Fox Valley Math League
Meet 5 - March 4th, 2013
Xavier High School
Event 1
Score
(No calculators)
Student Name_________________________________ School Name_______________________Team #___________
Print your name clearly
Every answer must be exact and completely simplified (including rationalized denominators).
1. Determine the number.
A multiple of eleven I be, Not odd, but even you see;
My digits (a pair), when multiplied there, Make a cube and a square out of me.
2. Solve the system
2x
y
xy 8
6
, writing your answers in ordered pair form.
3. A point P is located on segment AB in such a way that
AB
AP
AP
.
PB
1. __________
2. __________
3. __________
Given that AP=1, find the exact value of PB.
4. An athletic field with a perimeter of
1
4
mile consists of a rectangle with a semicircle
4. ____________
at each end. Express the area (square miles) of the field as a function of r (miles), the radius of the semicircle.
Fox Valley Math League
Meet 5 - March 4th, 2013
Xavier High School
Event 2
Score
(No calculators)
Student Name_________________________________ School Name_______________________Team #___________
Print your name clearly
Every answer must be exact and completely simplified (including rationalized denominators).
1. A pole is one third in the mud, one half in the water, and three feet out of the water.
What is the length of the pole (feet)?
1. __________
2. Find the value for k such that the equation has exactly one real root.
2. __________
kx
2
kx
1
0
x 2 a,
3. Given the function f ( x )
f ( x) is continuous at x
x
2
2ax 3, x
2
, what must the value of “a” be so that
3. __________
2.
4. Determine the slope m of the line shown in the figure where the area formed with
the axes is 4 square units. The point (2,1) is on the line.
4. __________
Fox Valley Math League
Meet 5 - March 4th, 2013
Xavier High School
Event 3
Score
(No calculators)
Student Name_________________________________ School Name_______________________Team #___________
Print your name clearly
Every answer must be exact and completely simplified (including rationalized denominators).
1. A cube has pyramids cut from each corner by passing planes through the
midpoints of the edges (adjacent to each vertex) of the original cube. The pyramids are
discarded and a new solid remains. How many edges does the new solid have?
1. __________
2. How many distinct triangles can be constructed by connecting three different vertices of a cube?
2.__________
3. Determine the area of the shaded region given that the radius of the circle is 1 unit and the
inscribed polygon is a regular polygon.
3. __________
4. Express the sum as a fraction (of whole numbers) in lowest terms.
4. __________
1
1
2 3 3 4
1
1
...
4 5
2012 2013
Fox Valley Math League
Meet 5 - March 4th, 2013
Xavier High School
Event 4
Score
(Calculators allowed)
Student Name_________________________________ School Name_______________________Team #___________
Print your name clearly
All answers must be exact and completely simplified or rounded to the hundredths place.
1. A particular bike wheel makes 1056 revolutions in one mile.
What is the radius of the wheel in feet?
1. __________
2. Suppose you have 600 meters of fencing with which to build two adjacent rectangular corrals.
The two corrals are to share a common fence on one side. Find the dimensions x and y in
meters so that the total area is as large as possible.
2. __________
3. If you randomly draw three of the following sticks from a paper bag, what is the probability
that a triangle can be formed? Sticks: 2 cm, 3 cm, 5 cm, 7 cm, 11 cm, 13 cm
3. __________
4. Each of the non-overlapping regions in the figure is an equilateral triangle (infinite pattern).
Find the sum of the areas of the shaded regions if the area of the largest triangle
shown is 1 square unit.
4. __________
Fox Valley Math League
Meet 5 - March 4th, 2013
Xavier High School
Team Event
Score
(Calculators allowed)
Student Name_________________________________ School Name_______________________Team #___________
Print your name clearly
Every answer must be exact and completely simplified (including rationalized denominators) unless specified otherwise in
the problem.
1. Determine all solutions to the system where x and y are real numbers. Give the answer(s)
in ordered pair form without rounding.
y
y
1. __________
3x
32 x
2
99
2. Expand the sum
log10
j 1
j
j 1
and simplify completely.
3. Determine the sum of the geometric series
3
3
3 1
3 3
2. __________
...
3. __________
4. A right circular cone (point side up) has a diameter of 8 cm and a height of 12 cm.
To what depth (exact cm.) must the cone be filled with liquid to be one-half full.
4. __________
5. Joe lives near a river where he goes swimming every day: he swims 1 mile upstream, 1 mile
downstream and exits the river the same place as he entered it. Recently Joe went on
vacation to a lake, where he noticed that during workouts lasting the same time, swimming
at the same constant speed, he is able to swim 2.2 miles each day. Give the positive ratio of
Joe’s velocity to the river’s velocity. This will express how many times faster Joe is
compared to the river.
5.__________
6. The number 1789, the year the French Revolution began, has three and no more than three
adjacent digits that are consecutive integers in increasing order. How many years between
1000 and 9999 have this property?
6. __________
Solution Key
Fox Valley Math League
Meet 5 - March 4th, 2013
Xavier High School
Event 1
(No calculators)
Student Name_________________________________ School Name_______________________Team #___________
Print your name clearly
Every answer must be exact and completely simplified (including rationalized denominators).
1. Determine the number.
A multiple of eleven I be, Not odd, but even you see;
My digits (a pair), when multiplied there, Make a cube and a square out of me.
1. __________
SOLUTION is 88
2. Solve the system
2x
y
6
xy 8
, writing your answers in ordered pair form.
SOLUTIONS
y 2 x 6 subs.
x (2 x 6) 8
( x 4)( x 1) 0
x 4, 1
2 x2 6x 8 0
(4,2)( 1, 8)
x
2
2. __________
3x 4 0
3. A point P is located on segment AB in such a way that
AB
AP
AP
.
PB
3. __________
Given that AP=1, find the exact value of PB.
Let AP 1 and PB y and AB 1 y
1 y 1
, y y 2 1 , so y 2 y 1 0
1
y
by the quadratic formula
y
1
(1) 2 4(1)( 1)
2(1)
1
5
2
, with y
4. An athletic field with a perimeter of
0,
1
4
AP
PB
1
y
2
1
5
1
1
5
5
1
5
2
mile consists of a rectangle with a semicircle
4. __________
at each end. Express the area (square miles) of the field as a function of r (miles), the radius of the semicircle.
¼ = 2πr + 2y, where y = length
y = 1/8 – πr, Area = πr2 + 2(1/8 – πr)r
Area = ¼ r – πr2
Fox Valley Math League
Meet 5 - March 4th, 2013
Xavier High School
Event 2
Solution Key
(No calculators)
Student Name_________________________________ School Name_______________________Team #___________
Print your name clearly
Every answer must be exact and completely simplified (including rationalized denominators).
1. A pole is one third in the mud, one half in the water, and three feet out of the water.
What is the length of the pole (feet)?
18 ft. let p
Then,3
1
3
length of the pole
1
2
P
P
3
1
6
P
P 18
2. Find the value for k such that the equation has exactly one real root.
kx
2
kx
1
b2 4ac 0
k (k 4) 0, k
0, k
f ( x) is continuous at x
2ax 2 when x
4a 3
k 2 4k
0,
x 2 a,
x
2
2ax 3, x
2
4
3. Given the function f ( x )
4 a
a 1
2. __________
0
one real root
x2 a
1. __________
, what must the value of “a” be so that
3. __________
2.
2
5a 1
5
4. Determine the slope m of the line shown in the figure where the area formed with
the axes is 4 square units. The point (2,1) is on the line.
y 1 m( x 2), x int. y
y int. x
Area
0, 1 m( x 2)
0, y 1 m( 2)
1
(base)(height )
2
4m 2 4m 1 0
m
y
x
2m 1
1 2m 1
( 2m 1) 4
2
m
1
2
4. __________
2m 1
m
Fox Valley Math League
Meet 5 - March 4th, 2013
Xavier High School
Event 3
Solution Key
(No calculators)
Student Name_________________________________ School Name_______________________Team #___________
Print your name clearly
Every answer must be exact and completely simplified (including rationalized denominators).
1. A cube has pyramids cut from each corner by passing planes through the
midpoints of the edges (adjacent to each vertex) of the original cube. The pyramids are
discarded and a new solid remains. How many edges does the new solid have?
1. __________
Solution: 4 edges for each of the 6 sides so 24 edges
2. How many distinct triangles can be constructed by connecting three different vertices of a cube?
8 7 6
56
8 C3
3 2
2.__________
3. Determine the area of the shaded region given that the radius of the circle is 1 unit and the
inscribed polygon is a regular polygon.
3. __________
Solution: Start by finding the measure of a central angle, 6-sided so
1
1
3
60
area of triangle
ab sin
(1)(1)sin 60
2
2
4
3
4 such triangles 4
3
4
4. Express the sum as a fraction (of whole numbers) in lowest terms.
1
1
2 3 3 4
First note
1
1
...
4 5
2012 2013
1
r ( r 1)
sum telescopes to
1
2
1
2013
1
r
1
2
2013 2
2 2013
1
r 1
1
3
1 1
3 4
2011
4026
1
4
1
5
...
1
2012
1
2013
4. __________
Fox Valley Math League
Meet 5 - March 4th, 2013
Xavier High School
Event 4
Solution Key
(Calculators allowed)
Student Name_________________________________ School Name_______________________Team #___________
Print your name clearly
All answers must be exact and completely simplified or rounded to the hundredths place.
1. A particular bike wheel makes 1056 revolutions in one mile.
What is the radius of the wheel in feet? (exact or nearest tenth)
5280 ft / mile
1056 revolutions / mile
5 ft / revolution
circum. 5 ft
radius
5
ft .8 ft
2
2. Suppose you have 600 meters of fencing with which to build two adjacent rectangular corrals.
The two corrals are to share a common fence on one side. Find the dimensions x and y in
meters so that the total area is as large as possible.
3x 2 y
600
y
A
x 300
3
x
2
xy
A( x )
300
1. __________
2. __________
3
x
2
3 2
x
2
300 x
3
( x 100) 2 15000
2
y
300
3
(100) 150
2
Thus by choosing x = 100 meters and y to be 150 meters
This could also be done graphically on the calculator.
3. If you randomly draw three of the following sticks from a paper bag, what is the probability
that a triangle can be formed? Sticks: 2 cm, 3 cm, 5 cm, 7 cm, 11 cm, 13 cm
3. __________
3 5 7,3 11 13,5 7 11,5 11 13,7 11 13
6!
3!3!
5
20
1
4
4. Each of the non-overlapping regions in the figure is an equilateral triangle (infinite pattern).
Find the sum of the areas of the shaded regions if the area of the largest triangle
shown is 1 square unit.
4. __________
The largest shaded triangle is ¼ of the largest triangle, the second largest shaded triangle is ¼
of ¼ of the largest triangle, and so on. Thus,
1
4
1
2
4
1
3
4
1
...
1
4
1
1
4
3
Solution Key
Fox Valley Math League
Meet 5 - March 4th, 2013
Xavier High School
Team Event
(Calculators allowed)
Student Name_________________________________ School Name_______________________Team #___________
Print your name clearly
Every answer must be exact and completely simplified (including rationalized denominators) unless specified in the
problem.
1. Determine all solutions to the system where x and y are real numbers. Give exact values
of x and y.
y (3x ) 2 2
y y2 2
x
y
y
3
32 x
2
0
y2
y
2, 1
99
2. Expand the sum
1
2
log10
log10
2
3
log10
1
2
1
100
2
3
3
4
4
5
log10
j 1
3
4
0 ( y 2)( y 1)
(9,2) and ( 1 3 , 1)
j
log10
j 1
log10
y 2
1. __________
and simplify completely.
... log10
2. __________
99
100
99
... 100
2
3. Determine the sum of the geometric series
3
geometric sum
a1
1 r
3 1
3 1
1
3 3
3
3
3 1
3 3
3. __________
...
3
3 1
2
3 3
3
3 3
2
3 1
3 3 3
2
3 1
3
3 1
2
3 1
3
2
4. A right circular cone (point side up) has a diameter of 8 cm and a height of 12 cm.
To what depth (exact cm.) must the cone be filled with liquid to be one-half full.
12 6 3 4 cm. Vol
64 cm 3
half is 32 cm 3
Since the unfilled has the same volume and is similar to the original cone,
2
h
1 h
r
and V
h 32 cm3 where h is the height of the unfilled portion.
3
3 3
Solving for h, h
6 3 4 cm . Hence the original cone should be filled to 12 6 3 4
4. __________
Solutions continued
5. Joe lives near a river where he goes swimming every day: he swims 1 mile upstream, 1 mile
downstream and exits the river the same place as he entered it. Recently Joe went on
vacation to a lake, where he noticed that during workouts lasting the same time, swimming
at the same constant speed, he is able to swim 2.2 miles. Give the positive ratio of Joe’s
velocity to the river’s velocity. This ratio will express how many times faster Joe is
compared to the river.
5.__________
5. Swimming one upstream and one mile downstream takes the same time (distance/velocity) as swimming in the lake
2.2 miles.
v1
Joe ' s velocity, v2
1
1
v1 v2
v1 v2
(v1
2v1
v2 )( v1 v2 )
2.2
v1
v1
v2
11 times faster
2.2
v1
2v12
(v12 v2 2 )2.2
10v12
( v12 v2 2 )11
11v2 2
v12
v1
v2
river ' s velocity
11
6. The number 1789, the year the French Revolution began, has three and no more than three
adjacent digits that are consecutive integers in increasing order. How many years between
1000 and 9999 have this property?
Two cases (1) the year starts with three adjacent digits with consecutive integer values; or
case (2) the year ends with three adjacent digits. For the first case, there are nine
combinations with 123 (excluding 1234). Likewise, there are nine cases for 234, 456, 567,
678 and ten combinations beginning with 789 for a total of 64 for case (1). For case (2) there
are nine combinations for years ending with 012 (e.g. 1012) and nine possible combinations
ending in 123 (e.g. 1123), with 0123 excluded. The digit one less than the second digit must
be excluded so there are only eight possible combinations ending with 234, 345, 456, 567,
678, 789. This nets 66 possible combinations for case (2), and 64 + 66 = 130 overall.
6. __________