2013 ictm saa algebra i - psd202.org

WRITTEN AREA COMPETITION ALGEBRA IICTM STATE 2013 DIVISION AA PAGE 1 OF 3

1. Let 3 5 12ax x bx x x for all values of x . Find the value of a b .

2. A standard pair of dice is rolled. What is the probability that the sum of the numbers facingup on the dice is a prime number? Write your answer as a reduced common fraction.

3. The ordered pair of the form ,x y that satisfies the system5 6 19

7 27

x y

x ky

is 1,4 . Find

the value of k .

4. The reciprocal of1

3is one of the roots for x of the quadratic equation: 2 6 0x kx . Find

the value of k .

5. Let k and w represent positive integers with k w . If 7 2k w , find the largest

possible value of the product kw .

6. Calvin has nine books labeled 1 through 9 in a row. In how many ways can he arrange these,if book number 5 has to be in the middle?

7. Assuming non-zero denominators,2

2 2 2

5 7 8 5

3 9 6 9 ( 3)( 9)

x kx w

x x x x x x

. Find the

value of k w .


8. Let a , b , c , and d be positive integers such that 1 a b c d and 1440abcd . Findthe largest possible value of d .

9. The line 10y intersects the curve 22 4 4y x x at two points. Find the distance

between these two points.

10. Some friends are throwing a party and decide to share expenses equally. The expenses willtotal $60. Another friend wishes to join. This will increase the total expenses by $10, butwill reduce each person’s share by $1. How many friends were initially going to throw theparty?

11. If x and y are positive integers such that 17 5 118 x y , find the value of y .

12. A given line has the equation of 3 5 7x y . The equation of a line that is parallel to the

given line and passes through the point 0, 6 can be written in the form kx wy p where

k , w , and p are positive integers. Find the smallest possible value of k w p .

13. Let x represent a positive integer. Let y represent a positive integer less than 14 such that

one more than twice x is equal to 8 less than the square of y . Find the sum of all possible

distinct values of x .

14. Cindy has 25 sticks of gum, Lee has 31 sticks of gum, and Jeffrey has 37 sticks of gum. Letk , w , and x be positive integers. The three discovered that if each sold from his/her supplyat the rate of x cents per stick, and then change their price to sell their remaining sticks at therate of k sticks for w cents, then each of the three would take in the same amount of money.

Find the smallest possible value of k w x . (Note: Each person took in a whole number

of cents and sold the “remaining” sticks in complete sets of k sticks.)


15. (Multiple Choice) For your answer write the capital letter which corresponds to thecorrect choice.

Karen is putting up posters for the Math contest. For each poster, Karen will use fourpieces of masking tape—each 10 inches long. Karen started with a 100-yard roll ofmasking tape. After putting up x posters, Karen still had part of her roll of tape left andhad not wasted any tape. The number of yards of tape that Karen had left on her roll afterputting up x posters was:

A) 100 10x B) 100 40x C)10

1009

x D)5

10018

x E) 100 360x

Note: Be certain to write the correct capital letter as your answer.

16. At College Prep High School, 96% of the graduating seniors went on to college. Of thosecollege-bound seniors, 65% went to colleges that were out of state. If that number who wentto college out of state was 312, how many seniors were in that graduating class from CollegePrep High School?

17. Find a , the smallest real number x , and b , the largest real number x , that are solutions for:

5 2 3 8x x x . Give your answer as ordered pair ,a b .

18. When written as a single fraction in simplest terms with k , w , p , and f positive integers,

( 1) ( 1) ( )

1 1 1 (2 )

2 4 2 2

x

x x x wx f

k p

. Find the value of k w p f .

19. If 2 1024x , find the smallest possible value of 3 5x .

20. Three farmers (Bob, Dick and Tom) rented a pasture and a large horse barn for a fixedamount per month. They agreed to pay in proportion to the number of horses each held. Forthe first month, Bob had 18 horses while Dick and Tom each had some horses. Dick paid atotal rent of $480 for the first month. For the second month, each put in 2 more horses, andTom paid a total rent of $300. For the third month, each put in 4 more horses than each hadthere in the second month, and Bob paid a total rent of $900. Find the number of dollars oftotal rent that Tom paid for the 3 months.

WRITTEN AREA COMPETITION GEOMETRYICTM STATE 2013 DIVISION AA PAGE 1 OF 3

1. A cube has an edge whose length is 10.1. Find the volume of this cube. Express youranswer as an exact decimal.

2. In the diagram, points 8,9A and B lie on the circle with

center at 6,4O . The equation of the circle on the diagram

can be represented by 2 2( 6) ( 4)x y k Find the value

of k .

3. Given: 4,0A , 10, 2B , 3,2C . In ABC , find the slope of the altitude from C to

AB .

4. The three vertices of a right isosceles triangle are 3,4 , 12, y , and 12, 4 . Find the sum

of all distinct possibilities for the value of y .

5. The diagrams show 4 squares and one equilateral triangle with side-lengths as indicated. Ifthe sum of the perimeters of the 4 squares is 5 times the perimeter of the equilateral triangle,find the value of x . Express your answer as an exact decimal.

6. Given ABE with EAB and EBA each trisected with the

segments shown in the diagram. If AE BE and 30AEB ,

find the degree measure of ADC .

7. A regular polygon of k sides has side-lengths that are integers. Each interior angle of thepolygon is an integral number of degrees. If 6 100k , find the sum of all possible distinctvalues of k .

y -axis

x-axis

O

A

B

5x

12810

A B

E

D

C


A B C

D

E

8. One of the interior angles of a rhombus has a degree measure of 30, and the length of one ofthe sides of the rhombus is 46. Find the area of this rhombus.

9. In the given figure, triangles ABE and BDC are equilateral. Points A , Band C are collinear, ABE has perimeter 24, and the ratio of the area of

ABE to the area of BDC is 4:1. What is the area of quadrilateralBCDE ?

10. In the diagram, points A , C , B , and D lie on the circle and diameter

AB is perpendicular to chord CD at E . If 4AE and 64EB , find

the length of CD .

11. Polygon ABCD is inscribed in a circle. 30AB , BC k , 73CD , and 55DA . If k is

a positive integer and the area of polygon ABCD is 30 11609 , find the value of k .

12. A rhombus with diagonals of lengths 12 and 16. A square has a perimeter of 28 2 . Allfour diagonals of the two quadrilaterals are contained in two perpendicular lines. Find thearea that is common to both the rhombus and the square.

13. In the diagram with coordinates of points A, B, and C as shown,point A is reflected across the y-axis to point 'A . Find the slope

of 'A C

. Express your answer as a common fraction reduced tolowest terms.

14. A circle is inscribed in a triangle with sides of 6, 12 and 14. The point of tangency dividesthe longest side of the triangle into two segments. What is the ratio of the smaller of thesetwo segments to the larger of the two segments? Give your answer in the form :k w .

E

B

A

C D

x-axis

y -axis

A(2,3)

B(10,7)

C(8,12)


15. A regular tetrahedron has an edge whose length is 24. Find the length of a radius of thecircumscribed sphere of this regular tetrahedron.

16. In the diagram, A-BCDE is a regular square pyramid as shown. Ifthe perimeter of the square base is 88, and the volume of thepyramid is 2420, find the length of an altitude of the pyramid.

17. ABC is a right triangle with right angle C . A circle with center at C is drawn so that thecircle contains both point A and the midpoint of the hypotenuse of ABC . What is thedegree measure of CBA ?

18. In the diagram, points A , B , D , and E lie on thecircle. Points A , B , and C are collinear, and points C ,D , and E are collinear. If 36ACE , and

94AE , find the degree measure of BD .

19. In the county of Dell, the River Sticks flows due east-west for many miles and is uniformly0.03 kilometers wide. The town of Hayden lies 1.36 kilometers due north of the north bankof the River Sticks. The town of Pluydon lies 3.43 kilometers due south and 5.75 kilometersdue east of Hayden and is on the other side of the river. The shortest possible road was builtconnecting these two towns, including a 0.03 kilometer bridge over the river, constructedperpendicular to its banks. Rounded to the nearest hundredth of a kilometer, how far would atraveler walk between Hayden and Pluydon if she stayed on this road? Express your answeras a decimal.

20. In scalene triangle ABC , ABC is obtuse. The lengths of all sides are integers, and thelength of the longest altitude is an integer. If the area of ABC is 90 square feet, find thesmallest possible number of feet in the perimeter of ABC .

D

B

C

A

E

B

A

CD

E

WRITTEN AREA COMPETITION ALGEBRA IIICTM STATE 2013 DIVISION AA PAGE 1 of 3

1. Let 1, 2,3, 4,5,6,7,8,9A . If k is an element of A , find the sum of all distinct values of

k such that the graph of ky x is symmetric with respect to the y-axis.

2. If x represents an integer, find the largest possible value of x such that2 3

5 206

x

.

3. Let 1i and let k and w represent real numbers. If (3 2 )(6 )i i x k wi is solved

for x , then 20 5x i . Find the value of k w .

4. The plane whose equation is 3x by cz contains the line of intersection of the planes

whose respective equations are 2 9x y z and 3 6x y z . Find the product bc .

5. Let 15,16, 22,23,36, 42,106,111,118, 214, 248B . Find the sum of all distinct members of

B that can be expressed as the sum of the squares of six (not necessarily distinct) oddintegers.

6. Let k represent a positive integer. For how many distinct values of k is the following

inequality true: 7 5 3678k .

7. Find the length of the latus rectum of the ellipse whose equation is2 24 2 2 16 22x x y y .


8. Let 1i . Let k and w represent positive integers with k w . If( 7 13 )( ) 2( 103 20 )i k wi i , find the value of w .

9. A set consists of 4 distinct positive integers—all of which are less than 41. How manydistinct such sets exist such that no subset of the set (including the set itself) has an integralmultiple of 6 as the sum of its members?

10. Find the ninth term of the geometric sequence whose first three terms are:25

3,

5

3and

1

3.

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

11. If 4f x x , then 1 2( )f x kx w , x p . Find the value of 2 3k w p .

12. In how many distinct points do the real graphs of 2 2 9103x y and 2 9102x y intersect?

13. Let k be a positive integer. If 1000k , find the number of distinct possible values of k

such that 1 2 3k k k k is the square of a positive integer.

14. A person is dealt four cards at random without replacement from a standard 52 card deck(four suits of 13 ranks each). Expressed as a decimal rounded to 4 significant digits, find theprobability that at least 2 cards are of the same rank.


15. Let x , y , z , and w be four distinct positive integers such that 2 2 2x y z w . If w is a

minimum, find the smallest possible value of x y z .

16. If k represents a positive integer greater than 1, find the smallest possible value of k such

that 2log 8 log 16k represents a positive integer.

17. Consider the set of 125 points in three-space whose coordinates are of the form , ,x y z ,

where x , y , and z are taken from the set 1, 2,3, 4,5 . How many lines are there that

contain five of these points?

18. If 2 4 2( 1) 5 9f x x x , then 2 4 2( 3)f x kx wx p where k , w , and p are integers.

Find the value of k w p .

19. Find the value of k if the graph of3 10

( )4

kxf x

x k

has a y-intercept of 25. Express your

answer as a reduced common or improper fraction.

20. In scalene ABC , ABC is obtuse. The lengths of all sides are integers, and the length ofthe longest altitude is an integer. If the area of ABC is 198, find the perimeter of ABC .

WRITTEN AREA COMPETITION PRECALCULUSICTM STATE 2013 DIVISION AA PAGE 1 OF 3

1. Find the fifth term of an arithmetic sequence whose first and second terms are respectively 1and 12.

2. If 2( ) 5f x x and 2( ) ( 5)g x x , find the value of (3) (4)f g .

3. Let 1i . Find the value of 7 4i .

4. Let ( , )k w represent the norm of the vector represented by ,k w . If ( ,9) 9 5k and

0k , find the value of k .

5. The degree measures of the three angles of ABC are in the ratio of 2 : 2 : 5 . The degreemeasures of the three angles of DEF are in the ratio of 5 :11:14 . One of the six anglesfrom these triangles is selected at random. Find the probability that the angle selected has adegree measure of 40. Express your answer as a common fraction reduced to lowest terms.

6. The first term of a geometric sequence of real terms is 2, and the fifth term of this sequence is18. If the fourth term of this sequence is negative, find the value of the fourth term of thisgeometric sequence.

7. Consider the sequence { }nx defined as ( 1)

6 5491

7 7n

n

n

xx

x for integral n such that 1n and

for which 1 11x . As n increases without bound, the limiting value of nx is k . Find the

value of k .


8. The polynomial equation 3 2 0x kx wx p with k , w , and p representing integers has

5 and 1 i as two of its roots for x . If 1i , find the value of p .

9. A plane contains points with coordinates 2, 1,1A , 0, 3,0B , and 1, 2,1C . A fourth

point not on this plane has coordinates 5,2, 4D . Find the volume of a tetrahedron with

vertices at A , B , C , and D .

10. If( )!

1!

x kx

x

for all possible values of x where x and k represent positive integers,

find the value of k .

11. Find the sum of the terms of an infinite geometric sequence whose first four terms are1 1 1

1, , , ,4 16 64

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

12. In a 3-dimensional rectangular coordinate system, we have the points 1, 4,3A , 2,6,8B ,

and , ,C k w f . If AB AC , find the value of 2 5k w f .

13. Team A plays Team B six times in a season. For each game, the probability that Team A

will win is3

5. Find the probability that Team A will win at least 3 consecutive games of the

six games in the season. Express your answer as a common fraction reduced to lowest terms.

14. Assume that a , b , c , and d are positive, 12a b c d , 2 2 2 2 50a b c d , and that

2c d . Then the greatest possible value of d can be expressed in the formk w p

f

where

k , w , p , and f are positive integers andk w p

f

is in simplest radical form. Find the

value of k w p f .


15. The tangent of one of the acute angles of a right triangle is 1. If one of the legs of this righttriangle has a length of 16, find the length of the other leg of this right triangle.

16. Sally wishes to produce an 8 digit code (sequence of digits) using only the digits 1, 2, 3, and4 for each code. How many distinct codes are possible if the number of times the digit 1 isused equals the number of times the digit 2 is used for each code?

17. In ABC ,89

sin( ) cos( )65

BAC BAC . If tan( ) 1BAC , find tan( )BAC Express

your answer as an improper fraction reduced to lowest terms.

18. (Multiple Choice) For your answer write the capital letter which corresponds to the correctchoice.

The graph of 324xy is symmetric with respect to the:

A) Origin.B) y-axis.C) x-axis.D) Both the origin and the x-axis.E) Both the origin and the y-axis.


19. Find the value of20 14

lim2 3x

x

x

.

20. Given that 2 1 0x x and1

( ) m

mP m x

x . Find the exact value of

(2013) (2014) (2015) (2016)P P P P .

FROSH-SOPH EIGHT PERSON TEAM COMPETITIONICTM STATE 2013 DIVISION AA PAGE 1 of 3

NO CALCULATORS

NO CALCULATORS

1. If 7k , for what value of w is the value of the fraction15.4

2 1k w undefined?

2. The perimeter of an equilateral triangle is 60. Find the length of a median of this equilateraltriangle.

3. Find the sum 3210 1132four four . Write your answer as a base eight numeral, using eight

as the subscript.

4. An equilateral triangle has vertices at the midpoints of alternate sides of a regular hexagon.The perimeter of the hexagon is 36. Find the perimeter of the equilateral triangle.

5. The perimeter of a square is 24 2 . Find the length of a diagonal of the square.

6. Two integers have a sum of 72 and are in a ratio of 4 to 5. When an integer k is added to thesmaller and subtracted from the larger of the original integers, the ratio is reversed. What isthe value of the integer k ?

7. The other day LeBron shot 100 free throws. The largest number of consecutive free throwshe made that day was 10. Out of LeBron’s 100 free throws, what is the largest number hecould possibly have made?

8. Given the system

193 4

2

72

4

y x xy

y x xy

. If 0xy , find the ordered pair ,x y that is the

solution to this system.


NO CALCULATORS

NO CALCULATORS

9. In the diagram, points A and B lie on the circle centered at

O whose equation is 22 28 52 18 0x x y . If

60AOB , the area of sector AOB can be expressed ask . Find the value of k .

10. Line p contains points 5, 3 and 1, 5 . Line q has an x-intercept of 1 and a y-intercept of

2 . Find the coordinates of the point of intersection of lines p and q . Write your answer

as an ordered pair ,x y .

11. Three measures of distribution of two genetic characteristics, A and B, in a population of 100subjects are: 17 have A and B, 23 have A but not B, and 15 have neither A nor B. The ratioof the number of people having characteristic A to the number of people having characteristicB can be expressed as :k w where k and w are positive integers. Find the smallest possible

value of value of k w .

12. (Multiple Choice) For your answer write the capital letter which corresponds to the correctchoice.

The locus of points in space that are equidistant from all points on a given circle is:

A) Two parallel lines. D) Two points. F) Four points.B) A line. E) Three points. G) A circle.C) A point.


13.

21 1

3 22 2 4

xx x

, find the smallest real solution for x .

14. A circle has an area of 200 . Find the area of a square inscribed in this circle.

15. 481 4x can be factored as 2 2ax bx c ax bx c where a , b and c are integers.

What is the absolute value of b ?

y -axis

x-axis

O

AB


NO CALCULATORS

NO CALCULATORS

O

D

C

B

A

16. In the diagram shown circle O has radius 8 and tangents AB

(at point B ) and AC (at point D ). If 15AD and 21AC ,how much closer to circle O is point C than point A ?

17. In the diagram, not drawn to scale, the line containing A ,B , and C is parallel to the line containing D , E , and F .If 138ABG and 47FEG , find the degreemeasure of BGE .

18. In the diagram, AB is tangent to the externally tangent circlesat A and B . If the lengths of the radii of the two circles are

respectively 18 and 8, find the length of AB .

19. How many integers are a solution for the inequality2

5 14 0x x ?

20. Tom has red, white, and blue flagstones for making a walk at his home. The flagstones willbe laid in single file. He has established the following three conditions:

a. no two consecutive stones can be the same color (such as RR)b. no consecutive pair of stones can have the same two colors in the same order (such as

RWRW)c. no non-overlapping foursome of stones can have the same four consecutive colors in

the same order (such as RWRBRWRB)Tom starts out laying a red stone first, a white stone second, and a red stone third. Howmany distinct orders now exist for the fourth, fifth, sixth, seventh, and eighth stones sothat Tom will not violate any of his three conditions.

A

B

G

FE

D

CBA

JUNIOR-SENIOR EIGHT PERSON TEAM COMPETITIONICTM STATE 2013 DIVISION AA PAGE 1 OF 3

NO CALCULATORS

NO CALCULATORS

1. If 3 2( ) 2 3f x x x , find the value of (7)f .

2. Let a

and b

represent vectors and let k represent a scalar with (3, 2)a

and b ka

. Find

the ordered pair representing b

if 14k .

3. If3 5 1 2 33 31

2 6 5x k w

and 2 3 2x k w , find the value of x . Express your answer

as a common or improper fraction reduced to lowest terms.

4. Let !

,!( )!

nC n k

k n k

where n and k represent positive integers. Find the positive

difference between the two distinct values of k such that 14, 2002C k .

5. Let be a radian measure. The first two terms of an arithmetic sequence, whose commondifference is greater than zero, are k and w where k and w are solutions for in the

interval [0,2 ] to the system3cos( )

1 cos( )

r

r

. Find the sum of the first five terms of this

arithmetic sequence.

6. Dealing only with real numbers, 7 7

1log ( 12) log ( ) 0

2x x . Find the value of x .

7. Find the value of the indicated sum5

2

1

( 3)k .


NO CALCULATORS

NO CALCULATORS

8. If the population of 100 bacteria doubles every fifteen minutes, how many minutes will ittake for the population of 100 bacteria to reach 12,800 bacteria?

9. The difference between two numbers is 48. If their arithmetic mean exceeds their positivegeometric mean by 18, find the larger of the two numbers.

10. In ABC , not necessarily drawn to scale, 8AC , 4AB , and

36CAB . Then BC can be expressed as k w f in

simplest radical form, where k , w , and f are positive integers.

Find the smallest possible value of k w f .

11. Find the value of2

5

14 95lim( )

5x

x x

x

.

12. Sammy has a batting average of 0.250 (on the average, 1 hit out of every 4 official at bats).On this basis, find the probability that on his next 5 official at bats, Sammy will get a hit onhis first official at bat, at least one hit out of his next 3 official at bats, and a hit on this fifthofficial at bat. Express your answer as a common fraction reduced to lowest terms.

13. The sum of two numbers is 17, and the sum of their reciprocals is 0.5 . Find the product ofthe two numbers.

4

8

A B

C


NO CALCULATORS

NO CALCULATORS

14. The graph of f consists of the union of two line segments. The first line segment connects

0,1 to 2, 2 , and the second line segment connects 2, 2 to 5,3 . The graph of

3 3 7y f x also consists of the union of two line segments. One line segment

connects 3, 10 to 1, 13 , and the other line segment connects 1, 13 to ,x y .

Find the ordered pair ,x y .

15. The arithmetic mean of 23, 30, 16, 33, and two other numbers that differ by 2 is 21. If thesmallest of these six numbers is discarded, find the arithmetic mean of the remaining fivenumbers.

16. Find the magnitude of the cross product of vector 1,3,5 and vector 2,1,3 .

17. An ellipse has a major axis whose length is 8 and has foci at 2,0 and 2,0 . The area of

this ellipse can be expressed in the form k . Find the exact value of k .

18. (Always, Sometimes, or Never) For your answer, write the whole word Always,Sometimes, or Never—whichever is correct.

If 3x represents an irrational number, then 3x represents a rational number.

19. In the diagram with coordinates as shown and P in theinterior of ABC , 25AC , and 26BC . The ratio of thearea of PAB to the area of PAC to the area of PBC is2 :3 : 5 . Find the ordered pair that represents point P .Express your answer as an ordered pair with each memberof the ordered pair expressed as an improper fractionreduced to lowest terms.

20. How many distinct integers k such that 1000 2000k with four different digits are theresuch that the absolute value of the difference between at least two digits is 3?

x-axis

y -axis

P

C

B(17,0)A(0,0)

CALCULATING TEAM COMPETITIONICTM STATE 2013 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. A scalene triangle has side-lengths of 67.83 , 93.41, and 96.22 . Find the perimeter of thisscalene triangle.

2. In the diagram, AB is the hypotenuse of right ABC and has alength of 26. The radius of the inscribed circle has a length of 5.Find the smallest possible value of BC .

3. At the beginning of May, Kim had $9876.51 in her checking account, and all checks writtenhad cleared. During May, Kim wrote checks of $27.43, $68.49, $111.50, $250.00, and$49.18. During May, she made deposits of $550.03 and $847.96. During May, she earnedinterest of $18.03. If all the checks written by Kim during May had cleared, what was theamount in her checking account at the end of May? Give the exact answer in dollars andcents. Do not use scientific notation.

4. If 3 2( ) ( 3) 4( 2.1) 13.76f x x x kx , find the value of k such that (1.413) 16.78f .

5. When 10( 3 )x y z is expanded and completely simplified, find the sum of the numerical

coefficients.

6. In the diagram of two intersecting lines with degreemeasures as shown, find the value of k .

7. The first three terms of a geometric sequence are respectively: 863.1, 604.17 , 422.919 .Find the fourth term of this geometric sequence. Express your answer as an exact decimal.Do not round off or use scientific notation.

8. Team A is playing Team B in a series of a maximum of 5 games. The series ends as soon asone of the two teams has won 3 games. Assume that the probability that Team A will winany game is x and that that probability is constant. Assume also that the probability thatTeam A will win the series is 0.6432. Find the value of x .

A C

B

(4.019k)

(w+81.02)

(2.333w+12.19)


9. In the diagram (not necessarily drawn to scale), 15AD ,13AB , and 24BC . 113DAB , and

75ABC . Find the area of quadrilateral ABCD .

10. Given the arithmetic sequence: 1.432, 2.544, 3.656, ,k . If the sum of the terms of this

arithmetic sequence is greater than 150, find the smallest possible value of k . Express youranswer as an exact decimal. Do not use scientific notation.

11. Find the largest possible integer that is smaller than the smallest root of3 214.8 19.61 111.228x x x . Write your answer as an exact integer. Do not use

scientific notation.

12. The smallest three digit positive integer ' 'XYZ such that 3 3 3X Y Z is equal to the three

digit integer ' 'XYZ is 153 since 3 3 31 5 3 153 . Find the next smallest three digit positiveinteger meeting these same given conditions. Express your answer as an integer. Do notuse scientific notation.

13. Tom just invested $1,000 in a savings account that pays 6.3% annual percentage rate (APR)compounded monthly. Once the interest has been credited after k months, Tom has morethan $1500 in his savings account. Find the smallest possible value of k . Express youranswer as an exact integer.

14. An equilateral triangle is inscribed in a circle whose equation is 2 2( 6) 10 41x y y .

Find the area of the equilateral triangle.

15. An investor deposits $1000 each month for seven months in her brokerage account topurchases shares of the ICTM Company. Each month she purchases as many whole numberof shares as possible for the money available. Any leftover amount is left in the account andavailable for the next month. For example, if she had $1011.98 available and the cost pershare was $21.01, she would be able to purchase 48 shares and have $1003.50 available fornext month (after her new deposit.) During the first month, she had exactly $1000 available.The cost per share for each of the seven months was respectively: $22.03, $19.67, $18.89,$18.98, $20.73, $22.08, and $23.11. How many total shares did she purchase during thisseven month period? Give your answer as an exact whole number. Assume nocommissions or any other costs are involved.

D C

B

A


16. The points 4.235,6.231A , 12.33, 5.678B , and 3.666, 4.221C are the vertices of

ABC . Find the length of the longest side of ABC .

17. In the diagram (not necessarily drawn to scale) withcoordinates as shown, 89AC , and 90BC . Theratio of the area of PAB to the area of PAC tothe area of PBC is 5 : 6 : 9 . Find the distance fromP to C .

18. Find the value of x if ( 1)3(8.164 864.2) 32.46x x .

19. In the diagram (not necessarily drawn to scale), ACD is

equilateral, B lies on AC , 90ABE , 15AB , 28BE ,

82AC . Find the distance from E to the closest point on AD .

20. Point A lies on both circles with centers at D and

B . Point E lies on D and on DC . The

segment containing A , B , and C is a diameter,

and AC

is tangent to D . The area of B is

three times the area of D . If 7EC , find the

area of the region bounded by AB , BE , and the

minor arc AE .

x-axis

y -axis

P

C

B(17,0)A(0,0)

D

C

A

B

E

E

CA

D

B

FROSH-SOPH 2 PERSON COMPETITION ICTM 2013 STATE DIVISION AA PAGE 1 OF 1 1. Find the area of a trapezoid with parallel sides of lengths 14 and 35 and legs of lengths 13 and 20.

2. If 15 55x , 5 15y , and 45 95z , let k be the greatest possible value of x z

y

. If

2 216 9 20a c and 4 3 10a c , let 4 3w a c . Report as your answer the value k w .

3. Let k be the smallest integer such that kx must be greater than 2x , if 0.6 1x . All the sides

of a right triangle have lengths that are integers. If 10 is the length of the smallest side, let w be the smallest possible length of the largest side. Find the value of k w .

4. Players stand in a circle. Player 1 stays in. Player 2 is knocked out. Player 3 stays in. Player 4 is

knocked out. This process continues, knocking every other Player out, until only one Player remains. Find the largest possible number of players from 14,15,16,17,18,19, 20, 21, 22, 23, 24

such that Player 13 is the last player remaining. 5. If x and y are integers such that 19 23x and 8 4y , let k be the largest possible value

of the product xy . Let the “Nelson” of a number be defined as 36 less than 3 times the number.

The number that is equal to its “Nelson” is w . Report as your answer the value k w .

6. Let k be the combined areas of the inscribed and circumscribed circles of a triangle whose side-

lengths are 12, 16, and 20. The three points 4,5 , 7,17 , and 95,w are collinear. Find the

value of k w .

7. Let k be the number of gallons of a 50% silver nitrate solution that are added to 15 gallons of a

30% silver nitrate solution to produce a 35% silver nitrate solution. Let w be the number of sides of a regular polygon whose degree measure of one of the exterior angles is 8. Find the value of

k w .

8. Let ,x y be the coordinates of the point that is 0.2 of the way from 4,16 to 31,56 . Let

( 2)2 8k w and ( 3)9 3w k . Find the value of k w x y .

9. Let n be the number of distinct integer values for x such that 20 3 17x . A cube has an edge

of length e and a volume of v . If each edge of the cube is increased in length by 25%, the volume of the new cube is kv . Find the value of n k . Express your answer as an improper

fraction reduced to lowest terms. 10. Let k be the sum of the distinct positive prime factors of 6840. A hole that is 60 units in diameter

is cut in a sheet of cardboard assumed to have no thickness. A sphere 68 units in diameter is set in the hole. Let w be the number of units the sphere extends below the surface of the cardboard. Find the value of k w .

2013 SAA School ANSWERS

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

Total Score (see below*) =

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.

Answer Score(to be filled in by proctor)

1.

2.

3.

4.

5.

6.

7.

8.(Must be this reduced improper fraction.)

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

294

32

31

22

82

229

50

33893

64

47

NOTE: Questions 1-5 onlyare NO CALCULATOR

ANS

ANS

ANS

ANS

ANS

FROSH-SOPH 2 PERSON COMPETITION ICTM 2013 STATE DIVISION AA LARGE PRINT QUESTION 1

1. Find the area of a

trapezoid with parallel sides of lengths 14 and 35 and legs of lengths 13 and 20.


2.If 15 55x , 5 15y ,

and 45 95z , let k be the greatest possible

value of x z

y

.

If 2 216 9 20a c and 4 3 10a c , let

4 3w a c .

Report as your answer the value k w .


3. Let k be the smallest integer

such that kx must be greater than 2x , if 0.6 1x . All the sides of a right triangle have lengths that are integers. If 10 is the length of the smallest side, let w be the smallest possible length of the largest side. Find the value of k w .


4. Players stand in a circle.

Player 1 stays in. Player 2 is knocked out. Player 3 stays in. Player 4 is knocked out. This process continues, knocking every other Player out, until only one Player remains. Find the largest possible number of players from 14,15,16,17,18,19,20,21,22,23,24

such that Player 13 is the last player remaining.


5. If x and y are integers such

that 19 23x and 8 4y , let k be the

largest possible value of the product xy. Let the “Nelson” of a number be defined as 36 less than 3 times the number. The number that is equal to its “Nelson” is w. Report as your answer the value k w .


6.Let k be the combined

areas of the inscribed and circumscribed circles of a triangle whose side-lengths are 12, 16, and 20. The three points 4,5 , 7,17 , and

95,w are collinear. Find the value of k w .


7. Let k be the number of

gallons of a 50% silver nitrate solution that are added to 15 gallons of a 30% silver nitrate solution to produce a 35% silver nitrate solution. Let w be the number of sides of a regular polygon whose degree measure of one of the exterior angles is 8. Find the value of k w .


8. Let ,x y be the

coordinates of the point that is 0.2 of the way from 4,16 to 31,56 . Let ( 2)2 8k w and

( 3)9 3w k . Find the value of k w x y .


9. Let n be the number of

distinct integer values for x such that 20 3 17x . A cube has an edge of length e and a volume of v. If each edge of the cube is increased in length by 25%, the volume of the new cube is kv. Find the value of n k . Express

your answer as an improper fraction reduced to lowest terms.


10. Let k be the sum of the

distinct positive prime factors of 6840. A hole that is 60 units in diameter is cut in a sheet of cardboard assumed to have no thickness. A sphere 68 units in diameter is set in the hole. Let w be the number of units the sphere extends below the surface of the cardboard. Find the value of k w .

JUNIOR-SENIOR 2 PERSON COMPETITION ICTM 2013 STATE DIVISION AA PAGE 1 OF 1

1. Let w be a positive integer less than 300 such that w k k k where k is a positive integer. Find the sum of all distinct possible values of w .

2. Let 1i . Find the value of 8log 8 8i . Express your answer as an improper fraction

reduced to lowest terms.

3. It is given that 1 2

( )3

xf x

and

1 3( )

2

xg x

. Find

7 8 9 10f g g f f g g f .

4. Let x be a positive integer such that 1 20x . Find the sum of all possible distinct values

of x such that the sum of the degree measures of the interior angles of some convex polygon is !x degrees.

5. The bases of a trapezoid have lengths of 13 and 34, and the non-parallel sides have lengths of

10 and 17. Find the length of the altitude of the trapezoid. 6. The roots for x of 3 2 935 847 0x kx x are three distinct positive integers. Let a be a

positive integer greater than one. Let w be the sum of all distinct possible values for a such that log (4096)a is a positive integer. Find the value of k w .

7. Let k be the length of the third side of a triangle if the angle included between the sides of

lengths 11 and 35 is 60 . Let the focus of the parabola whose equation is 2112 3

8y x x

be ,h w . Find the value of k h w .

8. Let 1 2 3 99

log( ) log( ) log( ) log( ) log( )2 3 4 1 100

nS

n

. The length k of a side of

an equilateral triangle is also a root of the quartic equation 4 2474 4840 0k k . Find the value of S k .

9. The quartic equation 4 3 2 2262 0x ax bx cx has four distinct positive integral roots for x . If a is NOT a positive prime number, find the sum of all possible values of a .

10. Let g be the number of distinct permutations of the letters in the word “geometry.” Let p

be the probability of drawing 2 hearts if 2 cards are selected (without replacement) at random from 4 hearts and 2 spades. Find the value of the product gp .

2013 SAA School ANSWERS

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

Total Score (see below*) =

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.

Answer Score(to be filled in by proctor)

1.(Must be this reduced improper fraction.)

2.

3.

4.

5.

6.

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

50497

6

34

175

8

4101

300

20

124

8064

NOTE: Questions 1-5 onlyare NO CALCULATOR

ANS

ANS

ANS

ANS

ANS

JUNIOR-SENIOR 2 PERSON COMPETITION ICTM 2013 STATE DIVISION AA LARGE PRINT QUESTION 1

1. Let w be a positive integer less than 300 such that w k k k where k is a positive integer. Find the sum of all distinct possible values of w.


2. Let 1i . Find the value of 8log 8 8i . Express your answer as an improper fraction reduced to lowest terms.


3. It is given that

1 2

( )3

xf x

and

1 3

( )2

xg x

. Find

7 8

9 10 .

f g g f

f g g f


4. Let x be a positive integer such that 1 20x . Find the sum of all possible distinct values of x such that the sum of the degree measures of the interior angles of some convex polygon is !x degrees.


5. The bases of a trapezoid have lengths of 13 and 34, and the non-parallel sides have lengths of 10 and 17.

Find the length of the altitude of the trapezoid.


6. The roots for x of

3 2 935 847 0x kx x are three distinct positive integers. Let a be a positive integer greater than one. Let w be the sum of all distinct possible values for a such that log (4096)a is a positive integer. Find the value of k w .


7. Let k be the length of the third side of a triangle if the angle included between the sides of lengths 11 and 35 is 60. Let the focus of the parabola whose equation is

2112 3

8y x x be

,h w . Find the value of

k h w .


8. Let

1 2 3log( ) log( ) log( )

2 3 499

log( ) log( )1 100

S

n

n

The length k of a side of an equilateral triangle is also a root of the quartic equation

4 2474 4840 0k k . Find the value of S k .


9. The quartic equation

4 3 2 2262 0x ax bx cx has four distinct positive integral roots for x. If a is NOT a positive prime number, find the sum of all possible values of a.


10. Let g be the number of distinct permutations of the letters in the word “geometry.” Let p be the probability of drawing 2 hearts if 2 cards are selected (without replacement) at random from 4 hearts and 2 spades. Find the value of the product gp .

FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND 1ICTM 2013 DIVISION AA STATE FINALS QUESTION 1

1. Set 68,56,56A and set 41,35,39, 49,56B . Find the average (or the

arithmetic mean) of the average of set A and the average of set B.


2. Let4

ANSk . Find the sum of the first k primes.


3. Twice the supplement of an angle is ANS larger than the complement of the

angle. Find the degree measure of the angle.


4. A triangle has vertices at 10,6 and 10, 6 , and ,0k . The triangle has an

area of ANS. Find the larger value of k . Give your answer as a reduced common orimproper fraction.


1. If 20132013 is written as an integer in standard notation, find the units digit of thatinteger.


2. An engineer said that he could finish a highway section in 3 days with his presentsupply of a certain type of machine. However, with ANS more of these machines thejob could be done in 2 days. The machines all work at the same rate. Find thenumber of days it would take to do the job with one machine.


3. Given ANS points in a plane such that no three are collinear, find the number oflines those points can determine?


4. The lateral area of Alex’s right cylindrical can of Dringles potato chips is

ANS square inches. Abby has a rectangular prism box of Lucky Nuggets cereal.

The volume of the box is 300 cubic inches. The height of the box is 15 inches. Thelength of the box is 5 inches more than the width of the box in inches. The lateralarea of the potato chip can is k times the total surface area of the cereal box. Giveyour answer for k as a decimal rounded to the nearest hundredth.


1. Merchant A and Merchant B each have an item for sale for $10,000. Merchant Aoffers three successive discounts of 20%, 20% and 10% on Monday, Tuesday, andWednesday. Merchant B offers three successive discounts of 40%, 5% and 5% onthe same three days. Find the positive difference in dollars between the two saleprices at the end of the day Wednesday.


2. A positive number is mistakenly divided by the sum of the digits in ANS insteadof correctly being multiplied by the sum of the digits in ANS . Let k be the absolutevalue of the difference between the correct answer and the incorrect answer. k is

%x of the correct answer. Find the value of x . Give your answer rounded to thenearest whole percent.


3. Circles of radius ANS and 25 are externally tangent and are circumscribed by athird circle, as show in the figure (not drawn to scale). Find the exact area of theshaded region.


4. ANS should be in the form .k Let m represent the sum of the digits in k . Thedifference in the areas of two similar triangles is m square feet, and the ratio of thelarger area to the smaller area is the square of an integer. The area of the smallertriangle, in square feet, is an integer and one of its sides is 3 feet. Find the length ofthe corresponding side of the larger triangle, in feet.


1. At his usual rate Sean rows 15 miles downstream in five hours less time than ittakes him to return. If he doubles his usual rate, the time downstream is only onehour less than the time upstream. Find the rate of the stream’s current in miles perhour. (Assume the stream flows at a constant rate.)


2. Amanda and Daoud working together can complete a job in ANS days. Daoudand Muhsen can do the same job in four days; while Amanda and Muhsen can

complete the job working together in2

25

days. In how many days can Amanda

complete the job if she works alone?


3. A cube is made by using twelve pieces of wire, each ANS inches long, for thetwelve edges of the cube. An ant starts at one vertex and then walks along the edgesand returns to its starting point without passing through any other vertex more thanonce and without retracing any edge (or part of an edge.) Find the maximum possibledistance the ant could walk under these conditions. Give your answer measured ininches.


4. The lateral surface area of a right circular cone is 10ANS square feet. If the

radius of the cone’s base is 12 feet, find the length of the altitude of the conemeasured in feet.


1. k and w are the roots of 2 0x kx w , 0k and 0w . Find the sum

k w .


2. a and b are single digit integers used as the ten’s digit in the following problem.

The three-digit number 2 3a is added to the number 327 ANS to give the three-

digit number 5 9b . If 5 9b is divisible by 9, find the value of a b .


3. Let 3.5k ANS . The area of a rectangle remains unchanged when it is made k

inches longer and2

3of an inch narrower, or when it is made k inches shorter and

4

3inch wider. Find the numerical area, in square inches, of the rectangle.


4. Two high school classes took the same test. One class of ANS students scored amean grade of 80% while another class of 30 students scored a mean grade of 70% .Find the mean grade for all students in both classes.

2013 SAA FR/SO RELAYCOMPETITION

PROCTOR ANSWER SHEET

ROUND 1

1. 52

2. 238

3. 32 (Degrees optional.)

4.46

3(Must be this reduced improper

fraction.)

ROUND 2

1. 3

2. 18 (Days optional.)

3. 153 (Lines optional.)

4. 1.38 (Must be this decimal.)

ROUND 3

1. 345 ($ optional.)

2. 99 (% optional.)

3. 4950 (Must be this exact answer.)

4. 6 (Feet optional.)

EXTRA ROUND 4

1. 2 (mph optional.)

2. 3 (Days optional.)

3. 24 (Inches optional.)

4. 16 (Feet optional.)

EXTRA ROUND 5

1. 1

2. 6

3. 20 (Square inches optional.)

4. 74 (% optional.)

JUNIOR-SENIOR RELAY COMPETITION ROUND 1ICTM 2013 DIVISION AA STATE FINALS QUESTION 1

1. Given3 4 10

2 3 9

x y

x y

, find the value of x y .


2. A right cylindrical can that has a radius of 10 centimeters and a height of 15 centimeters hasa volume that is 100 times the volume of a right cylindrical can that has a radius of ANScentimeters and a height of k centimeters. Find the value of k .


3. A stone building which is h feet tall casts a 750 foot shadow when the sun has an angle of

elevation of 35. On the top of the stone building is a vertical antenna whose height is ANS

feet. Find the distance, in feet, from the base of the building to the top of the antenna. Giveyour answer as a decimal rounded to the nearest tenth of a foot.


4. Better Sales sold a 40-inch HD TV for ANS dollars final total charge. The TV was on salefor 30% off the original price and the salesman gave an additional discount of 20% off thesale price, but then had to add the 7.5% Illinois sales tax charge on this agreed price for theTV. Find the original price of the TV. Give your answer rounded to the nearest cent innormal decimal dollar and cents notation.


1. The arithmetic mean of Julio’s three Algebra II test scores is 82. If Julio earned a 78 and 88on the first two tests respectively, find his score on the third test.


2. Vincent mixed 16 grams of a 50% hydrochloric acid solution with 20 grams of a %ANShydrochloric acid solution. How many grams of pure hydrochloric acid did Vincent’s finalmixture contain?


3. 2

log log 3x

ANS xANS

and 0x . Find the value of x that makes this equation

true. Give your answer rounded to the nearest integer.


4. In a right triangle one leg is 7 inches. The degree measure of the acute angle of the triangle

adjacent to the 7 inch side is24

tan7

Arc

. A circle is inscribed in the right triangle. Find

the numerical area, in square inches, of a sector of this circle which has a central angle of

ANS . Give your answer as a decimal rounded to the nearest hundredth.


1. If the pre-image line whose equation is 2 3 5x y is reflected across the line y x , find the

slope of the resulting image line? Give your answer as a reduced common or improperfraction.


2.a

ANSb

and k a b . If 23 12y x x k , find the minimum value of y .


3. For the graph of

3 2

2

6 11 19 6

2 25 13

x x xy

x x x ANS

, each vertical asymptote has an equation in

the form x a and each horizontal asymptote has an equation in the form y b . Find the

sum of all possible (not necessarily distinct) value(s) for a and b .


4. Use real number (radian) measures in this problem. Find the sum of the period, phase shift,

and amplitude of the graph of the function 5

6cos 103

y ANS x

. Give your

answer as a decimal rounded to the nearest thousandth.


1. Find the largest negative integer in the domain of the function 2( ) 3 7 6.f x x x


2. A circular stained glass window is divided into 5 sectors (not necessarily equal). The radius

of the window is ANS feet. The two yellow sectors have a total area of21

4

square feet.

Find the sum total of the degree measure of the central angles of these two sectors.


3. On the same day, Maria invested ANS dollars at 1.2% per year compounded continuouslyfor 10 years at Bank A and invested $250 at 1.1% per year compounded quarterly for 10years at Bank B. At the end of the 10 years, she cashed in each investment separately. Findthe total amount, rounded to the nearest cent, Maria received when she cashed in her twoinvestments. Give your answer in normal dollar and cent decimal notation. (Note: Banksround down to the nearest cent when determining the cash value of an investment.)


4. Let 100k ANS . The area of a residential lot is k square feet. The lot is shaped like a

trapezoid with a height of 160 feet. The longer parallel side is 52 feet longer than the shorterparallel side. Find the length, rounded to the nearest foot, of the longer parallel side.


1. Find the value of x that is a solution for 4

2 42 33 9 9 27x x

.


2. Points 3,5 , 1, ANS , and 6, k are collinear. Find the value of k . Give your answer as

a reduced common or improper fraction.


3. Find the integral value of x that is a solution for 16 16log 2 26 log 2x x ANS .


4. How many integral solutions does 5 17ANS x have?

2013 SAA JR/SR RELAYCOMPETITION

PROCTOR ANSWER SHEET

ROUND 1

1. 1

2. 15 (Centimeters optional.)

3. 540.2 (Must be this decimal, feet optional.)

4. 897.34 (Must be this decimal, $ or dollarsoptional.)

ROUND 2

1. 80

2. 24 (Grams optional.)

3. 155 (Must be this integer.)

4. 12.17 (Must be this decimal, square inchesoptional.)

ROUND 3

1.3

2(Must be this reduced improper fraction.)

2. 7

3. 9

4. 6.116 (Must be this decimal.)

EXTRA ROUND 4

1. 3

2. 210 (Degrees optional.)

3. 515.79 (Must be this decimal, $ or dollarsoptional.)

4. 348 (Must be this integer, feet optional.)

EXTRA ROUND 5

1. 10

2.5

4(Must be this reduced improper fraction.)

3. 3

4. 11 (Integers or solutions optional.)

ORAL COMPETITIONICTM STATE 2013 DIVISION AA

ICTM 2013 State Oral Division AA

1. Let O, A and B have coordinates (0, 0, 0), (2, 1, 0) and (-1, 3, 1), respectively.If the point (4, -5, r) is on the plane determined by O, A and B, find the value of r.

2. On page 169 of the reference we read:

Conclusion. If X is any point of the interior or boundary of triangle ABC, then

there are scalars r, s and t so that X = rA + sB + tC in which

r + s + t =1,

0 £ r £1,

0 £ s £1,

0 £ t £1.

ì

í

ïï

î

ïï

(i) What special point corresponds to the values r = s = t =1

3?

(ii) Give an analogous vector representation of the set of points in three-spacethat are on the boundary of or interior to tetrahedron ABCD. Also tellwhat restrictions on the coefficients apply to the faces of the tetrahedron.

3. The proof of Desargues’ theorem given on pages 183-184 of the reference makesuse of vectors and is valid in both two-space and three-space under theassumption that every two coplanar lines meet. The theorem is unusual in that asimpler proof can be given in the (typically more complicated) three-space case,as follows.

Assume now that triangles ABC and ¢A ¢B ¢C in Figure 94 (reference, page 181) arenot coplanar and that the sets {S, A, ¢A }, {S, B, ¢B } and {S,C, ¢C } are collinear

triples, as are the triples {A, B, P}, { ¢A , ¢B , P}, {B,C,Q}, { ¢B , ¢C ,Q}, {A,C, R} and

{ ¢A , ¢C , R}.Give an argument, not using vectors, that P, Q and R are collinear.



EXTEMPORANEOUS QUESTIONS

Judges: Give this sheet to the students at the beginning of the extemporaneousquestion period.

STUDENTS: You will have a maximum of 3 minutes TOTAL to solve and presentyour solution to these problems. Either or both the presenter and the oral assistantmay present the solutions.

1. Let A

, B

and C

be non-zero, non-coplanar vectors in three-space.

Consider the statement: A

· B

( ) ´ C

= A

· B

´ C

( )Under the specified conditions is this statement always, sometimes or never true?

2. If three points are given, as shown, in the xy-plane, the area of triangle P1P2P3 is

given by the formula Area =1

2·abs

1 x1 y1

1 x2 y2

1 x3 y3

æ

è

ççç

ö

ø

÷÷÷

(i) Without giving all of the algebraic details, explain how vectors can beused to arrive at this formula.

(ii) Is the “abs” function necessary? Explain.

(iii) If the value of the determinant is zero, what can you conclude?



SOLUTIONS FOR JUDGES

1. Let O, A and B have coordinates (0, 0, 0), (2, 1, 0) and (-1, 3, 1), respectively.If the point (4, -5, r) is on the plane determined by O, A and B, find the value of r.

SOLUTION

The vector OA ´ OB is perpendicular to the plane, so an equation of the plane canbe found as follows.

OA ´ OB =

i j k

2 1 0

-1 3 1

= i - 2 j + 7k.

An equation of the plane is 1(x - 0)- 2(y - 0)+ 7(z - 0) = 0 . This uses thecoordinates of O, but those of A or B would also work since all three yield thesame equation x - 2y + 7z = 0.Substitute 4, -5 and r to get 14 + 7r = 0, so r = -2.




2. On page 169 of the reference we read:

Conclusion. If X is any point of the interior or boundary of triangle ABC, then

there are scalars r, s and t so that X = rA + sB + tC in which

r + s + t =1,

0 £ r £1,

0 £ s £1,

0 £ t £1.

ì

í

ïï

î

ïï

(i) What special point corresponds to the values r = s = t =1

3?

(ii) Give an analogous vector representation of the set of points in three-spacethat are on the boundary of or interior to tetrahedron ABCD. Also tellwhat restrictions on the coefficients apply to the faces of the tetrahedron.

SOLUTION

(i) A careful sketch shows that the centroid of the triangle corresponds to thethree values all being 1/3. This is easiest to see if the free point O is takento be the centroid.

(ii) The points X on the boundary of or interior to the tetrahedron ABCD are

described by X

= rA

+ sB

+ tC

+ wD

, where

r + s + t + w =1,

0 £ r £1,

0 £ s £1,

0 £ t £1,

0 £ w £1.

ì

í

ïïï

î

ïïï

Face ABC corresponds to w = 0, and so on.




3. The proof of Desargues’ theorem given on pages 183-184 of the reference makesuse of vectors and is valid in both two-space and three-space under theassumption that every two coplanar lines meet. The theorem is unusual in that asimpler proof can be given in the (typically more complicated) three-space case,as follows.

Assume now that triangles ABC and ¢A ¢B ¢C in Figure 94 (reference, page 181) arenot coplanar and that the sets {S, A, ¢A }, {S, B, ¢B } and {S,C, ¢C } are collinear

triples, as are the triples {A, B, P}, { ¢A , ¢B , P}, {B,C,Q}, { ¢B , ¢C ,Q}, {A,C, R} and

{ ¢A , ¢C , R}.Give an argument, not using vectors, that P, Q and R are collinear.

SOLUTION

P lies on both planes ABC and ¢A ¢B ¢C since it lies on both lines AB

and ¢A ¢B

.

Q lies on both planes ABC and ¢A ¢B ¢C since it lies on both lines BC

and ¢B ¢C

.

R lies on both planes ABC and ¢A ¢B ¢C since it lies on both lines AC

and ¢A ¢C

.Since planes ABC and ¢A ¢B ¢C meet in a line, P, Q and R are collinear on this line.





1. Let A

, B

and C

be non-zero, non-coplanar vectors in three-space.

Consider the statement: A

· B

( ) ´ C

= A

· B

´ C

( )Under the specified conditions is this statement always, sometimes or never true?

SOLUTION

It is never true since the symbol on the left side is nonsense.The expression in parentheses is a scalar, and the cross-product operation requirestwo vectors in three-space.





2. If three points are given, as shown, in the xy-plane, the area of triangle P1P2P3 is

given by the formula Area =1

2·abs

1 x1 y1

1 x2 y2

1 x3 y3

æ

è

ççç

ö

ø

÷÷÷.




(i) Without giving all of the algebraic details, explain how vectors can beused to arrive at this formula.

(ii) Is the “abs” function necessary? Explain.

(iii) If the value of the determinant is zero, what can you conclude?

SOLUTION

(i) Find the cross-product of two vectors, say P1P2 and P1P3.The length of this cross-product vector is the area of the parallelogram ofwhich the given triangle is half. This length, which involves thesubscripted coordinates, is algebraically messy, but it apparently can bemade equivalent to the equally messy determinant expansion.This area formula is easier to verify than to derive.

(ii) The sign of the determinant depends on the order in which the coordinatesare placed in the determinant. Thus the value can be positive or negative.We need the absolute value to make the area come out positive.

(iii) The value of the determinant is zero if and only if the three points arecollinear.

2013 ictm saa algebra i - psd202.org

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