________
c
!
Score
November 2, 2009
Round 2
No calculators allowed. All answers must be in exact simplified form.
Name ____________________________________School ____________________________ Team ________
1. Laura jogs seven blocks the first day of her training program. She increases
her distance by two blocks each day. On the last day she jogs 25 blocks.
How many days was she in training?
10 days
1. _______________
25 – 7 = 18. So there were 9 increases of 2 blocks after the first day. She jogged for ten days.
2. Scores on a particular standardized test are approximately normally
distributed with a mean of 515 and a standard deviation of 90. To the
nearest whole percent, what percent of students score above 605?
16%
2. _______________
A score of 605 is one standard deviation above the mean. In a normal distribution, the empirical rule tells us
that about 68% of the scores will be within a standard deviation of the mean. So half of the remaining 32%
will be above 605. That’s 16%
3. The first two terms of a sequence are a, b. From then on, each term is equal
to the opposite of the previous term, plus the term before that. What is
the sixth term?
3. _______________
5b – 3a
a, b, –b + a, –(–b + a) + b = 2b – a, –(2b – a) + (–b + a) = –3b + 2a, –(–3b + 2a) + 2b – a = 5b – 3a
4. Compute 231base 5 . 42base 5 and give the answer in base 5 notation.
21302base 5
4. _______________
You multiply normally, but you must represent 5 as 10, 6 as 11, etc. and carry whenever you get a product of
5 or more.
231
x 42
1012
20240
21302
Score
November 2, 2009
Round 3
Name ____________________________________School ____________________________ Team ________
No calculators allowed. All answers must be in exact simplified form.
1.
In the figure shown, find x.
1. _____________
x = 13
3
12
4
d2 = 32 + 42 = 25, so d = 5
x2 = 52 + 122 = 169, so x = 13.
x
2.
3.
It rained on exactly 11 days during Jane’s holiday trip. On each rainy day
it rained either in the morning or in the afternoon but not both. There are
exactly 13 afternoons when it did not rain and exactly 16 mornings when it
did not rain. How many days did the trip last?
There are 3 more dry mornings than dry afternoons. That means that, out of the 11 rainy days, 3 more of
them must have had rain in the afternoon than morning. That’s 4 rainy mornings and 7 rainy afternoons.
7 rainy afternoons means 7 dry mornings, so 9 more dry days are needed to get to 16 dry mornings.
(That also makes a total of 13 dry afternoons.) So 11 rainy days plus 9 dry days makes 20 days.
9
37
Three people take turns drawing a card from a shuffled deck, returning the
3. _______________
card and reshuffling after each draw. They agree that whoever draws a heart
first will pay for dinner. If in the first set of drawings no one draws a heart,
they draw again, taking turns in the same order, and they continue in this manner
!
until eventually someone draws a heart. What is the probability that the third
person will have to pay for dinner?
3
1
probability of a ‘fail’ (non heart) and a probability of
4
4
" 3 % n(1 1
success. To get the first heart on the n-th draw you must ‘fail’ n–1 times then succeed. P = $ ' ) . For the
#4&
4
third person to get the first heart,!n = 3 or 6 or 9 or 12 or . . . so the probability
! of this happening is
2
5
8
11
" 3% 1 " 3% 1 " 3% 1 " 3% 1
9
and constant ratio
$ ' ( + $ ' ( + $ ' ( + $ ' ( . . . This is a geometric series with initial term
#4& 4 #4& 4 #4& 4 #4& 4
64
!
9
9
" 3 % 3 27
9
= .
$ ' = . The sum is 64 27 = 64
37
# 4 & 64
1" 64 64 37
!
This is a waiting time problem. There is a
!
!
20 days
2. _______________
!
4.
If real numbers x, y satisfy (x + 5)2 + (y – 12)2 = 142, then
the minimum value of x2 + y2 equals?
1
4. _______________
(x + 5)2 + (y – 12)2 = 142 is a circle of radius 14 centered at (–5, 12) [call this Circle 1]. This center point is
13 units from the origin. x2 + y2 is the distance of the point (x, y) from the origin.
Consider a radius of
Circle 1 that passes through the origin. Call the intersection of this radius with the circle Point A. Point A is 1 unit
from the origin. Because the origin is on this radius, the circle x2 + y2 = 1 is internally tangent to Circle 1 at
point A. No point of Circle 1 is inside the circle x2 + y2 = 1, so 1 is the closest distance Circle 1 gets to the
origin.
Score
November 2, 2009
Round 4
Name ____________________________________School ____________________________ Team ________
Calculators allowed! All answers must be in exact simplified form or rounded to the nearest hundredth
unless otherwise directed.
1.
x=7
1. _______________
Solve the equation for x:
x(x + 5) = (x ! 1)(x + 7)
x2 + 5x = x2 + 6x – 7, then subtracting x2 from both sides,
5x = 6x – 7, then subtract 6x from both sides.
–x = –7, or x = 7.
4
3
2. _______________
2. In ∆ABC with C at the right angle, if sin(A) = 3/5, find tan(B).
B
3
If sin(A) = , then the triangle is a 3-4-5 right triangle.
3
5
2
2
2
(5 –3 = 4 .)
C
4
So, tan(B) =
3
!
5
4
3. If log ab + log bc + log ac = 10 for positive a, b and c,
what is the
! value of log abc ?
log ab + log bc + log ac = log (ab.bc.ac) = log(abc)2 = 2log(abc)
So, 2log(abc) = 10, and log(abc) = 5
A
!
3. _______________
5
4. Here's a spooky problem to ponder -- if you dare! The ghoulish inhabitants
4. _______________
200 ft.
of a ghost town decide to build themselves a new home -- that is, a brand new
cemetery. They have scraped together $10,000 (by selling their gold fillings and
haunting their rich living relatives). They plan to use as much of the vacant fence
land north of Creepy Street as they need, but they will need to build a rectangular
around the new cemetery. On the three sides of the rectangle not facing Creepy Street
they will use materials costing $10 per foot of fence. But on the one side which borders
Creepy Street, they need more expensive materials costing $15 per foot. In order to make
their new home as large as possible in area, what should be the length of the fence
along Creepy Street? Remember, they have $10,000 to spend on the fence.
Suppose that the cemetery measures x feet along Creepy Street and y feet in the other dimension. Then the cost
for the fence is $10 per foot for 2y+x feet and $15 per foot for x feet. So, 10(2y+x) + 15x = 10000
Solving this for y we get y = (10000 - 25x)/20 = 500 - (5/4)x
The area of the cemetery will be A = xy = x[500 - (5/4)x] = 500x -(5/4)x2
"b
"500
=
= 200 . But if x = 200 then the other dimension is y = 500 2a "2(5 /4)
(5/4)x = 500 - (5/4)(200) = 250 . This means the dimensions of the cemetery should be 200 feet along Creepy
Street and 250 feet deep.
The vertex of the parabola is at x =
!
Score
November 2, 2009
Team Event
Name ____________________________________School ____________________________ Team ________
Calculators allowed! All answers must be in exact simplified form or rounded to the nearest hundredth
unless otherwise directed.
1. A five-digit number with all 5 digits different – let’s call it ABCDE – has
the following characteristics:
• The four-digit number DEAD is prime.
• The four-digit number BEAD is divisible by 9.
• The number B and the three-digit number ACE are both powers of E.
• C is less than A.
Find the five-digit number ABCDE.
1. _______________
58132
Starting with the third criteria, B is a single-digit power of E. (E isn’t 1 since B can’t be E.) E must be 2
or 3. If E = 2, B = 4 or 8. If E = 3, B = 9.
Consider the possibility that E = 3. ACE is a three-digit power of E. Three digit powers of 3 are 243 and
729. It needs to end in E, so ACE would be 243. But this violates the condition that C is less than A.
So E = 2. Three digit powers of 2 are 128, 256, and 512. ACE would be 512. B must be 4 or 8. Consider
the case that B = 4. Then BEAD is divisible by 9, so 4 + 2 + 5 + D is divisible by 9. D = 7. But then
DEAD = 7257 is not prime. It is divisible by 3.
This leaves B = 8. BEAD is divisible by 9, so 8 + 2 + 5 + D is a multiple of 9. D = 3. and DEAD = 3253 is
prime. ABCDE = 58132.
3143
9990
2. _______________
2. Write the repeating decimal 0.3146146146146146146146146146 . . .
as a simplified fraction a/b where a and b are integers.
Think of this as 0.3 + 0.0146146146146. . .
3
146
0.146 146
and 0.146 =
and 0.0146 =
0.3 =
=
10
999
10
9990
3 146 2997 146 3143
+
=
+
=
10 9990 9990 9990 9990
!
!
!
!
Alternatively,
write !
n = 0.3146146... Then 10n = 3.146146... and 10000n =
3146.146146...
Subtracting, we get 10000n - 10n = 3146.146146... - 3.146146...
9990n = 3143
So n = 3143/9990.
3. A bin contains 25 balls: 10 red, 8 yellow, and 7 blue. We draw three balls at
random from the bin, and we will say that we "win" if our three balls represent
exactly two colors. (That is, we "win" if we draw two balls of one color and another
ball of a different color.) What is the probability of winning this game?
1529
2300
3. _______________
!
There are C(25, 3) = 2300 ways to draw three balls from the bin.
In order to win we need to do one of three things: (1) draw exactly 2 red balls (with the third ball being some
other color), (2) draw exactly 2 yellow balls, or (3) draw exactly 2 blue balls.
To choose exactly two red balls amounts to choosing two balls from the set of 10 red balls, followed by picking
any of the 15 non-red balls as the third choice. So, the number of ways to choose exactly 2 red balls is
C(10,2)*15 = 45*15 = 675 . Similarly, the number of ways to choose exactly 2 yellow balls is C(8,2)*17 = 28*17
= 476 and the number of ways to choose exactly 2 blue balls is C(7,2)*18 = 21*18 = 378 .
So, the total number of winning choices is 675+476+378 = 1529.
1529
So, the probability of winning this game is
.
2300
4. A parabola opens downward and has its vertex on the line y = 2. It passes
through the points (0,1) on the y-axis and (2,0) on the x-axis. If its vertex
! of a.
is the point (a,2), find the value
2 2 "1 # 0.83
4. _______________
A parabola opening downward with vertex (a,2) will have an equation y - 2 = -b(x -!
a)2 for some constant b. We
can plug in our two points (0,1) and (2,0) to get two equations in the variables a and b
1 - 2 = -b(0 - a)2, or in other words 1 = ba2
and
2
0 - 2 = -b(2 - a) , or in other words 2 = b(2 - a)2
From the first equation we see that b = 1/a2, and substituting this into the second equation we get
2 = (1/a2)(2 - a)2
2a = (2 - a)2 = 4 - 4a + a2
a2 + 4a - 4 = 0
2
Using the quadratic formula, we find the solution is
!
"4 + 32
= 2( 2 "1) # 0.83
2
!
2 1" # $ 0.92
5. The circle and the square at the right have the
same center and the same area. If the circle
has radius 1, what is the length of segment AB?
4
5. _______________
1
"
2
A
!
B
The circle and square both have area π, so the length of a side of the square is
12 "
( )
#
2
2
= 1" #4 , so AB = 2 1" #4 $ 0.92
!
" . Half the length of AB is
!
!
1
= 0.25
4
6. _______________
6. Suppose line n is tangent to the curve y = f(x) at the point
(10, 6) as shown below. Find f '(10)
!
The f’(10) is the slope of the line tangent to the curve at x = 10. The slope of the line is
!
6"0
6 1
=
= = 0.25
"
10" 14 24 4