2014 ICTM RAA Algebra I -Final - Plainfield East High...

WRITTEN AREA COMPETITION ALGEBRA IICTM REGIONAL 2014 DIVISION AA PAGE 1 OF 3

1. Given 2x and 3y , find the value of the expression 2 3x y .

2. On an algebra test taken by all but one student in a class, the mean score is 82 and the medianscore is 81.5. All test scores were integers between 0 and 100 inclusive. The student whowas absent takes the test later and scores 86. Find the absolute value of the differencebetween the mean and the least possible median for all of the 25 students in the class whotook the test. Express your answer as an exact decimal.

3. A $20 item has a price increase of 35%. A second item has a price decrease of 20%. Afterthese changes, both items are now the same price. Find the original price of the second item.Express your answer as a decimal rounded to the nearest hundredth.

4. Find the y coordinate of the point of intersection of the graphs of 2 3 13x y and

2 2 1x y .

5. Find the value(s) of k so that the graph of the equation 2y x k has an x-intercept of

4.

6. The average of two values is 2x y . If a third value of 5 4x y is included, find the average

of all three of the values. Express your answer as a simplified polynomial expression interms of x and y .

7. When Mr. Worth died, he left an estate worth $325,000. His will specified that 35% of his

estate should be divided equally among his 5 children, and3

5of his estate should be divided

equally among his 12 grandchildren. The remainder was to be divided equally among his 8great-grandchildren. Find how much more, in dollars, one of his grandchildren will receivethan one of his great-grandchildren. Express your answer as a decimal rounded to the nearesthundredth.


8. Given that 0x , 0y , and4 2 10

3 2 3

x y

y x

, find the ratio of x to y . Express your answer

in the forma

bwhere a and b are integers and have no common factors.

9. A line m goes through the point 4, 3 and is perpendicular to the line 3 2 9x y . The

point , 3k k lies on this perpendicular line m . Find the value of k .

10. If 22 1 3 2x x ax bx c for all values of x , find the smallest solution for y for the

equation 2 1 2 0cy a y b .

11. Point A is located on a number line at5

28

and point B is located on a number line at1

53

.

Point C is located on the number line so that it is halfway between points A and B. Find thenumber line coordinate of point C. Express your answer as a mixed number.

12. If the ratio of a to b is 4 to 5, find the value of the expression2 3

4 3

a b

b a

. Express your

answer as a common or improper fraction reduced to lowest terms.

13. There are 12 elements in the intersection of sets A and B, and there are 20 elements in theunion of sets A and B. If set A has 4 fewer elements than set B, find the number of elementsthat are in set B, but not set A.

14. Given 2 1

3 2

xf x

x

. Find the value(s) of k for which 2 4f k is undefined.


15. Given2 1

3

xy

, find the numeric value of the expression

10

6 9 1x y .

16. Given that 2 8nx , find the value of the expression 4 23 2n nx x .

17. Given that3

32

a b

a b

, find the value of

3

2

a b

a b

. Express your answer as an integer or

common or improper fraction reduced to lowest terms. Do not use decimals.

18. Given the function 22 3 1f x x x , find the value(s) of k so that 2 2f k .

Express your answer(s) as an integer or decimal. Do not use fractions.

19. If x y and 3 4 12 3 2 8 cax y x y x y , find the value of 2a c .

20. When solved for x , the solution set for 7 3 2 5x x is :x x k or x w . Find the

sum k w . Express your answer as a common or improper fraction reduced to lowest

terms.

2014 RAA Name ANSWERS

Algebra I School(Use full school name – no abbreviations)

Correct X 2 pts. ea. =

Note: All answers must be written legibly in simplest form, according to the specificationsstated in the Contest Manual. Exact answers are to be given unless otherwisespecified in the question. No units of measurement are required.

(Must be thismixed number.)

1. 11.

(Must be this (Must be this reducedexact decimal.) improper fraction.)

2. 12.

(Must be thisdecimal, $ optional.) (Elements optional.)

3. 13.

4. 14.

(Must have both valuesin any order.)

5. 15.

6. 16.

(Must be this (Must be this reduceddecimal, $ optional.) common fraction.)

7. 17.

(Must be this reduced (Must be these exactcommon fraction.) answers in any order.)

8. 18.

9. 19.

(Must be this reducedimproper fraction.

10. 20.3

26

3

4

14218.75

3 2x y OR 2 3y x

2, 6

4

33.75

0.16 OR .16

108

62

45

7

3OR

12

3OR 2.3

3, 2.5

3

13

182

5

32

6

23

8

473

48

WRITTEN AREA COMPETITION GEOMETRYICTM REGIONAL 2014 DIVISION AA PAGE 1 OF 3

D

C

B

A

D

C

B

A

1. The area of a square is 36. If a side of the square is increased by 2, find the area of theresulting square.

2. The degree measure of angle A is 2 3x and the degree measure of the complement ofangle A is 5 4x . Find the numeric degree measure of angle A .

3. A rectangle with perimeter 16 has a diagonal of 2 10 . Find the length of the longer side ofthe rectangle.

4. In a regular 15-gon, each interior angle measures k , and w total diagonals can be drawn.

Find the value of k w .

5. If AB AC , 30DAC , 3AB and 4AC , the

length of AD can be written, in simplified form, as

a b c

d

. Determine the value of a b c d .

6. Find the number of times each day that the smaller angle between the minute hand and thehour hand of a clock is equal to 30 .

7. AB is tangent to the given circle at point A and BD is a secant

line to the circle. Find the length of AB if 1BC and 8CD .


E

D

B

C

D

CB

A

O

8. Points A and B are located on a cube. If the length of a diagonal of a face of this cube is12, find the exact maximum possible distance between A and B .

9. The interior of rectangle ABCD is located entirely inside a circle where 24AB , 7BC .

AB lies on the diameter of the circle and points C and D lie on the circle. If a point isselected at random from inside the circle, find the probability that the point is NOT inside therectangle. Express your answer as a decimal rounded to the nearest hundredth.

10. In circle O , AC is the perpendicular bisector of OB . If

14OD , find the exact length of AC .

11. A triangle has vertices 5,4A , 4,4B and 4,13C . Find the coordinates of the point

of intersection of the median from A to BC and the altitude from B to AC . Express your

answer as an ordered pair ,x y .

12. In the given circle, 42BC and 60DCE . The

measure of BE is 46 less the measure of CD . Find the

degree measure of BE .

13. ICTM is a convex quadrilateral with 60IC , 7CT , 24TM and 65MI . CTM is

a right angle. Find the area of this quadrilateral.

14. Two quadrilaterals ABCD with coordinates 5, 2A , 1, 4B , 2, 3C , and 1,D k

exist with numerical area 18.5. One quadrilateral is convex and one is concave. Find thesum of the possible values of k . Express your answer as a common or improper fractionreduced to lowest terms.


F ED

C

B

A

D

CBA

D

C

BA

15. In the given diagram, ABD and ACD are isosceles

triangles, and AD BC . The length of BC is k times the

length of AB . Find the exact value of k .

16. The length of the hypotenuse of a right triangle is 20. The radius of an inscribed circle is 4,and the ratio of the area of the circle to the area of the right triangle is k . Find the value ofk . Express your answer as a common or improper fraction reduced to lowest terms.

17. A 10 cm length stick is broken in one place. Find the probability that the longer piece is atleast twice as long as the shorter piece and no more than 5 cm longer than the shorter piece.Express your answer as a common fraction reduced to lowest terms.

18. In ABC , DE BC , BF bisects ABC and

CF bisects ACB . If 26AB , 34AC , and

40BC , find the length of DE .

19. In the diagram shown, triangle ABD is isosceles

with AB BD . The degree measure ofCBD = 96 . Find the degree measure of A .

20. An inverted cone-shaped paper cup is filled to the top with water. The slant height of thecone is 6 cm and the radius of the top (base) of the cone is 4 cm. There is a small hole in thebottom of the cone, and the water is slowly dripping out of the cup. After 5 minutes, thewater in the cone has lost 25% of its volume. Find the distance, in cm, from the top (base) ofthe cone to the water at that time. Express your answer as a decimal rounded to the nearesthundredth of a cm.


Geometry School(Use full school name – no abbreviations)



(Must be this

( ordered pair.)1. 11.

(Degrees optional.) (Degrees optional.)

2. 12.

3. 13.

(Must be this reducedimproper fraction.)

4. 14.

(Must be thisexact answer.)

5. 15.

(Must be this reducedcommon fraction.)

6. 16.


7. 17.


8. 18.

(Must be thisdecimal.) (Degrees optional.)

9. 19.

(Must be this (Must be thisexact answer.) decimal, cm

10. 20. optional.)14 3

0.72 OR .72

6 6

3

44

160

246

6

29

64

0.41 OR .41

1

6

48

24

1

6

2

60

7

834

76

1,7

WRITTEN AREA COMPETITION ALGEBRA IIICTM REGIONAL 2014 DIVISION AA PAGE 1 OF 3

1. For all real numbers x , 22 6f x f x x . Find the value of 3f .

2. Find the values of x for which1

1x

x

is true. Write your answer as an inequality with x

as one side of the inequality.

3. In right ABC with right angle at C , 2 3AB , logBC x , and 2 logAC x . Given

that 1x , find the exact value of cos A .

4. Let 2lnf x x and 2lng x x . Choose the best response and give as your answer

the capital letter of your choice.

A. The graphs of f and g are identical.

B. The graphs of f and g have no points in common.

C. The graphs of f and g have a finite number of common points.

D. The graphs of f and g have infinitely many points in common.

5. Determine the value of w iflog 1

log 2

k w

w k .

6. The point 7, 5 is on the graph of y f x . Find the corresponding point that must be on

the transformed graph 3 2 1y f x . Write your answer as an ordered pair ,x y .

7. A parabola has an equation of the form 2y ax bx c . The parabola passes through the

points 0,0 , 4,0 and 2,8 . Find the value of the coefficient a .


8. Find the area of the region bounded by the graph of 20x y .

9. Find the y-intercept of the line that is tangent to the circle 22 2 25x y at the point

3,6 . Write your answer as an exact decimal.

10. Point P is located in the first quadrant and lies on the line 8y . Point P is also 10 units

away from the point 4,14 . Find the coordinates of point P. Write your answer as an

ordered pair ,x y .

11. Find the minimum value of

2

22

2

1

11

1 1

xxf x

xx

x x

over the interval 1 2x . Write

your answer as an exact decimal.

12. The face diagonals of a box in the shape of a rectangular prism have lengths 3, 5 and 6. Findthe exact length of the diagonal of this rectangular prism.

13. Several married couples attend a party. Everyone shakes hands once with each person at theparty except themselves and their spouse. If 40 handshakes take place, how many couplesare at the party?

14. Given a , b , and c are real numbers. The average of a and b is c . The average of a andc is one more than b . The average of b and c is two more than twice a . Find the value ofc . Write your answer as a common or improper fraction reduced to lowest terms.


15. Suppose 0b c a and b a c . Find the largest solution for x in the equation2 0ax bx c .

16. Given the function 3 1f x x , find all value(s) of x where x f x . Write your

answer(s) as common or improper fraction(s) reduced to lowest terms.

17. Carly has 369 markers. Out of these, 130 are black, 100 are blue, and the others are neitherblue nor black. She chooses two markers at random. Find the probability that one of the twois black and the other is neither blue nor black. Write your answer as a decimal rounded tothe nearest thousandth.

18. The letters a and b represent integer digits in the six-digit number 322 4a b . This number is

divisible by 99. Find the value of a b .

19. Find the ordered pair of exact positive numbers ,x y that satisfies both 2 2x y x y and

xy x y . Write your answer as an ordered pair ,x y .

20. Evaluaten

1 2 3 4 n... ...

3 9 27 81 3 . Write your answer as a common or improper

fraction reduced to lowest terms.


Algebra II School(Use full school name – no abbreviations)



(Must be this exact decimal.)

1. 11.

Not possible2. 12.

(Must be thisexact answer.) (Couples optional.)

3. 13.

(Must have thiscapital letter.)

4. 14.

5. 15.

(Must be this (Must have both reducedordered pair.) common fractions in any order.)

6. 16.

(Must be thisexact decimal.)

7. 17.

(Square units optional.)

8. 18.(Must be this ordered

(Must be this decimal but pair with these or

accept 0,8.25 .) equivalent exact entries.)

9. 19.

(Must be this (Must be this reducedordered pair.) common fraction.)

10. 20. 4, 8

8.25

800

2

4,15

1

4OR 0.25 OR .25

D

6

3OR 1

63

0x OR 0 x

3

3

4

1 5 3 5,

2 2

7

0.266 OR .266

1 1,

2 4

1

11

3 OR

11

3

5

(Note: original answer

of 35 was also accepted)

1.5

WRITTEN AREA COMPETITION PRE-CALCULUSICTM REGIONAL 2014 DIVISION AA PAGE 1 OF 3

1. One of the solutions for x in the equation 2 0x bx c where b and c are integers is

1 2x . Find the value of the sum b c .

2. 1i . Solve for b if 1 04 3 4 3

b b

i i

.

3. The graph of 2 24 8 2 1x y x y is rotated 90 clockwise about the origin. Find the

coordinates of the center of the new conic. Express your answer as the ordered pair ,x y .

4. Find the number of polar points below that are coincident (occupy the same position) as the

polar point 5,6

?

I.7

5,6

II.5

5,6

III.13

5,6

IV.11

5,6

V. 5,6

5. Find the value(s) of x for which 4 3 7 7x x .

6. Simplify the expression 6 4 2 2 4 6cos 3 cos sin 3 cos sin sinx x x x x x .

7. Let ( ) 0P x be a third degree polynomial equation with integer coefficients and leading

coefficient of 1. Let the polynomial equation ( ) 0P x have at least one root that is an

integer. If (6) 11P , (4) 3P , and ( 4) 341P , find (10).P


8. A bag contains red and blue marbles. When two marbles are drawn, the probability that theyare both red is equal to the probability they are both blue. The probability that one of each

color is drawn is4

7. Find the least number of marbles that are in the bag.

9. The distance between the lines3

4y x b and 3 4 7x y is 2. If 0b , find the value of

b .

10. If 1i , find the sum of an infinite geometric series whose first term is 2 i and common

ratio is1

2i .

11. Determine the numerical coefficient of 2 2 2x y z in the simplified expansion of

5 4 3

1 1 1x y z

12. The solutions for x in the equation 2 52 ln ln 3x x are of the form ae where e is the

base of a natural logarithm. Find the sum of all possible values of a . Write your answer as acommon or improper fraction reduced to lowest terms.

13. Given 4 3 8 1f x x , 1f x ax b . Find the sum a b .


14.2

2

2

cos2sin

cos2sin

cos2sin

2sin

xu x

xx

xx

x

, where 0 cos sinx x and1

cos3

x . Find the

exact value of u. Write your answer as a single reduced fractional expression.

15. Find the number of distinct real values for x so that2 5

2 3

x x

x x

will not have an

inverse.

16. When David and Jim play pool, the probability that David wins exactly two games out offour is equal to the probability that he wins exactly three games out of four. Find theprobability that Jim wins any given game. Write your answer as a common fraction reducedto lowest terms. Assume that the outcome of a game cannot affect the probabilities in othergames, and that both players have a non-zero chance of winning a game.

17. A parabola is described by2

2

1

x t

y t

. Find the exact distance between the vertex and the

y-intercept of this parabola.

18. Determine the exact sum of all values of , 0 2 for which 2 3cos

5 .

19. The function y f x is periodic with period 3. If 6 4f , 8 2f and 10 9f , find

the value of 2 1 3 2 4 3f f f .

20. In a triangle with sides a , b , and c , 3a b c a b c ab . Find secC , where angle

C is the angle opposite side c .


PreCalculus School(Use full school name – no abbreviations)



1. 11.

(Must be this exact answer only (Must be this reducedor exact algebraic equivalent.) improper fraction.)

2. 12.

(Must be thisordered pair.)

3. 13.

(Must be this exactsingle fract’l expr.)

4. 14.

5. 15.


6. 16.


7. 17.


8. 18.

9. 19.

10. 20.2

3

4OR 0.75 OR .75

8

177

1

3

2 OR TWO

1, 1

19 3

6

i

3

2

40

4

2 5

2

5

2

2 2 3

3

OR 3 2 2

3

4

5

2

180

FRESHMAN/SOPHOMORE 8 PERSON TEAM COMPETITIONICTM REGIONAL 2014 DIVISION AA PAGE 1 OF 3

NO CALCULATORS

NO CALCULATORS NO CALCULATORS NO CALCULATORS

D

B

E

C

A

1. The point , 5x is the midpoint of the segment connecting the points 8,2 and 10, y .

Find the sum x y .

2. The length of one leg of a right triangle is 4 and the length of one side of a square is 6. Thearea of the right triangle is half the area of the square. Find the exact numerical length of thehypotenuse of the right triangle.

3. Given that 0xy and

34 2 3

23 5 3

8 2

4

b cx x y

ax yx y xy

, find the sum a b c .

4. The angles of a pentagon have degree measures that are in a ratio of 2:3:3:5:5. If one of theangles is selected at random, find the probability that the angle is obtuse. Write your answeras a common fraction reduced to lowest terms.

5. In the given diagram, 6AC CE and

2AB BD . Find the exact length of AB .

6. Set A has 5 elements and set B has 4 elements. The elements of set S are all of thepossible numbers of elements that could be in A B . Find the sum of the elements in set S .

7. ABC has vertices at 2, 4A , 6,2B and 8, 4C . The line containing the median

from A to BC has a y-intercept of b . Find the value of b . Write your answer as the y-intercept and not the coordinates for the point that is the y-intercept on a graph.)


NO CALCULATORS


8. If 1.5 1.45k

w where the fraction

k

wis an improper fraction reduced to lowest terms, find

the value of k .

9. Evaluate the expression

2 2

2 2

3 3 3 33 10

4 4 8 8

3 3 3 32 3

8 4 8 4

.

10. A circle is inscribed in a triangle that has sides of lengths 5, 6 and 9. The point of tangencyof the circle with the side of length 9 divides that side into two segments of lengths k and w .If k w , find the ratio :k w . Write your answer in the form of the ratio :k w .

11. x y is defined as ax by . If 7 2 4 and 3 2 4 , find the value of 9 2 .

12. If the product 0 1 2 3 4 2014a a a a a a is equal to 2014ka , find the value of k .

13. Let a and b be the lengths of legs of a right triangle with hypotenuse of length 2.

Determine the value of 8 6 2 4 4 2 6 84 6 4a a b a b a b b .

14. Find all integral value(s) of x so that the expression6

2 1x represents an integer.


NO CALCULATORS


D

C

BA

F

E

D

C

B

A

15. The graphs of 2 2 1y x x and 3y x intersect at a point with coordinates ,x y in the

first quadrant. Find the ,x y coordinates of this point. Write your answer as an ordered

pair ,x y .

16. ABC is isosceles with side AC congruent to side AB . The

measure of ACB is twice the measure of CAB . CE bisects

C and BD trisects B so that the measure of CBD is lessthan the measure of DBA . Find the degree measure of CFD .

17. If 2 24xy and 3

60xy , find the ratio of x to y . Write your answer in the form :x y

where x and y are positive integers with no common factors.

18. Find the simplified value of the expression

34

3 5

6 10

2 10 9 10

. Write your answer in

scientific notation.

19. In the given figure, 2AB , 1BC , AD BD and CD

is tangent to the circle at D . Find the exact length of AD .

20. If 0a and 2 2 2 28 3 3 2 9 2x ax a x a ax , find the largest solution for x as anexpression in terms of a .

2014 RAA School ANSWERS

Fr/So 8 Person Team (Use full school name – no abbreviations)



1. 11.


2. 12.

3. 13.

(Must be this reduced (Must be these four integerscommon fraction.) only, in any order.)

4. 14.

(Must be thisordered pair.)

5. 15.

(Degrees optional.)

6. 16.

(Must be this ratioand in this form.)

7. 17.

(Must be this answerin scientific notation.)

8. 18.


9. 19.

(Must be this ratio (Or exactand in this form.) simplified

10. 20. equivalent.)5 : 4

1

298

2

3OR 0.6 OR .6

35

4 3

2

5

3

97

17

3

2

a OR

3

2a

OR 1.5a

3

51.2 10

5 : 2

60

1, 4

2, 1, 0,1

256

2015

2OR

11007

2OR 1007.5

8

JUNIOR/SENIOR 8 PERSON TEAM COMPETITIONICTM REGIONAL 2014 DIVISION AA PAGE 1 OF 3

NO CALCULATORS


1. Find the coordinatex of the point that is the solution to the system

13 14

45 12

yx

yx

.

2. Let 1i . Find the polynomial function P x with real coefficients, of smallest degree,

and with leading coefficient of 1 that has roots 2 i and 5. Write as your answer the sum of

the numerical coefficients of P x .

3. Let 1i . A point 0x is called a fixed point of function f if 0 0f x x . Find all

complex numbers which are fixed points of 2

1 82

3f x x

x

. Express your answer(s) in

standard a bi form.

4. If 21 7 2f x x x and 21f x ax bx c , find the ordered triple , ,a b c .

5. Four distinct numbers are chosen from the first nine positive integers. Find the probabilitythat 3 is the smallest of the numbers chosen. Express your answer as a common fractionreduced to lowest terms.

6. Find the least integer that is a solution for x in 2 5 8x x .

7. Four children were playing in a back yard when one of them broke a window. Whenconfronted, they said:

“John did it,” said Ann.John said, “It was Gail who broke it.”“Anyway, it wasn’t me,” Sally declared.Gail said “John’s a liar when he says I did it.”

If only one of the children told the truth, determine who broke the window. Write the fullname of the person who broke the window for your answer.


NO CALCULATORS


8. One of the foci of the ellipse 2 2

4 1 2 16x y lies in the first quadrant. Find the

exact coordinates of this focus. Express your answer as an ordered pair ,x y .

9. The arithmetic mean of two numbers is 20. Find the maximum value of the geometric meanof the two numbers.

10. Find the sum of all x on the interval 0 2x for which sin 3 cos 6x x . Express your

answer as a common or improper fraction reduced to lowest terms.

11. Find the largest possible value of the product xy if 2 1 9x any 2 9 1y .

12. Find the value of sin 2 given that5

sin cos4

. Express your answer as a common

fraction reduced to lowest terms.

13. Find all value(s) of x for which 2

23log 1 12x .

14. In ABC with angles measured in degrees, 2C B , 12AB and 8AC .

Find cos C . Express your answer as a common fraction reduced to lowest terms.


NO CALCULATORS


15. ABCDEF is a regular hexagon of side length 3. Find the ratio of the area common toACE and BDF to the area of hexagon ABCDEF . Express your answer in the form :k w

where k and w are integers with no common factors.

16. Let 1i . Find the value of 8

0

1n

n

i

.

17. Find the value of 4

1024 1024 1024 1024log ...

0 1 1023 1024

. (Note:

,k w

kC C k w

w

is combination or “choose” notation.)

18. A coin is to be flipped five times resulting in heads or tails. Find the probability that withinthe five flips, the longest sequence of heads will be exactly two heads in length. Expressyour answer as a common fraction reduced to lowest terms.

19. The solutions for x of the equation 3 23 13 0x x x k form an arithmetic sequence. Findthe value of k .

20. When a polynomial P x of degree 2n is divided by 3x , the remainder is 5. When

P x is divided by 1x , the remainder is 1. Find the remainder when P x is divided by2 4 3x x .


Jr/Sr 8 Person Team (Use full school name – no abbreviations)



1. 11.


2. 12.(Must have allthree values in (Must have both valuesany order.) in either order.)

3. 13.

(Must be this (Must be this reducedordered triple.) common fraction.)

4. 14.

(Must be this reduced (Must be this ratiocommon fraction.) in this form.)

5. 15.

6. 16.

(Must be thisfull name.)

7. 17.(Must be this orderedpair or with exact algebraic (Must be this reducedequivalent entries.) common fraction.)

8. 18.

9. 19.

(Must be this reducedimproper fraction.)

10. 20.

17

2

OR

17

2

20

1, 2 2 3

Sally

0 OR zero

10

63

1, 3, 8

2, 1 3, 1 3i i OR

2 0 , 1 3, 1 3i i i

8

1

2OR 0.5 OR .5

2 1x OR 1 2x

15

11

32

512

16 15i OR 15 16i

1:3

1

8

5, 3

9

16

20

CALCULATING TEAM COMPETITIONICTM REGIONAL 2014 DIVISION AA PAGE 1 of 3

Round answers to four significant digits and write in either standard or scientific notationunless otherwise specified in the question. Except where noted, angles are in radians. Nounits of measurement are required.

1. Assume the cost of gasoline is 359.9 cents per gallon and assume that a car averages 27.83miles per gallon. Find the number of cents the gasoline cost to drive this car 148.1 miles.

2. In the diagram (not necessarily drawn to scale), AD CD and

AB AC . 15AB , 4AD , and 3CD . Find the numeric

length BD .

3. Find all positive root(s) (or zero(s)) for the equation 3 20.682 2.767 2.449y x x x .

4. Point 3.433,P y lies on a line whose equation is 1.244 2.113y x . At point P , a line is

drawn perpendicular to the line with the given equation. If this perpendicular passes through

5.664, k , find the value of k .

5. Let A and B be the foci of the ellipse whose equation is2 2( 1) ( 2)

1100

x y

k

. Let M be

one of the endpoints of the minor axis of the ellipse. Let the minor axis be parallel to the x-axis. If 113.26AMB , find the value of k .

6. The diagonals of a parallelogram have lengths 89.19 and 26.19 and intersect at an angle of39.7 . Find the area of this parallelogram.

7. The measures of an interior angle of a regular k sided polygon is 156 . The measure of an

interior angle of a regular 3k sided polygon is %w more than 156 . Find the value of

w . Write the value of w only as your answer. Do not use the % sign.

4

3

15

A

D C

B


8. A frustum of a solid right circular cone has a bottom base whose area is twice the area of itsupper base. A hemisphere whose flat portion is the upper base of the frustum is carved intothe frustum and has its nearest point to the bottom base located 1 cm from the center of thebottom base. The frustum has the hemispherical portion removed from the frustum, and the

remaining portion of the frustum has a volume of 10 3cm . Find the total numerical surface

area in 2cm for this remaining portion of this solid frustum.

9. If 2x , find the value of x such that 2( 2) 5 3x .

10. Bob ran from point A to point B at a constant rate of 7.243 feet per second. After arriving atpoint B, Bob immediately turned around and ran from point B to point A at a constant rate ofx feet per second. In regard to his total distance and his total time, Bob’s average rate was6.964 feet per second. Find the value of x .

11. Find the slope of the line whose equation is 798.3 13.25 76.51x y .

12. In ABC , 26.35CAB , 81.43CBA , and the area of ABC is 3526.37 . Find thenumeric length AB .

13. If 1x solve for x when 5log log 7xx .

14. If the vector 3.483, 6.815, 14.34p w k is equal to the vector

3 14.66,2 16.03,5k 2.444p w , find the value of p k w .


15. Let x represent the length of the third side of a right triangle whose other two sides havelengths of 145 and 408, in some order. Find the absolute value of the difference between thesmallest possible value of x and the largest possible value of x .

16. Find the real value of x such that

3 7

2 5

201

14

x

x

17. A circle, represented by the equation 2 26 8 144x x y y , has its center at P . Point A

lies in Quadrant I, lies on the circle, and has an x-coordinate of 15. Point B lies on thepositive x-axis and also lies on the circle. Find the area of the segment of the circle that isbounded by the chord from A to B and the minor arc from A to B .

18. If 0k , find the smallest possible value of k such that cos( ) 0.4399k .

19. If a dealer could purchase his goods for 7.684% less while keeping his selling price fixed,his profit, based on the cost of his purchased goods, would be increased from his present

profit of %k to 9.843 %k . Find the value of k . Write your answer as the value of k

only. Do not use the % sign.

20. Susan is playing Theresa in a best of 7 games in a table tennis match. The first player to win4 games will be the winner of the match. The probability that Theresa will win any particulargame after the first game is constant. Susan now leads the match by a score of 1 game to

none. If the probability that Theresa will win the match is now2

3, find the probability that

Theresa will win any particular game from now on.


Calculator Team (Use full school name – no abbreviations)


Note: All answers must be written legibly. Round answers to four significant digits and writein either standard or scientific notation unless otherwise specified in the question. Exceptwhere noted, angles are in radians. No units of measurement are required.

(Centsoptional.)

1. 11.

2. 12.

3. 13.

(Trailing zeronecessary.)

4. 14.

5. 15.(Trailingzeronecessary.)

6. 16.(Must be thisvalue only,without % sign.)

7. 17.

(Degrees

( 2cm optional.) optional.)

8. 18.(Must be this valueonly, without the% sign.)

9. 19.(Feetper secondoptional.)

10. 20.

1915 OR 31.915 10

17.69 OR 1.769 10

OR 11.769 10

1.732 OR 01.732 10

4.590 OR 04.590 10

330.5 OR 23.305 10

746.0 OR 27.460 10

2.564 OR 02.564 10

35.94 OR 3.594 10

OR 13.594 10

3.885 OR 03.885 10

6.706 OR 06.706 100.6598 OR .6598

OR 16.598 10

18.25 OR 1.825 10

OR 11.825 10

116.1 OR 21.161 10

4.865 OR 04.865 10

9.211 OR 09.211 10

51.64 OR 5.164 10

OR 15.164 10

27.72 OR 2.772 10

OR 12.772 10

5.869 OR 05.869 10

123.7 OR 21.237 10

60.25 OR 6.025 10

OR 16.025 10

FROSH-SOPH 2 PERSON COMPETITIONICTM 2014 REGIONAL DIVISION AA PAGE 1 OF 3

1. The given diagram shows 3 congruentregular hexagons between parallel linescontaining sides and pairwise sharing avertex. ABGHIF has numerical area 36.Find the area of hexagon ABCDEF .

2. Find the sum of all real solution(s) for the equation 2 5 1x x

3. Let k be the sum of the roots of the equation 22 7 15 0x x . Let w be the slope of the

line having an equation 2 5 6x y . Find k w . Express your answer as a common or

improper fraction reduced to lowest terms.

4. k is a two-digit integer in which the ten’s digit exceeds the unit’s digit by 2 and the sum ofthe ten’s digit and twice the unit’s digit is 17. w is Jack’s age now if five years ago Bennywas 5 times as old as Jack was at that time and four years from now Benny will be twice as

old as Jack will be. Find k w .

5. Let 3k k represent the numerical area of a triangle with sides length 12, 6 6 3 , and

6 2 . Let 2w represent the length of the diagonal of a square with perimeter 12. Find

k w .

6. Let x and y be positive integers such that x y . Several distinct ordered pairs of the form

,x y are solutions for 2

2 3 6084x y . Let k be the sum of the x coordinates and let w

be the sum of the y coordinates of these solutions. Find k w .

7. Let5 5 5 5 5 5

5 5 5 5 5 5 5

7 7 7 7 7 7

6 6 6 6 6 6 6k

. A 5 gallon jar with 1.5 gallons of antifreeze and the

rest water and a 4 gallon jar with 0.5 gallons of antifreeze and the rest water are combined ina large 15-gallon empty tank. Let w be the fractional part of this new mixture that is

antifreeze. Find kw . Express your answer as a common or improper fraction reduced to

lowest terms.

8. k is the number of distinct integral solutions for x if 2 1x . Equilateral triangle ABC

is inscribed in a circle and has area 64 3 . Let w be the exact area of the circumscribing

circle. Find the exact value of kw .

C

EF

A

B G

I

H D

FROSH-SOPH 2 PERSON COMPETITIONICTM 2014 REGIONAL DIVISION AA PAGE 2 OF 3

9. In ABC , AB BC and 5AB BD . The perimeter ofABD equals the perimeter of ACD . Find the length of

AC . Express your answer as an improper fraction reduced tolowest terms.

10. Let 2 24 4x kx k where k represents the length of the altitude drawn to the longest sideof a triangle with sides of length 3, 4, and 5. Find the positive solution for x . Express youranswer as a common or improper fraction reduced to lowest terms.

5

5B

A

CD

FROSH-SOPH 2 PERSON COMPETITION EXTRA QUESTIONS 11-12ICTM 2014 REGIONAL DIVISION AA PAGE 3 OF 3

11. Set A contains 15 elements, set B contains 9 elements, and set C contains 13 elements. A B

( A and B ) contains 6 elements, A C 8 elements, B C 5 elements, and A B C contains 2

elements. The Universal set containing A , B , and C contains 50 elements. How manyelements are in the universal set but not in any of A , B , or C ?

12. A dart board is a set of concentric circles as shown with eachannulus (circular band) having the same width as the radius of theinnermost circle. Find the probability that, when three darts arethrown, the score achieved is exactly 70. Answer as a commonfraction reduced to lowest terms. (Assume all three darts hit theboard, stick in one of the bands, and the order in which the dartshit the board does not matter.)

20

40

30

10

ICTM Math Contest

Freshman – Sophomore

2 Person Team

Division AA

FROSH-SOPH 2 PERSON COMPETITION LARGE PRINT QUESTION 1ICTM 2014 REGIONAL DIVISION AA NO CALCULATORS ALLOWED

1. The given diagram

shows 3 congruent regularhexagons between parallellines containing sides andpairwise sharing a vertex.ABGHIF has numericalarea 36. Find the area ofhexagon ABCDEF .

C

EF

A

B G

I

H D


2. Find the sum of allreal solution(s) for the

equation 2 5 1x x


3. Let k be the sum of theroots of the equation

22 7 15 0x x . Let wbe the slope of the linehaving the equation2 5 6x y . Find k w .

Express your answer as acommon or improperfraction reduced to lowestterms.


4. k is a two-digit integer inwhich the ten’s digit exceedsthe unit’s digit by 2 and thesum of the ten’s digit andtwice the unit’s digit is 17.w is Jack’s age now if fiveyears ago Benny was 5 timesas old as Jack was at that timeand four years from nowBenny will be twice as old asJack will be. Find k w .


5. Let 3k k representthe numerical area of atriangle with side lengths

12, 6 6 3 , and 6 2 .

Let 2w represent thelength of the diagonal of asquare with perimeter 12.Find k w .

FROSH-SOPH 2 PERSON COMPETITION LARGE PRINT QUESTION 6ICTM 2014 REGIONAL DIVISION AA CALCULATORS ALLOWED

6.Let x and y be positiveintegers such that x y .Several distinct ordered pairsof the form ,x y are

solutions for

2

2 3 6084x y .

Let k be the sum of the xcoordinates and let w be thesum of the y coordinates of

these solutions. Find k w .


7. Let5 5 5 5 5 5

5 5 5 5 5 5 5

7 7 7 7 7 7

6 6 6 6 6 6 6k

.

A 5 gallon jar with 1.5 gallons ofantifreeze and the rest water and a4 gallon jar with 0.5 gallons ofantifreeze and the rest water arecombined in a large 15-gallonempty tank. Let w be thefractional part of this new mixturethat is antifreeze. Find kw .

Express your answer as a commonor improper fraction reduced tolowest terms.


8. k is the number ofdistinct integral solutionsfor x if 2 1x .

Equilateral triangle ABCis inscribed in a circle and

has area 64 3. Let w bethe exact area of thecircumscribing circle.Find the exact value of kw .


9. InABC ,

AB BCand 5AB BD . Theperimeter of ABD equalsthe perimeter of ACD .Find the length of AC .Express your answer as animproper fraction reducedto lowest terms.

5

5B

A

CD


10. Let 2 24 4x kx k where k represents thelength of the altitudedrawn to the longest sideof a triangle with sides oflength 3, 4, and 5. Findthe positive solution for x.Express your answer as acommon or improperfraction reduced to lowestterms.

FROSH-SOPH 2 PERSON COMPETITION EXTRA LARGE PRINT QUESTION 11ICTM 2014 REGIONAL DIVISION AA CALCULATORS ALLOWED

11. Set A contains 15 elements,set B contains 9 elements, andset C contains 13 elements.A B (A and B) contains 6elements,A C contains 8 elements,B C contains 5 elements, andA B C contains 2 elements.The Universal set containing A,B, and C contains 50 elements.How many elements are in theuniversal set but not in any ofA, B, or C?

FROSH-SOPH 2 PERSON COMPETITION EXTRA LARGE PRINT QUESTION 12ICTM 2014 REGIONAL DIVISION AA CALCULATORS ALLOWED

12. A dart board is aset of concentriccircles as shownwith each annulus(circular band)having the samewidth as the radius of the innermostcircle. Find the probability that,when three darts are thrown, thescore achieved is exactly 70.Answer as a common fractionreduced to lowest terms. (Assumeall three darts hit the board, stick inone of the bands, and the order inwhich the darts hit the board doesnot matter.)

20

40

30

10


Fr/So 2 Person Team (Use full school name – no abbreviations)

Total Score (see below*) =


Answer Score(to be filled in by proctor)

1.

2.(Must be this reduced improper fraction.)

3.

4.

5.

6.(Must be this reduced common fraction.)

7.(Must be this exact answer.)



10.

TOTAL SCORE:(*enter in box above)

Extra Questions:


12.

13.

14.

15.

* Scoring rules:

Correct in 1st minute – 6 points

Correct in 2nd minute – 4 points

Correct in 3rd minute – 3 points

PLUS: 2 point bonus for being firstIn round with correct answer

132

239

10

83

21

24549

162

25625

324

5

3039

256

NOTE: Questions 1-5 onlyare NO CALCULATOR

JUNIOR-SENIOR 2 PERSON COMPETITIONICTM 2014 REGIONAL DIVISION AA PAGE 1 OF 2

1. The inequality 1 7x has the solution :x k x w . Find the sum k w .

2. In right ABC with right angle at C , 4AC and 8AB . The expressiontan cos 2sinA B C has a value that can be written in simplified and reduced form as

k w p

q

. Find the sum k w p q .

3. Let 3 3log ( ) log (4) 2k . Let S be the sum of the arithmetic sequence consisting of the first

23 positive integers. Find the value of the product kS .

4. The solutions of the equation 3 2 22 40 0x x x are k , w , and p . Find the value of

2 2 2k w p .

5. Points A , B , and C lie on circle O with A and B endpoints of a diameter. The area of

circle O is 27 and the altitude of ABC to side AC is 9. cosk ABC . If

1 2 3f x x then 6w f . Find the sum k w .

6. Assume Matt Garza pitches exactly 200 innings in a season. By August of 2013, he hadallowed 8 home runs in 71 innings in the National League before being traded and allowing 7home runs in 45 innings in the American League. At these rates, find the difference in homeruns, rounded to the nearest whole number, he would have allowed if he had pitched theentire season in one league or the other.

7. Let k be the sum of the solutions for22 5 33 1x x . Let w be the sum of the distinct rational

zeros of the function 4 3 23 16 35f x x x x . Find the sum k w . Express your answer

as a common or improper fraction reduced to lowest terms.

8. Find the value of x if 314 213nine eleven sixx .

9. Find the coefficient of the term containing 7 6 5x y z in the expansion of 18

x y z .

10. The graph of a parabola opens upward and has endpoints of the latus rectum at 5,5A and

11,5B . The equation of the circle passing through the endpoints of the latus rectum and

the vertex of the parabola can be written as 2 2 2x h y k r . Find the sum

h k r .

JUNIOR-SENIOR 2 PERSON COMPETITION EXTRA QUESTIONS 11-12ICTM 2014 REGIONAL DIVISION AA PAGE 2 OF 2

11. Find the exact sum of the solution(s) for x if2 2 2

1 2

3 4 6 8 2

x x x

x x x x x x

. (Assume

non-zero denominators.)

12. If a , b , and c are the lengths of the sides of right ABC with a b c . The area of the

right ABC is 2 3 and 2c a . Find the numerical length of the shorter leg.

ICTM Math Contest

Junior – Senior

2 Person Team

Division AA

JUNIOR-SENIOR 2 PERSON COMPETITION LARGE PRINT QUESTION 1ICTM 2014 REGIONAL DIVISION AA NO CALCULATORS ALLOWED

1. The inequality1 7x has the solution

:x k x w .

Find the sum k w .


2. In right ABC withright angle at C , 4AC and 8AB .The expressiontan cos 2sinA B C has avalue that can be written insimplified and reduced

form ask w p

q

. Find

the sum k w p q .


3. Let

3 3log ( ) log (4) 2k .Let S be the sum of thearithmetic sequenceconsisting of the first 23positive integers.Find the value of theproduct kS .


4. The solutions of theequation

3 2 22 40 0x x x are k , w, and p.

Find the value of

2 2 2k w p .


5. Points A, B, and C lieon circle O with A and Bendpoints of a diameter.The area of circle O is27 and the altitude of

ABC to side AC is 9. cosk ABC .

If 1 2 3f x x

then 6w f .

Find the sum k w .

JUNIOR-SENIOR 2 PERSON COMPETITION LARGE PRINT QUESTION 6ICTM 2014 REGIONAL DIVISION AA CALCULATORS ALLOWED

6. Assume Matt Garza pitchesexactly 200 innings in a season.By August of 2013, he hadallowed 8 home runs in 71innings in the National Leaguebefore being traded and allowing7 home runs in 45 innings in theAmerican League. At these rates,find the difference in home runs,rounded to the nearest wholenumber, he would have allowed ifhe had pitched the entire seasonin one league or the other.


7. Let k be the sum of the

solutions for22 5 33 1x x .

Let w be the sum of thedistinct rational zeros of thefunction

4 3 23 16 35f x x x x .

Find the sum k w .

Express your answer as acommon or improperfraction reduced to lowestterms.


8. Find the value of x if314 213nine eleven sixx .


9. Find the coefficient ofthe term containing

7 6 5x y z in the expansion

of 18

x y z .


10. The graph of a parabolaopens upward and hasendpoints of the latus rectumat 5,5A and 11,5B .

The equation of the circlepassing through theendpoints of the latus rectumand the vertex of theparabola can be written as

2 2 2x h y k r .

Find the sum h k r .

JUNIOR-SENIOR 2 PERSON COMPETITION EXTRA QUESTIONS 11-12ICTM 2014 REGIONAL DIVISION AA PAGE 11 OF12

11. Find the exact sumof the solution(s) for xif

2 2

2

1

3 4 6 8

2

2

x x

x x x x

x

x x

(Assume non-zerodenominators.)

JUNIOR-SENIOR 2 PERSON COMPETITION EXTRA QUESTIONS 11-12ICTM 2014 REGIONAL DIVISION AA PAGE 12 OF12

12. If a, b, and c are thelengths of the sides ofright ABC witha b c and 2c a .The area of right

ABC is 2 3.Find the numericallength of the shorterleg.


Jr/Sr 2 Person Team (Use full school name – no abbreviations)

Total Score (see below*) =


Answer Score(to be filled in by proctor)

1.

2.

3.

4.

5.(Must be this integer.)


7.

8.

9.

10.

TOTAL SCORE:(*enter in box above)

Extra Questions:

11.

12.

13.

14.

15.

* Scoring rules:

Correct in 1st minute – 6 points

Correct in 2nd minute – 4 points

Correct in 3rd minute – 3 points

PLUS: 2 point bonus for being firstIn round with correct answer

2

12

621

45

18

14,702,688 OR 14702688

2212

947

6 OR 47

6

24

1

6 OR 1

6

OR 0.16 OR .16

2

NOTE: Questions 1-5 onlyare NO CALCULATOR

ORAL COMPETITIONICTM REGIONAL 2014 DIVISION AA

Problem 1:

The mail typically arrives at Jane's house at a random time between 3PM and 5PM. Jane alwaysarrives home from work at 4PM and then leaves to takes her dog out for an hour long walk at 4:30PM.What is the probability that Jane will be at her home when her mail arrives?

Problem 2:

The telephone company has notified their customers that there will be a 1 hour service outagesometime after 1PM which will be over sometime before 5PM today. Billy and his grandmother plan tobe on the phone for 30 minutes from 2:30PM to 3:00PM today. What is the probability that theirconversation will be affected by the outage?

Problem 3:

Suppose a point with coordinates ,A B is chosen randomly from within a square whose vertices have

coordinates (0,0), (10,0), (10,10) and (0,10).

a) What is the probability that 8B ?

b) What is the probability that 2B A ?

c) What is the probability that 3 2A ?

d) What is the probability that 2 2 4A B ?

e) What is the probability that 2 21 2A B ?

ORAL COMPETITIONICTM REGIONAL 2014 DIVISION AA

EXTEMPORANEOUS QUESTIONS

Give this sheet to the students at the beginning of the extemporaneous question period.

STUDENTS: You will have a maximum of 3 minutes TOTAL to solve and present your solutionto these problems. Either or both the presenter and the oral assistant may present the solutions.

Extemporaneous Problem 1:

The diagram below consists of two concentric circles, a larger (outer) circle of radius 10 meters and ansmaller (inner) circle of radius 5 meters. Suppose a point is randomly chosen somewhere on the interiorof the larger circle.

a) What is the probability that the point chosen is also inside of the smaller circle ?

b) What is the probability that the point chosen is not inside of the smaller circle ?


A random point is chosen from within a square with sides of length 1. A smaller square with sides oflength s lies inside this square. For what value of s does the point have a probability of 0.75 of beingoutside of the smaller square?


Dan and Tom have decided to meet for lunch. They are approaching the restaurant from oppositedirections and are 10 miles apart. The restaurant is somewhere between them but neither knows where.If Dan is biking 20 miles per hour and Tom is biking 10 miles per hour what is the probability that Tomwill reach the restaurant first?

10 m

5 m

1

1

s

s

ORAL COMPETITIONICTM REGIONAL 2014 DIVISION AA – Judges Solutions

Problem 1:

The mail typically arrives at Jane's house at a random time between 3PM and 5PM. Jane alwaysarrives home from work at 4PM and then leaves to takes her dog out for an hour long walk at 4:30PM.What is the probability that Jane will be at her home when her mail arrives ?

Solution:The probability that Jane will be home is the probability that the mail will arrive between 4PM

and 4:30PM which is:30 minutes 1

2 hours 4 .

Problem 2:

The telephone company has notified their customers that there will be a 1 hour service outagesometime after 1PM which will be over sometime before 5PM today. Billy and his grandmother plan tobe on the phone for 30 minutes from 2:30PM to 3:00PM today. What is the probability that theirconversation will be affected by the outage ?

Solution:The crucial idea for this problem is to base the analysis on when the outage begins. Billy'sconversation with his grandmother will only be affected by the outage if it begins sometimebetween 1:30PM and 3:00PM. Furthermore, the outage will begin sometime after 1PM, butbecause the outage has to end before 5PM, it must begin before 4PM. Hence the probability weare looking for is:

1.5 1

3 2

Span of start time which will affect Billy hours

Span of all possible start time hours

Problem 3:

Suppose a point with coordinates ,A B is chosen randomly from within a square whose vertices have

coordinates (0,0), (10,0), (10,10) and (0,10).

a) What is the probability that 8B ?

b) What is the probability that 2B A ?

c) What is the probability that 3 2A ?

d) What is the probability that 2 2 4A B ?

e) What is the probability that 2 21 2A B ?

Solution:

a) The region 8B looks like:

The probability of a point falling in the shaded region is

8 10 4

10 10 5 .

8B

b) The region 2B A cuts the square as illustrated:

(10,8)

(2,0)

So the probability is: .5 8 8 32 8

100 100 25

area of triangle

area of square .

c) The region 3 2A means that 1 5A which looks like:

1A 5A

The probability of a point falling in the shaded region is

4 10 2

10 10 5 .

d) The region 2 2 4A B is a quarter circle centered at (0,0) with radius 2:

(0,2)

(2,0)

So the probability is 212

4

100 100

area of quarter circle

area of square

.

e) The region 2 21 2A B is a sector of a circle with center at 1,0 and radius 2 and the

triangle with vertices at (0,0), (0,1) and (1,0) which appears as follows:

(0,1)

(1,0)

The circle sector has a radius 2 and sweeps3

4

radians or 135 . This area can be computed

by finding the area of the sector. The sector is135 3

360 8

of the circle and has area

23 3 32

8 8 4Area of circle . The triangle has area

1 11 1

2 2 .

The probability is then

3 1

3 24 2

100 400

area of sector + area of triangle

area of square

(or exact equivalent).

ORAL COMPETITIONICTM REGIONAL 2014 DIVISION AA – JUDGES SOLUTIONS

EXTEMPORANEOUS QUESTIONS

Give this sheet to the students at the beginning of the extemporaneous question period.

STUDENTS: You will have a maximum of 3 minutes TOTAL to solve and present your solutionto these problems. Either or both the presenter and the oral assistant may present the solutions.


The annular region illustrated below consists of two concentric circles, a larger (outer) circle of radius10 meters and an smaller (inner) circle of radius 5 meters. Suppose a point is randomly chosensomewhere on the interior of the larger circle.

a) What is the probability that the point chosen is also inside of the smaller circle ?

b) What is the probability that the point chosen is not inside of the smaller circle ?

Solution:

a) The probability that the point is inside the smaller circle is:

2

2

5 1

410

area of inner circle

area of outer circle

.

b) The probability that the point is outside the smaller circle is:

1 – the probability from (a), so the answer is3

4.

Note that an alternate solution would be to find the difference in the areas of the larger and

smaller circles and then divide by the area of the larger circle:100 25 75 3

100 100 4

.

10 m

5 m


A random point is chosen from within a square with sides of length 1. A smaller square with sides oflength s lies inside this square. For what value of s does the point have a probability of 0.75 of beingoutside of the smaller square?

Solution:

The probability that the point is outside the smaller square is 21 s . Thus, the value of s that

meets the conditions is the solution to 21 0.75s which is 0.5s .

Extemporaneous Problem 3:Dan and Tom have decided to meet for lunch. They are approaching the restaurant from oppositedirections and are 10 miles apart. The restaurant is somewhere between them but neither knows where.If Dan is biking 20 miles per hour and Tom is biking 10 miles per hour what is the probability that Tomwill reach the restaurant first ?

Solution:

The specific distance of 10 miles between Dan and Tom is not used in the solution. By the timeDan and Tom would hypothetically reach each other Dan will have covered twice as much groundas Tom. Therefore, Tom has a 1 in 3 chance of reaching the restaurant first. So the probability

that Tom will reach the restaurant first is1

3.

1

1

s

s

2014 ICTM RAA Algebra I -Final - Plainfield East High...

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