WRITTEN AREA COMPETITION ICTM STATE 2015 DIVISION A PAGE 1 OF 3. of the number, determine that number. 3 x 2x

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1 WRITTEN AREA COMPETITION ALGEBRA I ICTM STATE 015 DIVISION A PAGE 1 OF 3 1. The operation is defined as 11 k 015. a b 3a b. Determine the value of k such that. Mary can work 10 similar problems in 15 minutes. Matt can work 6 of the same problems in 15 minutes. Determine the number of minutes it will take them working together to work 015 of these similar problems. Express your answer as an improper fraction reduced to lowest terms. 3. If 3 4 more than 3 4 of a number is equal to the opposite of 3 4 of the number, determine that number. 4. Determine the vertex of the parabola given by ordered pair k, w. y x x. Express your answer as the x y z x y z x y z k w p. k w p where k, w, and p are integers. Determine the sum 6. Determine the sum of the solution(s) for the equation your answer as a common or improper fraction reduced to lowest terms. 3 x x. Express x 9 x 5x 6 7. For all real values of x, a k c. a x 3 x 4 c 4x 1 kx 101. Determine the sum

2 WRITTEN AREA COMPETITION ALGEBRA I ICTM STATE 015 DIVISION A PAGE OF 3 8. Determine the positive difference of the solutions for x in the equation x x 9. In a group of 100 adults, 18 have neither a Visa or Master Card credit card. 5 of these adults have Visa cards and 37 have Master Cards. One adult is selected at random. The probability this person has a Visa and a Master Card is k %. Determine the value of k. Report your answer as k only. 10. The equation of the line perpendicular to the line 15x y 10 0 passing through the point 15,6.5 can be written as Ax By C 0 where A, B, and C are relatively prime integers with 0 A. Determine the sum A B C. 11. Determine the smallest positive integer that gives the same non-zero remainder when divided by each of the elements from the set,3, 4,5, 6, 7,8,9. 1. All ages are whole numbers of years. Tom, Dick, and Harry s ages add to 91. Harry is one year younger than Dick and Tom s ages combined. Dick is one year more than two times Tom s age. Determine Harry s whole number of years age. 13. Determine the sum of the roots for the equation 8x 5x 63 x k w 3 3 where w is a positive prime integer. Determine the sum k w.

3 WRITTEN AREA COMPETITION ALGEBRA I ICTM STATE 015 DIVISION A PAGE 3 OF Xavier and Evie start at noon at points 60 kilometers apart on a trail riding mopeds directly towards each other, and meet at 1:30 pm. Xavier s speed was 4 km/hr greater than Evie s speed. Determine Xavier s speed in km/hr. 16. The positive integer x is the product of 5, k, and w with w 0. k is a positive two-digit integer such that the tens digit is two more than the ones digit. w is the positive two-digit integer whose tens digit is the same as the ones digit of k and whose ones digit is the same as the tens digit of k. Determine the value of the integer x when w is as small as possible. 17. Determine the value of the expression a b b c when a 6, b, and c When x 3 7x 4x k value of k. is divided by x, the remainder is 15. Determine the exact 19. Determine all value(s) for x such that 3x Determine the sum of all integral solutions for the equation x 8 1.

4 015 SA Name ANSWERS Algebra I Correct X pts. ea. = School (Use full school name no abbreviations) Note: All answers must be written legibly in simplest form, according to the specifications stated in the Contest Manual. Exact answers are to be given unless otherwise specified in the question. No units of measurement are required (Must be this reduced improper fraction, minutes optional.) OR 0.5 OR (Must be this exact ordered pair.) ( years optional.) ( km/hr optional) 10 OR 9 1 OR.5 1, (Must be this reduced Improper fraction.) (Must be this integer only, with no % used.) OR OR OR ITEM ANALYSIS Div 1A Ques % correct Div A Ques % correct

5 WRITTEN AREA COMPETITION GEOMETRY ICTM STATE 015 DIVISION A PAGE 1 OF 3 1. Determine the exact perimeter of an isosceles right triangle with a hypotenuse of length 6.. Circle O is externally tangent to circle P with respective radii of lengths 4 and 8. A common tangent line is tangent to circle O at point A and tangent to circle P at point B. Determine the exact length of AB. 3. Lines l and m are parallel in the figure shown. K 50 and P 80. Determine the degree measure of B. l P B 4. In the diagram shown, AE and AD are secant segments drawn to the circle. AB 5, CE 4, and BD 3. Determine the exact length of AE. E m C B K A 5. In the diagram, BAC is a right angle. From point A, an ant C travels to point D on the hypotenuse of ABC along AD, the altitude drawn to BC. When it reaches point D, the ant turns and heads toward side AC along DE, the altitude to the hypotenuse of F G ADC. When it reaches point E on side AC, the ant turns and heads toward BC again along EF, the altitude to the hypotenuse of D E DEC. When it reaches point F on side BC, it turns for the last time and heads toward side AC along FG, the altitude to the B A hypotenuse of EFC. When it reaches point G on side AC, the ant stops. Assume the ant travels in straight lines and does not retrace its steps. Determine the total distance the ant traveled if AB 1 and AC 16. Express your answer as an improper fraction reduced to lowest terms. D 6. A certain rectangle has length twice its width. Determine the exact area of a circle circumscribed about this rectangle when the width of the rectangle is 8 units. 7. ABCD is a rhombus with BAD 60. Let E be a point above the plane of the rhombus such that CE is perpendicular to both CB and CD. BD 1 and BE. Determine the exact length of AE. B E C A D

6 WRITTEN AREA COMPETITION GEOMETRY ICTM STATE 015 DIVISION A PAGE OF 3 8. A right equilateral triangular prism and a right circular cylinder have the same volume and the same height. If the radius of the cylinder is 5, determine the length of each side of the equilateral triangle base. Express your answer as a decimal rounded to four significant digits. 9. AB is tangent to circle O at point B. The area of ABO is 4 square units and AB 8. Determine the exact numeric area of circle O. B D 8 A O 10. A 10 foot wide alley is level and has parallel vertical walls on either side. Two adjacent poles lean against opposite walls, one on each wall, with their bases against the foot of the other wall. One pole reaches to a vertical height of 30 feet on its wall, and the other reaches 0 feet up its wall. The sides of the poles touch each other. Determine the vertical height, in feet, above the alley where these poles touch. 11. Let P, Q, R, and T be coplanar points such that Q lies on TR and 51 PRQ TPQ TRP. If TR 17 3 and PQ, then the sum of the two possible 4 lengths of TP can be expressed in the simplified and reduced form k w p where k, w, f p, and 0 f are integers. Determine the sum k w p f. 1. Let A, B, C, D, E, and F be points equally spaced around a circle. GF is tangent to the circle at point F. Determine the degree measure of GFA. G F A B C 13. A right circular cylinder with a lower base circle of radius 4 is truncated at an angle to the vertical axis so that the top base is an ellipse. The longest height of this cylinder is 9 while the shortest height is 7. Determine the exact numerical volume of this truncated cylinder. E D

7 WRITTEN AREA COMPETITION GEOMETRY ICTM STATE 015 DIVISION A PAGE 3 OF ABC is inscribed in a circle. AB 16, BC 30, and AC 34. A point is chosen at random from inside the circle. Determine the probability that point is also in the interior of the triangle. Express your answer as a decimal rounded to four significant digits. 15. Given the diagram with angle measures as shown. Determine the degree measure of obtuse angle DBC. D 3x In the rectangular solid shown, not drawn to scale, CG 8, GH 1, and CB 9. Determine the exact length of segment BH A E x B H D 5-x F G C C A B 17. A regular hexagon with sides of length 8 has both a circumscribed circle about the hexagon and an inscribed circle within the hexagon. Determine the ratio of the area of the inscribed circle to the area of the circumscribed circle. Express your answer as k : w where k and w are relatively prime positive integers. 18. Two congruent squares with the same center are positioned so the diagonal of one of the squares is rotated 45 from the other, producing shaded triangles with equal areas.. The numeric area of one of these triangles is. Determine the exact numeric area of the octagonal region that remains unshaded inside the squares. P 19. In PQR, median QT and median PS are concurrent at point C. PC 4x 6 and CS x. Determine the numeric length of PS. Q C S T R 0. Determine the exact numeric length of the altitude to the hypotenuse of a right triangle with legs of 8 and 8 3.

8 015 SA Name ANSWERS Geometry Correct X pts. ea. = School (Use full school name no abbreviations) Note: All answers must be written legibly in simplest form, according to the specifications stated in the Contest Manual. Exact answers are to be given unless otherwise specified in the question. No units of measurement are required. 6 6 OR 61 (Must be this exact answer.) OR (Must be this ("degrees" or exact answer.) " " optional.) ("degrees" or " " optional.) OR OR 1 11 OR 11 1 OR 11 (Must be this exact answer.) (Must be this exact answer, "cubic units" optional.) (Must be this exact decimal.) (Must be this reduced ("degrees" or improper fraction.) " " optional.) (Must be this exact answer, "sq. units" optional.) (Must be this exact answer, "units" optional.) (Must be this exact decimal.) (Must be this exact answer, "sq. units" optional.) ("ft." or "feet" optional.) OR : 4 (Must be this exact ratio using this notation.) (Must be this exact answer.) ITEM ANALYSIS Div 1A Ques Ques % correct Div A % correct

9 WRITTEN AREA COMPETITION ALGEBRA II ICTM STATE 015 DIVISION A PAGE 1 of 3 1. Determine the value of the determinant Determine the median of the zeros of 3 f x x 10x 9x Determine the solutions for the inequality 5x 7x 4. Express your answer in interval notation. 4. Determine the ordered pair solution x, y answer as an ordered pair x, y x y 8 64 for the system y x 1. Express your using integers or common or improper fractions for entries. k p w 5. When simplified, x reduces to x where k, w, p, and q q are integers with q positive. Determine the value of k w p q. 6. The second term of a geometric sequence is 0.5 and the 13th term is 104. The last term of this finite sequence is 819. Determine the sum of the terms of this sequence. Express your answer as an exact decimal.

10 WRITTEN AREA COMPETITION ALGEBRA II ICTM STATE 015 DIVISION A PAGE of 3 3i 8 k wi 7. i 1. where k, w, and p are relatively prime integers and p 0. 7i 5 p Determine the sum k w p. 8. The Octad family of 8 persons walked in a parade with 30 other walkers. Before the parade, all of the walkers shook hands with each other exactly once. The percentage of handshakes among just the Octad family out of the total number of handshakes is k %. Determine the value of k. Express your answer as a decimal rounded to four significant digits. Do not include the percent symbol % in your answer. 9. Determine all solutions for the equation x 4 x 1 when solved over the set of Complex Numbers. 10. A 10 gallon container is filled with a solution that is 0% alcohol and 80% water. Determine the number of gallons that must be drained from this container and replaced with pure alcohol so that the container is filled with a solution that is 50% alcohol. Express your answer as a common or improper fraction reduced to lowest terms. 11. Determine the possible value(s) of k such that when x 5x 7 remainder is 3. is divided by x k, the 1. During a school car wash, it was determined that Evie could wash a car in 8 minutes and Aune could wash the same type car in 6 minutes. At these rates, determine the number of minutes it would take Evie and Aune working together to wash 7 of these type cars i. a, b, c, and d are real valued coefficients. i and i 4 3 x ax bx cx d 0. Determine the value of d. are two roots for

11 WRITTEN AREA COMPETITION ALGEBRA II ICTM STATE 015 DIVISION A PAGE 3 of f x x 8 and g x x 5. Determine the value of g f g Let A Determine a,1, the,1 entry in matrix A f x x kx wx p, where k, w, and p represent real numbers. One zero 3 i. of f x is i. k w 39. Determine the real zero of f x. 17. On a nine-player softball team, the pitcher must bat first and the shortstop must bat sixth. Determine the number of possible nine-player batting orders for this team. 18. Determine the sum of all possible integer solution(s) for 4x x 6 6 x 6 8. Determine the product of all possible solution(s) for this equation. 0. Radar shows a thunderstorm at a distance of 15 miles. You see a bolt of lightning followed by the clap of thunder 30 seconds later. Fifteen minutes later you again see a bolt of lightning followed by the clap of thunder, only this time 15 seconds apart. Assume your distance from the storm is directly proportional to the time gap between lightning and thunder and that the storm is moving on a straight line path from lightning strike to lightning strike towards you. Determine the rate at which the storm is approaching you in miles per hour.

12 015 SA Name ANSWERS Algebra II Correct X pts. ea. = School (Use full school name no abbreviations) Note: All answers must be written legibly in simplest form, according to the specifications stated in the Contest Manual. Exact answers are to be given unless otherwise specified in the question. No units of measurement are required. (Must have both answers, either order.) (Must be in interval notation.) (Must be this ordered pair using improper fractions.) (Must be this decimal, comma use optional.) (Must be this decimal, no % sign.) (Must be this reduced improper fraction, "gal." or "gallons" optional.) , ("minutes" or "min." optional.), , (Must be this value only.) (Must have all four answers,, any order, may use or 0's, i.e. 0i i 3, i 3.) ("mph" or "miles per hour" optional. ITEM ANALYSIS Ques Div A Ques Div 1A % correct % correct

13 WRITTEN AREA COMPETITION PRECALCULUS ICTM STATE 015 DIVISION A PAGE 1 OF 3 1 cos The period of the graph of y x is k. Determine the exact value of k.. Determine the focus of the graph of the parabola y 3 4 x 7 as an ordered pair x, y.. Express your answer 3. At Merry Math Academy, there are 0 juniors. Of these juniors, 140 are enrolled in Pre- Calculus, 110 are enrolled in Chemistry, and 80 are enrolled in both courses. Determine the probability that a junior enrolled in Chemistry is also enrolled in Pre-Calculus. Express your answer as a common fraction reduced to lowest terms. 4. Points 1,8 P and Q,1. of the vector 3 PQ are used to form a vector PQ. Determine the exact magnitude 5. Determine the coefficient of the term in the expansion of x 1 that contains 4 x. 6. Determine the value of x when 9 1log log x The graph of y f x is shown. Determine the coordinates of the absolute (global) minimum of the graph of y f x 1 5. Express your answer as an ordered pair x, y. y-axis (-1,5) (3,4) (-5,3) (-4,0) (-,0) (,0) x-axis (-3,-1) (1,-4)

14 WRITTEN AREA COMPETITION PRECALCULUS ICTM STATE 015 DIVISION A PAGE OF 3 8. The magnitude of vector u 5, k r is 13 and the directional angle of vector v r 3, 4 is 30. Determine the largest value of k. i. When converted to standard form, cos 60 isin (sometimes written as cis60 ) may be written in reduced and simplified radical form as a bi with a and b real numbers. Determine the exact values of a and b. Express your answer as the ordered pair a, b. 10. f x 15x 7x and g x x 3. Determine the value of 1 f g. 11. Determine the value of, , when 9sin 3cos x 8x 18x kx w is divisible by x x 3 for all valid replacements for x. Determine the ordered pair k, w. 13. Two ships leave the same port at the same time. The first, the Minnow travels at 15 knots on a bearing of 0. The second, the Guppy, travels on a bearing of 80. Exactly three hours later, the navigator of the Minnow sights the Guppy at a bearing of 60. Determine the rate of the Guppy in knots, rounded to four significant digits. 14. Triangle ABC is isosceles with AB AC. D lies on AB, and E lies on AC. CAB 0, EBC 50, and DCB 60. Determine the degree measure of EDC. D A E B C

15 WRITTEN AREA COMPETITION PRECALCULUS ICTM STATE 015 DIVISION A PAGE 3 OF Determine the positive degree measure of the acute angle between the lines y 3x 1 and y 5 x. 16. The graph of y 3x 5x 5x kx 8 has a removable discontinuity ("hole") at x. Determine the y-coordinate of this point of discontinuity. Express your answer as a common or improper fraction reduced to lowest terms. 17. Vector v 3i j and vector w i 4 j where i and j are the standard unit vectors. Determine the dot (or inner) product v w. 18. The circle x y r is tangent to the graph of x y 1. Determine the exact numeric radius length r. 19. The rational function f x x 7x 1 has an oblique ("slant") asymptote y kx w. 7x 63 Determine the exact sum k w. Express your answer as a common or improper fraction reduced to lowest terms. sin x sin x 1 cos x 0. Determine the exact value for tan x when. Express your answer as a common or improper fraction reduced to lowest terms.

16 015 SA Name ANSWERS Pre-Calculus Correct X pts. ea. = School (Use full school name no abbreviations) Note: All answers must be written legibly in simplest form, according to the specifications stated in the Contest Manual. Exact answers are to be given unless otherwise specified in the question. No units of measurement are required. 1 4 OR 0.5 OR ("degrees" or optional.) (Must be this (Must be this 8,3 ordered pair.) ordered pair.) 19,1. 1. (Must be this reduced common fraction.) (Must be this exact answer.) (Comma optional.) (Must be this ordered pair.) , ,70 4 0, (Must be this exact ordered pair.) (Must be this decimal, "knots" optional.) ("degrees or optional.) ("degrees" or optional.) (Must be this reduced improper fraction.) (Must be this exact answer.) (Must be this reduced improper fraction.) (Must be this reduced common fraction.) OR OR 1 4 OR 0.5 OR OR 4 9 ITEM ANALYSIS Div 1A Ques Ques % correct Div A % correct

17 FROSH-SOPH EIGHT PERSON TEAM COMPETITION ICTM STATE 015 DIVISION A PAGE 1 of 3 NO CALCULATORS 1. When simplified completely, the algebraic expression a b a b written as ka wb p. Determine the sum k w p can be. The radius of the circle inscribed in the triangle is 5. B 90 and AC 30. Determine the exact perimeter of ABC. A 3. If b a b a and a b b a, determine the value of B C 4. Determine the positive difference between x and the reciprocal of x when 5x 57x 5 0. Express your answer as a common or improper fraction reduced to lowest terms. 5. ABCD is a parallelogram as marked with x EC and y FC. Determine the ratio of x to y. Express your answer in the ratio form k : w where k and w are relatively prime positive integers. B 10 A E 1 x 15 C y F D 6. A certain convex polygon has the same number of sides as it does diagonals. Determine the number of sides for this polygon. 7. In the diagram as marked, a b. x and y are complementary angles. Determine the value of y x y a b Determine the sum of the solutions for x when 7x x x 1 x 3x 65x. Express your answer as a common or improper fraction reduced to lowest terms. NO CALCULATORS NO CALCULATORS NO CALCULATORS

18 FROSH-SOPH EIGHT PERSON TEAM COMPETITION ICTM STATE 015 DIVISION A PAGE of 3 NO CALCULATORS 9. Each integer from 1 to 1000, inclusive, is printed on one of 1000 ping pong balls and placed in an urn. The balls are mixed and one ball is drawn from this urn. Determine the probability that the integer drawn is a multiple of or 5. Express your answer as a common fraction reduced to lowest terms. 10. X and Y are the midpoints of their respective sides in ABC as shown. CB 4, XB 1 and CY 18. Determine the exact perimeter of XYQ. X A Y C Q B 11. All ages are in whole numbers of years. Sixty-three years ago, Dave was twice as old as Daisy was. Thirty-eight years ago, Tom was one tenth of Daisy's age at that time. Dave is thirty-two years older than Tom. Determine the sum of Dave and Daisy's current ages. 1. i 1. When completely simplified, the solutions for x in the equation x 5x 8 can be written as k wi p f k w p f. using positive integers k, w, p, and f. Determine the sum 13. In the diagram, PT is tangent to circle O at T and PY is a secant segment. PL 5, PT 10, CA 3, and AY 9. LA k and AT w. Determine the exact sum k w. L C A O Y P T 14. For every 00 pounds of weight added to a car, the shock absorbers compress (go down) 1 centimeter (cm). When Jason and John get into the car at the same time, the shock absorbers go down 1.5 cm. When John and Michael get into the car at the same time, the shock absorbers go down cm. When all three get into the car at the same time, the shock absorbers go down.5 cm. Determine the exact weight of Michael in pounds. NO CALCULATORS NO CALCULATORS NO CALCULATORS

19 FROSH-SOPH EIGHT PERSON TEAM COMPETITION ICTM STATE 015 DIVISION A PAGE 3 of 3 NO CALCULATORS 15. A series of m n "checkerboards" where n m 1 are made by arranging adjacent congruent squares m rows tall and n columns wide. All the squares except those that form the outside edges or borders are then shaded. The first board is a 3 arrangement and has no shaded squares. The second board is a 3 4 arrangement and has interior shaded squares. The third board is a 45 arrangement and has 6 interior shaded squares. Determine the number of interior shaded squares in the 100th board in this sequence. 16. In the diagram as shown (but not drawn to scale), BD 13, BC 5, and ED 0. Determine the numerical area of quadrilateral ABCE. A E D C B 17. Convex polygon ABCDEFG is graphed in the coordinate plane. The vertices are A 0,0, B 0,15, C 1,0, D 8,15, E 10,1, F 9,8, and G 5,. Determine the exact area of polygon ABCDEFG. 18. Determine the sum of the solution(s) for x If k and w are positive integer number bases such that 56k 65 possible value of k w. w, determine the least 0. Point P1, 4 and Q, determine PQ. The equation for the line that is the perpendicular bisector of PQ can be written as Ax By C 0 where A, B, and C are relatively prime integers with 0 A. Determine the sum A B C. NO CALCULATORS NO CALCULATORS NO CALCULATORS

20 015 SA School ANSWERS Fr/So 8 Person (Use full school name no abbreviations) Correct X 5 pts. ea. = Note: All answers must be written legibly in simplest form, according to the specifications stated in the Contest Manual. Exact answers are to be given unless otherwise specified in the question. No units of measurement are required. ("years" or "yrs." optional) (Must be this reduced improper fraction.) (Must be this ratio in this format.) ("degrees" or " " optional.) OR : (Must be this reduced common fraction.) (Must be this reduced common fraction.) ("pounds" or "lbs." optional.) ("boards" and comma usage optional.) ("square units" or "sq. un." optional.) ("square units" or "sq. un." optional.) 4 ITEM ANALYSIS Div 1A Ques Ques % correct Div A % correct

21 JUNIOR-SENIOR EIGHT PERSON TEAM COMPETITION ICTM STATE 015 DIVISION A PAGE 1 OF 3 NO CALCULATORS 1. Determine the value of k in the matrix equation 1 3 k The sum of two numbers is 5 and the product of these two numbers is 1. Determine the sum of the squares of these two numbers. 3. Determine the number of people who must be present in a large auditorium to guarantee that more than 10 people have the same birthday in any given year. 4. Determine the sum of the negative zeros of the function f x x x 6x 6x 5x i. Exactly zeros for the function f x 8x 3 1 x 3 x x 5. 1 have imaginary components. The others are pure real numbers. These two roots can be written in the simplified and reduced form k wi with real valued k, and real values w 0, and p 0. p Determine the exact ordered triple k, w, p. Express your answer as this ordered triple. 6. At a certain college, 45% of the students are women and 55% are men. 35% of all students are over 5 years of age. 40% of the women are over 5 years of age. The probability that a random student chosen from this college is a woman or someone over the age of 5 is k %. Determine the value of k. Report your answer as k only with no percent sign. 7. A bag contains 10 balls. On each ball is written a different integer between 1 and 10, inclusive. Three balls are drawn from the bag without replacement. Determine the probability that these three balls will have either three consecutive even integers or three consecutive odd integers written on them. Express your answer as a common fraction reduced to lowest terms. NO CALCULATORS NO CALCULATORS NO CALCULATORS

22 JUNIOR-SENIOR EIGHT PERSON TEAM COMPETITION ICTM STATE 015 DIVISION A PAGE OF 3 NO CALCULATORS 8. An integer has four digits when written in base seven and three digits when written in base eight. Let the smallest such integer, when written in base ten, be k and the largest such integer, when written in base ten, be w. Determine the value of k w eleven. Report as your answer the integer only. You need not include the subscript "eleven". 9. k and w are positive integer number bases with 4 the difference between k and w. 0.6k 0.46w k w.. Determine the sum 10. Point P is located on the conic represented by 14x 36y 16. Determine the sum of the distances from point P to each of the foci. 11. A right circular cylinder has radius k and height w. Determine all ordered pair(s) of integral values for k and w such that the numeric volume of this cylinder is equal to the numeric total surface area of this cylinder. Express your answer(s) as ordered pairs k, w. 15. n 4 1. Determine the value of the product log n x 4x 1 k wx p 3 x x x 1 x 1 x 1. Determine the exact sum k w p 14. Determine the exact value of sin when sin and cos 0. 5 NO CALCULATORS NO CALCULATORS NO CALCULATORS

23 JUNIOR-SENIOR EIGHT PERSON TEAM COMPETITION ICTM STATE 015 DIVISION A PAGE 3 OF 3 NO CALCULATORS 15. In the diagram shown, B lies on AC, AB 1, BC 1, ADB 15, and BDC 30. In simplest radical form, AD k w f where k, w, and f are positive integers. Determine the sum k w f. D A B C 16. Line is perpendicular to the line x 5y 7 Determine the exact distance from point P3, 4 to line.,8. and passes through the point 17. Determine the least value for x such that x 3 x 1 1 x is true. 18. Determine the number of distinct arrangements of all eight letters of the word "MATHLETE" if each vowel must be directly between two consonants. 19. The sum of the distinct exact real roots for cos x 1 sin x, when solved over the interval 0 x, is k. Determine the exact value for k. Express your answer as a common or improper fraction reduced to lowest terms. 0. The determinant k. Determine the value of k. NO CALCULATORS NO CALCULATORS NO CALCULATORS

24 015 SA Name ANSWERS Pre-Calculus Correct X pts. ea. = School (Use full school name no abbreviations) Note: All answers must be written legibly in simplest form, according to the specifications stated in the Contest Manual. Exact answers are to be given unless otherwise specified in the question. No units of measurement are required. 1 4 OR 0.5 OR ("degrees" or optional.) (Must be this (Must be this 8,3 ordered pair.) ordered pair.) 19,1. 1. (Must be this reduced common fraction.) (Must be this exact answer.) (Comma optional.) (Must be this ordered pair.) , ,70 4 0, (Must be this exact ordered pair.) (Must be this decimal, "knots" optional.) ("degrees or optional.) ("degrees" or optional.) (Must be this reduced improper fraction.) (Must be this exact answer.) (Must be this reduced improper fraction.) (Must be this reduced common fraction.) OR OR 1 4 OR 0.5 OR OR 4 9 ITEM ANALYSIS Div 1A Ques Ques % correct Div A % correct

25 CALCULATING TEAM COMPETITION ICTM STATE 015 DIVISION A PAGE 1 of 3 Round answers to four significant digits and write in standard notation unless otherwise specified in the question. Except where noted, angles are in radians. No units of measurement are required. (NOTE: DO NOT USE SCIENTIFIC NOTATION UNLESS SPECIFIED IN THE QUESTION) k 1. e 1000 and w. Determine the difference k w.. Point Pk, w lies on AB with A117.4, 38.5 and 986.7, the value of the sum k w. B. AP 4 PB 9. Determine 3. An unfair coin lands on either heads or tails. The probability that it lands on heads is k where k 0.5. The probability of getting the result of exactly heads in five tosses is 0.3. Determine the value of k Let matrices A 7 4, B 5 8, and C 3. Matrix X is such that A X B C. Determine the sum of the elements in matrix X. Express your answer as an exact decimal. 5. Determine the shortest distance from the point 5,4 to the graph of 3 y x 3x. 6. Determine the exact value of the determinant for the matrix Quadrilateral ABCD has vertices with coordinates A 1.,.3, B 9.6,3.8, 7.9, 7.8 and D 4.8, 6.1. Determine the area of quadrilateral ABCD. C, 8. Let x tx and let t be a randomly chosen real number between 0 and 100. The probability that there are two real-valued solutions for x for this quadratic is k %. Determine the value of k. Report as your answer only the value of k without the percent sign.

26 CALCULATING TEAM COMPETITION ICTM STATE 015 DIVISION A PAGE of 3 9. A horse is tied with a 10 foot tether to the outside corner of a 6 foot by 8 foot rectangular storage shed in the middle of a grazing field. Determine the numeric area, in square feet, of grazing area the horse can reach. 10. The ratio of male births to female births is 1.06 :1. Determine the probability that a family of five children has two or more boys. 11. A solid block of wood is in the shape of a cube with edges of length 0. A "hole" in the shape of a right circular cone is bored out of the block of wood such that the base of the cone is inscribed in one face of the cube and the vertex of the cone is a point in the opposite face of the cube. All surfaces of this block of wood with the "hole" are to be painted. Determine the numeric value of the area, in square units, to be painted. Express your answer as a decimal rounded to the nearest hundredth of a unit. 1. A right triangle has legs of lengths 4.57 and Determine the sine of the smallest angle of this triangle. 13. The points 1,7,, 1, and 5,3 lie on the graph of the quadratic y ax bx c. Determine the exact value for the sum a b c. Express your answer as an integer or common or improper fraction reduced to lowest terms. 14. Determine the value of log 31 ln sin Let A B 6.137, A 3B 9.164, and A BC. Determine the value of sin cos A tan sec B. Arccos sin C

27 CALCULATING TEAM COMPETITION ICTM STATE 015 DIVISION A PAGE 3 of Let Determine the sum of all distinct degree measures for such that 1 sin. 17. Let x and y be positive integers. k 30x 56y x and x 1 y 617 x x 1 x x 3 1. Determine the value of k. y y 1 y y In the diagram shown, but not drawn to scale, D, E, and F are right angles. AD 6, DB 5.16, BE 5, EC 6 5, FC 6, 15 and FA.Determine the numeric area of ABC. A F C E D B 19. Let a1, a 0, a3 1, and a 4 5. For all integers n 4, an an4 an3 an an 1. Determine the value of a 015. Express your answer in scientific notation. 0. Michael Mathman found that the probability of him getting a specific type of problem correct was 0.6. If Michael uses a calculator, the probability of getting the correct answer becomes Michael works one of these problems both ways. Determine the probability Michael got the correct answer at least one time. Express your answer as an exact decimal.

28 015 SA School ANSWERS Calculator Team (Use full school name no abbreviations) Correct X 5 pts. ea. = Note: All answers must be written legibly. Round answers to four significant digits and write in standard notation unless otherwise specified in the question. Except where noted, angles are in radians. No units of measurement are required. (NOTE: DO NOT USE SCIENTIFIC NOTATION UNLESS SPECIFIED IN THE QUESTION) (Must be this decimal, "square units" optional.) OR (Must be this exact decimal.) (Must be this decimal, trailing zeros necessary) OR (Must be this reduced common fraction.) 35.6 ITEM ANALYSIS Div 1A Ques % correct (Must be this integer, (Must be this comma usage and " " exact value.) "degrees" optional.) (Must be this decimal with no % sign.) ("square feet" or "sq. ft." optional.) OR (Must be in scientific notation.) OR OR (Must be this exact decimal.) Div A Ques % correct

29 FROSH-SOPH PERSON COMPETITION ICTM 015 STATE DIVISION A PAGE 1 OF 1. Concentric circles with center O have radii of 4 and 7. AOB 45. The shaded region between the concentric circles and interior to AOB has area k. 5w 18w 1w 9. k Determine the value of. Express your answer as a common or w improper fraction reduced to lowest terms. A B O. Let y y k and k 7w 13 9w 5 y. Determine the value of w. 3. In right ABC, C is a right angle. AD DE EF FG GC and CP PQ QR RS SB. AQ 15 and BF 105. Determine the length of AB. A D E F G C P Q R S B 4. Determine the sum of all possible integral values of k such that roots. 5x kx 9 0 has rational 5. Let A,8, B1, 3, and,0 C k be such that the area of ABC is 34. Determine the sum of all possible values for k. Express your answer as a common or improper fraction reduced to lowest terms. 6. A right circular cone with base radius 5 and a right square pyramid with base sides of length 5 both have height of 1. Determine the positive difference between the volumes of these solids. Express your answer as a decimal rounded to the nearest hundredth. 7. Let k, w, where k and w are positive integers, represent solutions to the equation 3x y 500. Determine the median of all possible values for w. x 8. The graph of the line containing the point that is the x intercept of the graph of y 8 and the point that is the y intercept of the graph of y x 3x also contains the point k,1. Determine the value of k. Express your answer as an exact decimal.

30 FROSH-SOPH PERSON COMPETITION ICTM 015 STATE DIVISION A PAGE OF 9. Two concentric circles have numerical areas of 65 and 576. Let k be the numerical length of a chord of the larger circle that is tangent to the smaller circle. Let w be the largest zero of f x x 49x 63x 015 k w. 3. Determine the sum 10. Determine the value of

31 FROSH-SOPH PERSON COMPETITION LARGE PRINT QUESTION 1 ICTM 015 STATE DIVISION A 1. Concentric circles with center O have radii of 4 and 7. AOB 45. The shaded region between the concentric circles and interior to A AOB has area k. B O 5w 18w 1w 9. Determine k the value of. Express your w answer as a common or improper fraction reduced to lowest terms.

32 FROSH-SOPH PERSON COMPETITION LARGE PRINT QUESTION ICTM 015 STATE DIVISION A. Let 3 y 4 y 7 k and k 7w 13 9w 5 y. Determine the value of w.

33 FROSH-SOPH PERSON COMPETITION LARGE PRINT QUESTION 3 ICTM 015 STATE DIVISION A 3. In right ABC, C is A D E a right angle. F G C P Q R S B AD DE EF FG GC and CP PQ QR RS SB. AQ 15 and BF 105. Determine the length of AB.

34 FROSH-SOPH PERSON COMPETITION LARGE PRINT QUESTION 4 ICTM 015 STATE DIVISION A 4. Determine the sum of all possible integral values of k such that 5x kx 9 0 has rational roots.

35 FROSH-SOPH PERSON COMPETITION LARGE PRINT QUESTION 5 ICTM 015 STATE DIVISION A 5. Let A,8, B1, 3, and C k,0 be such that the area of ABC is 34. Determine the sum of all possible values for k. Express your answer as a common or improper fraction reduced to lowest terms.

36 FROSH-SOPH PERSON COMPETITION LARGE PRINT QUESTION 6 ICTM 015 STATE DIVISION A 6. A right circular cone with base radius 5 and a right square pyramid with base sides of length 5 both have height of 1. Determine the positive difference between the volumes of these solids. Express your answer as a decimal rounded to the nearest hundredth.

37 FROSH-SOPH PERSON COMPETITION LARGE PRINT QUESTION 7 ICTM 015 STATE DIVISION A 7. Let k, w, where k and w are positive integers, represent solutions to the equation 3x y 500. Determine the median of all possible values for w.

38 FROSH-SOPH PERSON COMPETITION LARGE PRINT QUESTION 8 ICTM 015 STATE DIVISION A 8. The graph of the line containing the point that is the x intercept of the graph of y 8 and the point that is the y intercept of the graph of y x 3x also contains the point k,1. Determine the value of k. Express your answer as an exact decimal. x

39 FROSH-SOPH PERSON COMPETITION LARGE PRINT QUESTION 9 ICTM 015 STATE DIVISION A 9. Two concentric circles have numerical areas of 65 and 576. Let k be the numerical length of a chord of the larger circle that is tangent to the smaller circle. Let w be the largest zero of f x x 3 49x 63x 015. Determine the sum k w.

40 FROSH-SOPH PERSON COMPETITION LARGE PRINT QUESTION 10 ICTM 015 STATE DIVISION A 10. Determine the value of

41 015 SA School ANSWERS Fr/So Person Team (Use full school name no abbreviations) Total Score (see below*) = NOTE: Questions 1-5 only are NO CALCULATOR Note: All answers must be written legibly in simplest form, according to the specifications stated in the Contest Manual. Exact answers are to be given unless otherwise specified in the question. No units of measurement are required Answer OR "zero" 4 OR (Must be this reduced improper fraction,) (Must be this exact answer.) (Must be this reduced common fraction.) (Must be this decimal.) (Must be this exact decimal.) Score (to be filled in by proctor) TOTAL SCORE: (*enter in box above) Extra Questions: * Scoring rules: Correct in 1 st minute 6 points Correct in nd minute 4 points Correct in 3 rd minute 3 points PLUS: point bonus for being first In round with correct answer

42 JUNIOR-SENIOR PERSON COMPETITION ICTM 015 STATE DIVISION A PAGE 1 OF 1 1. Let k u v when u i j and v 0i 15 j. The roots of the equation. Let are p, q, and w. Determine the sum k p q w. 3 x x x 3. Let x, y N Determine the units digit of N. be an ordered pair of real numbers and a solution to the system 3 x x y xy x 3 0. Report as your answer all possible value(s) for the 3 x y 3x y xy xy x y 0 x quotient. y 7 4. Let log k log k 1 1. Let w logn n 1. Determine the sum k w 5. Let f x x kx w. n. f x has minimum values at x 4 and x 1. Let p be the 1th term of a geometric sequence whose 7th term is 88 and whose nd term is 816. Determine the sum k w p. 6. Let k be the number of trailing zeros in the number 65! Let w be the sum of the coefficients of all terms in the expansion of x y z 4. Determine the sum k w Trailing zeros are all the zeros right of the last non-zero digit in a number.). (Note: 7. Let 10 n 1 n 1 k w n 1 n1 3 n 1 p q. Determine the sum k w p q 8. In ABC, side a is opposite A, side b is opposite B, and side c is opposite C. If the measure of A is 30, b 10, and a 7, find the smallest possible length of side c. Express your answer as a decimal rounded to the nearest hundredth of a unit. 9. Let x represent a base ten positive integer. Let a and b represent single non-zero digits from base seven such that the three-digit base seven number ab seven x x. Determine all ordered pair(s) for which the aforementioned is possible. List all pair(s) in the form a, b. 10. Let r be the correlation coefficient of the least-squares regression line determined by 69.99,850. Let f ( x) 3.x 16.1 and let 39.99,350, 49.99,500 and g( x) 5x Determine the value of the sum r f g 1. as a decimal rounded to the nearest ten-thousandth... Express your answer

43 JUNIOR-SENIOR PERSON COMPETITION LARGE PRINT QUESTION 1 ICTM 015 STATE DIVISION A 1. Let k uv when and u i j v i j The roots of the equation 3 x 4x 17x 60 0 are p, q, and w. Determine the sum k p q w.

44 JUNIOR-SENIOR PERSON COMPETITION LARGE PRINT QUESTION ICTM 015 STATE DIVISION A. Let N Determine the units digit of N.

45 JUNIOR-SENIOR PERSON COMPETITION LARGE PRINT QUESTION 3 ICTM 015 STATE DIVISION A 3. Let x, y be an ordered pair of real numbers and a solution to the system 3 x x y xy x x y x y xy xy x y 3 0 Report as your answer all possible value(s) for the quotient x y.

46 JUNIOR-SENIOR PERSON COMPETITION LARGE PRINT QUESTION 4 ICTM 015 STATE DIVISION A 4. Let log k log k 1 1. Let w log n 1 7 n. n Determine the sum k w.

47 JUNIOR-SENIOR PERSON COMPETITION LARGE PRINT QUESTION 5 ICTM 015 STATE DIVISION A 5. Let f x x kx w. f x has minimum values at x 4 and x 1. Let p be the 1th term of a geometric sequence whose 7th term is 88 and whose nd term is 816. Determine the sum k w p.

48 JUNIOR-SENIOR PERSON COMPETITION LARGE PRINT QUESTION 6 ICTM 015 STATE DIVISION A 6. Let k be the number of trailing zeros in the number 65! Let w be the sum of the coefficients of all terms in the expansion of x y z 4. Determine the sum k w. (Note: Trailing zeros are all the zeros right of the last nonzero digit in a number.)

49 JUNIOR-SENIOR PERSON COMPETITION LARGE PRINT QUESTION 7 ICTM 015 STATE DIVISION A 7. Let 10 n1 1 n 1 k w n1 3 n 1 p q n. Determine the sum k w p q

50 JUNIOR-SENIOR PERSON COMPETITION LARGE PRINT QUESTION 8 ICTM 015 STATE DIVISION A 8. In ABC, side a is opposite opposite opposite A, side b is B, and side c is C. If the measure of A is 30, b 10, and a 7, find the smallest possible length of side c. Express your answer as a decimal rounded to the nearest hundredth of a unit.

51 JUNIOR-SENIOR PERSON COMPETITION LARGE PRINT QUESTION 9 ICTM 015 STATE DIVISION A 9. Let x represent a base ten positive integer. Let a and b represent single non-zero digits from base seven such that the threedigit base seven number ab x x. seven Determine all ordered pair(s) for which the aforementioned is possible. List all pair(s) in the form a, b.

52 JUNIOR-SENIOR PERSON COMPETITION LARGE PRINT QUESTION 10 ICTM 015 STATE DIVISION A 10. Let r be the correlation coefficient of the least-squares regression line determined by 39.99,350, 49.99,500 and 69.99,850. Let f ( x) 3.x 16.1 and let g( x) 5x Determine the value of the sum r f g 1.. Express your answer as a decimal rounded to the nearest ten-thousandth.

53 015 SA School ANSWERS Jr/Sr Person Team (Use full school name no abbreviations) Total Score (see below*) = NOTE: Questions 1-5 only are NO CALCULATOR Note: All answers must be written legibly in simplest form, according to the specifications stated in the Contest Manual. Exact answers are to be given unless otherwise specified in the question. No units of measurement are required Answer 9 8 OR.8 OR OR 1.75 OR (Must be this decimal.) 1,3, 3,1, 4, Score (to be filled in by proctor) (Must have all 3 ordered pairs in any order.) (Must be this decimal.) TOTAL SCORE: (*enter in box above) Extra Questions: * Scoring rules: Correct in 1 st minute 6 points Correct in nd minute 4 points Correct in 3 rd minute 3 points PLUS: point bonus for being first In round with correct answer

54 FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND 1 ICTM 015 DIVISION A STATE FINALS x x x 1) Determine the value for x when ) Alex received a kit to build a Star Wars spacecraft for his birthday. The directions said that the scale was 1 inch ANS feet. After he built the spacecraft he measured the length of the wing and found that it was inches.. Determine the length, in feet, of the wing of a full size spacecraft. 5 3) A square is cut into three congruent rectangles along two lines parallel to a side, as shown in the figure. The perimeter of each of the three rectangles is ANS. Determine the area of the original square. 4) Let k be the sum of the degree measures of the interior angles in a regular octagon. Let w represent the sum of the degree measures of the interior angles in a regular polygon with ANS sides. Determine the value of k w. Answers 1) 1 ) 4 ("feet" optional.) 3) 81 ("sq. units." optional.) 4) 15300

55 FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND ICTM 015 DIVISION A STATE FINALS 1) Determine the value of x when x 1. ) Determine the number of integers n between 1 and ANS, inclusive, such that the product of two linear factors with integer coefficients. 3) RSTV is an isosceles trapezoid with RS ANS, RV 1, and ST 18 perpendicular segment from W to RV. x x n can be written as. Determine the exact length of the 4) The dimensions of a rectangular solid are in the ratio of 3: 4 : 5. If the diagonal is ANS, determine the sum of the three dimensions. Express your answer as an improper fraction reduced to lowest terms. ANSWERS: 1) 100 ) 9 ("integers" optional.) 3) 1 (Must be this exact answer.) 4) 144 (Must be this reduced improper fraction.) 5

56 FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND 3 ICTM 015 DIVISION A STATE FINALS 1) For any three distinct numbers a, b, and c, define c a a b c. Determine the value of 4 5. c b ) The points k, ANS and 1,k lie on a line with slope k where k 0. Determine the exact value of k. 3) Five large equilateral triangles, each with side of numeric length ANS, are arranged in a plane so they are all on the same side of a line that contains one side of each triangle. Along this line, the midpoint of the base of one triangle is a vertex of the next triangle. Determine the area of the region of the plane that is covered by the union of the five large triangular regions. 4) Squares ABCD and EFGH are congruent. AB ANS and G is the center of square ABCD. Determine the numeric area of the region in the plane covered by the two squares. ANSWERS 1) 3 ) 3 3) 1 3 ("sq. units." optional.) 4) 756 ("sq. units." optional.)

57 FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND 4 ICTM 015 DIVISION A STATE FINALS 1) A rectangle is partitioned into four rectangles with integral length sides by two segments parallel to its sides. The areas of three of the resulting rectangles are shown. Determine the numeric area of the fourth rectangle. ) In the xy plane, the segment with endpoints 5,0 and k, ANS lies on the circle. Determine the value of k. 5,0 is the diameter of a circle. The point 3) In the arrow shaped polygon shown below, A, BCD, D, E, and EFG are right angles. BC FG 5, CD FE 0 DE ANS, and AB AG. Determine the numeric area of the arrow shaped polygon., 4) Abby measured some of the interior angles in a dodecagon. When she added them, she had a sum of ANS degrees. Determine the sum of the degree measures in the angles that she did not measure. ANSWERS 1) 15 ("sq. units." optional.) ) 10 3) 300 ("sq. units." optional.) 4) 1500 ("degrees" optional.)

58 FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND 5 ICTM 015 DIVISION A STATE FINALS 1) Last year a bicycle cost $160 and a cycling helmet cost $40. This year the cost of the bicycle increased by 5%, and the cost of the helmet increased by 10%. The percent increase in the combined cost of the bicycle and helmet is k %. Determine the value of k. Express your answer as k only. Do not use the % sign. ) If 34x 5 P, then ) ABC ANS x kp. Determine the value of k. is inscribed in a circle with B C ANS A. B and C are adjacent vertices of a regular polygon of k sides inscribed in this circle. Determine the value of k, the number of sides of this regular polygon. 4) In ABC, AB 5, 7 BC, AC ANS, D is on AC, and BD 5. Determine the ratio AD : DC. Express your answer as the ratio k : w where k and w are relatively prime positive integers. ANSWERS 1) 6 (Must be this answer only, no % used.) ) 4 3) 9 ("sides" optional.) 4) 19 :8 (Must be this exact ratio in this form.)

59 FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND 1 ICTM 015 DIVISION A STATE FINALS QUESTION 1 x x x 1) Determine the value for x when

60 FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND 1 ICTM 015 DIVISION A STATE FINALS QUESTION ) Alex received a kit to build a Star Wars spacecraft for his birthday. The directions said that the scale was 1 inch ANS feet. After he built the spacecraft he measured the length of the wing and found that it was inches.. Determine the length, in feet, of the wing of a full size spacecraft. 5

61 FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND 1 ICTM 015 DIVISION A STATE FINALS QUESTION 3 3) A square is cut into three congruent rectangles along two lines parallel to a side, as shown in the figure. The perimeter of each of the three rectangles is ANS. Determine the area of the original square.