spc conference math contest 6 12 2 ax a x axpehs.psd202.org/documents/jyoung1/1509993992.pdfspc...

SPC Conference Math Contest2013

Freshman Competition: Factoring

1. Factor completely: 2 26 12 2ax a x ax

2. Factor completely: 2 2 26 16 6x axy a y

3. Factor completely: 3 218 27 8 12a a a

4. Factor completely: 338 2x x


Freshman Competition: Linear Equations

1. The solution for x in the equation 5 3 2 3 3x x k is 8x . What is

the value of k?

2. Solve for x:2 37 3 3 8 5

23 40 5 5 9 12

x x

3. If2 3 2

3 1 5

x

x

and

2 3

4 1 2 3y y

, what is the value of 4 2x y ?

4. If 3a , solve for x in terms of a and/or y:

2 6 3 2 2a x y y y x ay

SPC Conference Math Contest

2013Freshman Competition: Radicals and Radical Equations

1. Simplify: 108 12 75 27

2. Solve for x: 2 3x x

3. If 3 2 2 4 2 2a , what is the value of a?

4. If 8 63

4 2 3 3 2a b

, what is the sum a b ?


Sophomore Competition: Coordinate Geometry - Lines

1. The point 4, k lies on the line with slope of2

3 and y-intercept of

14

3.

What is the value of k?

2. A line contains the points 5,2 and 1, 8 . At what point does this line

intersect the horizontal line through the point 1,3 ?

3. The median of a triangle connects a vertex to the midpoint of the oppositeside. If triangle ABC has A 4, 2 , B 8, 4 and C 2, 6 and y mx b

is the equation of the median from B, what is the value of b? Write youranswer as a single fraction.

4. Point P lies in the first quadrant on the line 2y x and is 58 units away

from the point 4,2 . What are the coordinates of point P?

12

10

30


Sophomore Competition: Areas

1. In the diagram, the two smaller circles are the samesize, tangent to each other and to the larger circle,and their diameters are contained in the diameterof the larger circle. If the radius of one of the smallercircles is 4, what is the shaded area?

2. A right triangle has a smaller leg of length 15 and a hypotenuse of 25. Asquare is then drawn so that one of its sides lies on the longer leg. What isthe total area of the figure?

3. A regular hexagon has a perimeter of 120. What is its area?

4. The prism shown has three rectangular sides and ends that are isoscelestriangles with base 10. The height of the figure is 12 and the bottomrectangular base of the figure has dimensions of 10 X 30 as shown.What is the total surface area of the prism?

O

F

E

D

C

B

A

H

G

F

E

D

C

BA

E

O

D

C

B

A

G

F E

D

C

B

A


Sophomore Competition: Circles

1. In circle O, the measure of BC is 86 and the measure

of DEC is 64 . Find the degree measure of arc AB

2. Given that AB is tangent to the given circle at B,AB 6 , AF 3 , CD 5 and GC 2 , what is the

length of CE ?

3. In circle O, the measure of AEC is 24 more than the

measure of ABC . What is the degree measure of ECD ?

4. Given circles A and B with CE 12 and GH 30 .

If EG 8 , find the length of internal tangent DF .


Junior Competition: Systems of Equations

1. Find the y-coordinate in the solution to the system:

29

2

x yx

44 3

x y y

2. The point 1,4 is the intersection of the line 14ax by and the curve2 11bx ay . What is the ordered pair ,a b ?

3. The system of equations 4 60x ay and 10 75bx y has infinite

solutions. What is the ordered pair ,a b ?

4. Find the coordinates ,x y of all points of intersection of the graphs of

2

2

214x

y and

82

xy .

D C

B

A


Junior Competition: Geometry and Trigonometry of a Right Triangle

1. In right triangle ABC with right angle C, A 30 and AC 12 . What is

the length of the hypotenuse of this triangle?

2. In triangle ABC with altitude BD as shown,AD 4 and DC 8 . What is the perimeter oftriangle ABD?

3. In triangle ABC, A 75 , B 45 and AC 8 . What is the length of

AB ?

4. Triangle ABC has A = 42 and C = 24 . The length of the altitude

from B to AC is 8. Rounded to 4 significant digits, what is the area oftriangle ABC?


Junior Competition: Logarithmic and Exponential Equations

1. Solve for x:

10 2

21

2 816

4

x x

x

x

2. Solve for k:2 1

ln 64 ln16 ln 2 ln83 2

k

3. An investment of $5000 was made 4 years ago and has earned $470.41 ininterest over that time. If the interest in the account has been compoundedmonthly, what is the annual interest rate the investment has earned duringthe 4 years? Give your answer as a percent, rounded to the nearesthundredth of a percent. Do not include the % sign in your answer.

4. Solve for x: 3log23 33 log log xx x


Senior Competition: Trig Applications

GIVE ALL ANSWERS ROUNDED TO 4 SIGNIFICANT DIGITS.

1. A fire department needs to ensure that they have a ladder that they can use toreach the top of the tallest building in their town, which is 180 feet tall. Thestandard angle with the ground that a ladder is set at is 65 . How many feetlong is the ladder that will meet their needs?

2. Boat A leaves a port and travels 175 miles to City C. At the same time, boatB leaves the same port and travels 228 miles to City D. If the angle betweenthe paths of boat A and boat B is 48 , how many miles apart are cities C andD?

3. Two sightings are taken to find the height of a building. The first angle ofelevation is 42 and the second angle of elevation, from 48 feet furtherback, is 35 . How many feet high is the building?

4. A pilot was flying from Antia to Charleston, located northeast of Antia. Dueto a storm as he left Antia, he flew on a flight path that was 26 south of thedirect path. After 160 miles he was past the storm, and corrected his flightpath to go directly to Charleston, which was another 380 miles. If the speedof the aircraft was a constant 270 miles per hour, how many minutes longerdid the trip take than it would have if he had been able to fly directly fromAntia to Charleston?


Senior Competition: Optimization

1. The function 22 12y x x k has a minimum value of 15 when x a .

What is the sum a k ?

2. At a football game, the concessions department finds that each of the 30 hotdog vendors sells an average of 92 hot dogs per game. For each additionalhot dog vendor added, each vendor averages sales of 2 less hot dogs pergame. How many vendors should the concessions department have workingto maximize hot dog sales?

3. A two liter can is to be made in the shape of a right circular cylinder. Thecan has both a closed top and closed bottom. What is the number of cm inthe diameter of the can that holds two liters and has the least surface area?Give your answer rounded to 4 significant digits. (Note: 1 liter = 1000 cm3)

4. A window has the shape of a rectangle with a semicircle at the top, so thatthe diameter of the semicircle is the width of the rectangle. If the perimeterof the entire window is 30 feet, what is the number of square feet in the areaof the window with maximum glass? Give your answer rounded to thenearest whole number.


Senior Competition: Conic Sections

1. The focus of a parabola with equation 2

x a y k h is at 6,1 and the

vertex lies on the line 8x . What is the equation of the directrix for thisparabola?

2. An ellipse written in the form

2 2

2 21

x h y k

a b

has vertices at 3, 2

and 3,12 . If the minor axis is 4 units long, find the values of a and b .

Write your answer as the ordered pair ,a b .

3. A circle has a center at 4, 1 and is tangent to the line 6y . If the

equation of the circle is 2 2 0x y ax by c , what is the sum a b c ?

4. A hyperbola has the equation

2 22 3

13 12

x y . One of the

asymptotes of this hyperbola has a negative slope. What is the equation ofthis asymptote? Give your answer in the form y mx b .


Freshman-Sophomore 8 Person Team

NO CALCULATORS ALLOWED IN THIS COMPETITION.

1. A letter is chosen at random from the letters in the word MATHEMATICS.What is the probability that the letter chosen is NOT a vowel? Give youranswer as a simplified fraction.

2. If 2x and 4y , what is the value of the expression2 2

1 2 2 1

x y

x y x y

?

3. One of the exterior angles of an isosceles triangle is 124 . What is thedegree measure of the largest possible interior angle of this triangle?

4. A positive number is divisible by 2, 3, 4, 5 and 6. What is the smallest suchnumber?

5. From point P, a secant and a tangent are drawn to circle O. The tangentintersects circle O at point Q; the secant intersects circle O and points Rand S. The ratio of PR to RS is 4 to 5. The length of PQ is 12. What isthe length PS?

6. A circle is inscribed in a regular hexagon. If the area of the circle is 36 ,what is the area of the hexagon?

7. The cube of a varies directly with the square of b . If 2a when 8b ,

what is the real number value of a when 3 3b ?

8. If 0, 0x y , 6x y xy and 2x y xy , what is the numeric value of

4 12x y ?

9. Using only the integers 1 through 10, how many groups of three differentintegers have a sum of 20?

10. Points A, B, C, D and E are located on a number line in that order such that

AB 5 , CE 2 BC , BD DE , and CD AB 3 . If point C is located at

a coordinate of 9, what is the coordinate of point E?

11. A region is bounded by the graphs of 4y x and 8y x . What is

the area of this region?

12. If 8x

y , what is the value of

2x y

y

?

13. Circle O has area 60 . If a chord AB in this circle has the same length asthe radius of the circle, what is the area of triangle AOB ?

14. In base b , 44 20 104b b b . What is the value of b?

15. If512

22

x

y , what is the sum of x and y ?

E

D

C

B

A

16. Given that one of the solutions for x in the equation 26 12 0x bx is1.5x , what is the value of b?

17. Reduce to lowest terms:2013 2014

2014 2013

12 12

12 12

. Write your answer as a single

fraction.

18. Circle A has a center at 5, 2 and a radius of 4. Circle B has a center at

0,10 and a radius of 6. Circle C is drawn so that it is externally tangent to

both circle A and circle B. What is the length of the smallest possible radiusof circle C?

19. In the following grid, the sum of the values in each row, column anddiagonal are equal. What is this sum?

5 02x

x 4 1

20. In the given diagram, AE 9 , CB 10 ,BD 6 , DE 7 and CDB EAB .

What is the length of CD ?


Junior-Senior 8 Person Team

NO CALCULATORS ALLOWED IN THIS COMPETITION.

1. Solve for x , if2 1

1x x

2. Set A has 32 subsets. How many elements are in Set A?

3. If 3 220xy xy and 4x xy , what is the smallest possible value of x?

4. Find the distance from the point P 4, 7 to the point on the circle

2 2

2 1 16x y that is nearest point P.

5. Solve for x if 424 1

xx

.

6. If 3 2f x x and 2 3g f x x , what is the value of 5f g ?

7. One of the linear asymptotes of the graph of 3 2

3 2

7 7 15

3 4 12

x x xf x

x x x

is

chosen at random. What is the probability that it is a vertical line?

8. For what value(s) of x does 2 4 3 5x x x x ?

9. What is the largest value of , 0 360 , for which

2sin 3cos6

4cos 6sin

?

10. If 12ab , 6bc and 9ac , what is the value of 2 2 2a b c ?

11. A point is moving along the path described by 2cosx t and

4sin 3y t . What is the Cartesian coordinate ,x y of the point at6

t

?

12. If log 0.5

4 3log log 10x , what is the value of x?

13. Three fair dice are tossed. What is the probability that the faces of exactlytwo of the die show a 6? Give your answer as a simplified fraction.

14. If 4 1f x x and 2 3g x x , for what value(s) of x does

1 1f g x g f x ?

15. A bag contains 5 red balls, 10 yellow balls, 13 green balls, and 2 blue balls.One ball at a time is drawn from the bag and its color noted. The ball is notreplaced before another ball is drawn. What is the minimum number ofdraws necessary to guarantee that balls of three different colors are drawn?

16. If 4 5

sin 29

x , find the value of 4 4sin cosx x .

17. What is the units digit in the sum 2013 201320 13 ?

18. The given diagram is made up of 12 squares.What is the degree measure of ABC ?

19. The expression1 1

1 1

3 3k k can be simplified to

3k b

a

where b is the smallest

possible positive integer. What is the sum a b ?

20. Given 3f x x ax b . When f x is divided by 1x , the remainder is

6, and when f x is divided by 2x , the remainder is 2. What is the

remainder when f x is divided by 3x ?

A

B

C


Calculator Team

GIVE ALL ANSWERS ROUNDED TO 4 SIGNIFICANT DIGITS UNLESSOTHERWISE SPECIFIED IN THE PROBLEM.

1. When 2013 is decreased by 3 2013 , the result is x more than 4 2013 .What is the value of x?

2. A segment contains all points that are on the line 3 5 17x y in the first

quadrant. What is the length of this segment?

3. The weight of an object in space is given by the function

2

2

4000

4000

sw

h

,

where h is the height above sea level in miles and s is the weight on earth atsea level in pounds. An astronaut orbiting 450 miles above the earth weighs150 pounds. How many pounds would the astronaut weigh on earth at sealevel?

4. In triangle ABC, 8a , 5b and 11c . What is the area of triangle ABC?

5. Solve for x: 2 22 3 5xe e

6. A bushel of corn can produce1

22

gallons of ethanol fuel. An acre of

farmland can produce 120 bushels of corn. In 2012, 13.948 billion gallonsof ethanol were produced. How many billion acres of farmland were neededto grow the corn to produce this ethanol?

7. The number of units in the circumference of circle O is 823.9. When theradius of circle O is increased by x%, the new circumference is 1262.3.What is the value of x?

8. The graphs of 2 2 3y x x and 2 5 2y x x meet at a point in the first

quadrant. What is the sum of the x and y coordinates of this point?

9. When20

13is written as a decimal, what is the digit in the 2013th place after

the decimal point? Give an exact answer.

10. How many square units are inside the graph of 2 2

2 1 20x y and

outside the graph of 2 2

2 1 13x y ?

11. A geometric sequence has 1 20a with a common ratio of 13r . What

is the sum of the first 10 terms of this sequence?

12. If 4 19x , what is the value of 2 13 x ?

13. The points 5,14 , 8,41 and 11,120.5 are on the graph of xy ab . What

is the value of b?

14. Find the largest angle , 0 360 , such that 24sin 5cos 2 .

15. What is the length of the median from A in the triangle ABC where A 5,2 ,

B 3, 7 and C 8, 1 .

16. A 10 question multiple choice quiz has 4 choices for each answer. What isthe probability that a student randomly guesses each answer and gets a scoreof 80% or above?

17. A triangle has a base of length b and a height of length 2.83. A secondtriangle is created with a base that is 4.2 shorter than the base of the first anda height that is k units longer than the height of the first. The area of thesecond triangle is 2.1k less than the area of the first. Find the product bk .

18. A circle is tangent to the lines 3 4 20x y and 3 4 13x y . What is the

area of this circle?

19. The half-life of a certain drug is 8.6 hours. If a patient is given a dose of 50mg at 1:00 pm, how many mg of the drug will still be in the patient’s systemat 10:45 pm?

20. The measure of the acute angle between the lines 2 5 8x y and

3 7x ky is 42 . Find the value(s) of k.

spc conference math contest 6 12 2 ax a x axpehs.psd202.org/documents/jyoung1/1509993992.pdfspc...

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