WRITTEN AREA COMPETITION ICTM REGIONAL 2017 DIVISION AA PAGE 1 OF 3

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1 WRITTEN AREA COMPETITION ALGEBRA I ICTM REGIONAL 2017 DIVISION AA PAGE 1 OF 3 y y 1. If 2x 4x, determine the ratio x. Express your answer as an integer or as a 2 4 y common or improper fraction reduced to lowest terms. 2. A rope 48 feet long is cut into 4 pieces whose lengths are in the ratio of 2 to 3 to 5 to 8. Determine the number of feet in the second shortest piece of this rope Let a b a 3ab b 5. Determine the sum of the solution(s) for the equation x Three integers are selected such that the sums of these integers taken two at a time are 20, 25 and 27. What is the product of these three integers? 5. 5x 2 25x 20 k x 1wx p the value of k w p. where k, w, and p are integers with k 1. Determine 6. Tom rode his motor scooter from home to a campground averaging 30 miles per hour. He traveled home along the same route averaging 20 miles per hour. The total time on the scooter for the two trips was 10 hours. Determine the distance in miles from home to the campground. 7. The perimeter of the quadrilateral ABCD can be written as the expression kx w. Determine the ordered pair k, w. A(-3,4) D(-3,-2) y-axis B(x,4) C(x,-2) x-axis

2 WRITTEN AREA COMPETITION ALGEBRA I ICTM REGIONAL 2017 DIVISION AA PAGE 2 OF k 8. Let 2 a 8 1 where a b with b a positive prime integer. Determine the value of k. Express your answer as an integer or as a common or improper fraction reduced to lowest terms. 9. Jo's school grades on a normal 4.0 scale with all courses weighted the same. That is, an A 4 points, a B 3 points, and so forth. After taking 15 courses, Jo has an exact grade point average (GPA) of 3.2. She takes one course in Summer School and gets another A. The following semester she takes five courses and has a goal to raise her overall GPA up to at least 3.4. Determine her overall grade point average if she indeed accomplishes her goal. Express your answer as a decimal rounded to the nearest hundredth x varies inversely as the square root of y. x 20 when when 1 y. 9 1 y. Determine the value of x Tickets to Universal Studios cost $136 for adults and $130 for children. One day, 26,473 guests (adults and children combined) attend Universal Studios and $3,496,948 is collected at the ticket office. Determine the number of children that attended that day. 12. The numerical area of a rectangle with consecutive sides of length 3x 1 and Determine the numerical perimeter of this rectangle. x is Determine the units digit for the number The number 1a2b 6, where a and b are digits, is evenly divisible by 12. Determine the number of possible values 1a2b 6 could represent.

3 WRITTEN AREA COMPETITION ALGEBRA I ICTM REGIONAL 2017 DIVISION AA PAGE 3 OF When solved for x, the equation possible value(s) for k. 2 5x kx 6 has exactly one real solution. Determine the 16. Four consecutive odd integers have the property that the sum of the smallest and twice the largest is 99. Determine the product of the smallest and largest of these four integers. 17. Given that a, b, and c are integers with no common factors such that a 0 and a a, determine the value of the expression. Give your answer as an b c 3 b c integer or as a common or improper fraction reduced to lowest terms. 18. At Electric City, the price of a basic phone has been reduced by 20% to a sale price of $24. Determine the original price in dollars for this phone Let a b 2000, a b ab 400, and Determine the value(s) of k. 2 2 a b kab for real numbers a, b, and k. 20. A pair of standard 6-sided die are rolled. Determine the probability of rolling either a number that is more than 4 on at least one dice or a sum that is more than 4 shown on the face-up die. Express your answer as a common fraction reduced to lowest terms.

4 2017 RAA Name ANSWERS Algebra I Correct X 2 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 reduced (Comma use and common fraction.) "children" optional.) ("feet" or "ft." optional.) (Must have both values, may use, either order, and may use "and", "or", or sol'n set.) ("miles" or "mi." optional.) ,18 ordered pair.) reduced common fraction.) exact decimal.) reduced improper fraction.) ("dollars" or "$" optional.) answer only.) reduced common fraction.)

5 WRITTEN AREA COMPETITION GEOMETRY ICTM REGIONAL 2017 DIVISION AA PAGE 1 OF 3 1. A circle is represented by the equation this circle. 2 2 x 6y 6x y 0. Determine the exact radius of y-axis A(-3,4) B(x,4) 2. The numeric area of the quadrilateral ABCD shown is 48. Determine the perimeter of quadrilateral ABCD. x-axis D(-3,-2) C(x,-2) 3. In ABD, AB 10, ABD 75, and BAD 60. Determine the exact length of BD. 4. A triangle has two sides of lengths 5 and 8. The restrictions for the numeric perimeter of this triangle can be expressed in interval notation as k, w. Determine these restrictions. Express your answer in interval notation. 5. The sum of the lengths of the two diagonals in quadrilateral ABCD is 44. Determine the perimeter of the quadrilateral formed by joining the midpoints of the consecutive sides of quadrilateral ABCD. 6. In ABC ABC, point D lies on ray AC. AB 52 2x BCD x BC, 6 4. Determine the, and degree measure of ABC. B A C D 7. In the diagram shown, O is the center of a circle and points A, B, and D lie on the circle with points A, O, B, and C collinear and CD tangent to the circle at D. DC 12 and BC 6. The circumference of Circle O is k. Determine the value of k. A O B C D B 8. (Note: Not drawn to scale.) Given Circle O as shown. The radius of Circle O is 6. AB and BC are tangent to the circle at points A and C. B 30 and D 55. The exact length of arc EF k. Determine the value of k. Express your answer as an integer or as a common or improper fraction reduced to lowest terms. D E F A O C

6 WRITTEN AREA COMPETITION GEOMETRY ICTM REGIONAL 2017 DIVISION AA PAGE 2 OF 3 9. Distinct points A, B, and C are collinear with B between A and C. AB 7 k, BC 2k 1, and AC 4k 3. Determine all possible value(s) for k. 10. In ABC, A 30, C 90, and point D lies on AC such that BDC 45. AB 8 3. Determine the exact area of ABD. 11. In ABC, the median from point A lies on the line bisector of BC is on the line y x 4 exact length of the median from A y x and the perpendicular 5 5 2,5. Determine the. Point A has coordinates 12. In Circle K, AKE 60, AT TE, HT 8 and the area of sector AKE is 24. Determine the exact area of quadrilateral KATE. K A H T E 13. In ABC, AC CB, and points P, R and S lie on their respective sides of ABC as shown such that PR AC and PS BC. PR 6, PS 8, and AB 28. Determine the exact length of PB. A R C P S B 14. The diagram shows a right triangle ABC with CD an altitude to hypotenuse AB. AC 10 and BD 15. Determine the exact length of BC. A D C B

7 WRITTEN AREA COMPETITION GEOMETRY ICTM REGIONAL 2017 DIVISION AA PAGE 3 OF In the diagram shown (but not drawn to scale), PA and PC are tangent segments to the circle and secant segment PD intersects the circle at point B such that AB CD. Arc measures are in the ratio AB : BC 7 : 4 and AD : DC 6 : 5 (where AD does not contain point B ). The ratio of the measures of the angles APD : CPD k : w where k and w are relatively prime positive integers. Determine the sum k w. D A B C P 16. A square has side of 5 and a circle has a diameter of 14. Determine the positive difference between the area of this square and the area of this circle. 17. ABCD is a parallelogram with points E, D, and C collinear. AE EC and FG BC. FG 60, AE 24, and the perimeter of ABCD is 182. Determine the numeric area of ABCD. A E F D B G C 18. A quadrilateral with consecutive sides of 28, 80, 60, and 96 is inscribed in a circle with radius 50. Determine the numeric area of this quadrilateral. 19. O and P are centers of circles that intersect at A and B to form common chord AB. OP intersects AB at point T. The circles have radii of 10 and 17 and OP 21. Determine the length of the common chord AB. O T A B P 20. A rhombus has diagonals of length 10 and 20. The area of the circle inscribed in this rhombus is k. Determine the value of k.

8 2017 RAA Name ANSWERS Geometry Correct X 2 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. exact answer.) exact answer.) exact answer.) ,26 open interval.) exact answer.) ("degrees" or " " optional.) reduced improper fraction.) (Must be one of the following:) OR OR OR OR

9 WRITTEN AREA COMPETITION ALGEBRA II ICTM REGIONAL 2017 DIVISION AA PAGE 1 of 3 1. Determine the exact largest zero of the function 2 f x 3x x 10. Express your answer as an integer or as a common or improper fraction reduced to lowest terms f x 1 3x 5x 6. Determine the value of 2 f. 3. Let k represent the sum of the roots and w the product of the roots for the polynomial equation. 7x 5x 3x 21x 19x k w. Express your. Determine the sum answer as an integer or as a common or improper fraction reduced to lowest terms Determine the exact value of x so that log4 x Three integers are such that when the smallest integer of the three is doubled, the middle integer of the three is increased by 3, and the largest integer is left unchanged, the arithmetic mean of the three new values is 2 more than the arithmetic mean of the original three integers. Determine the smallest of the original three integers. 1 3x4 6. Determine the value for x so that 125. Express your answer as an integer or as a 25 common or improper fraction reduced to lowest terms. 7. The sum of the first 6 terms of a geometric sequence is 28 times the sum of the first 3 terms of that sequence. Determine the exact common ratio for this sequence. Express your answer as an integer or as a common or improper fraction reduced to lowest terms.

10 WRITTEN AREA COMPETITION ALGEBRA II ICTM REGIONAL 2017 DIVISION AA PAGE 2 of 3 8. The endpoints of the latus rectum (focal diameter) of a parabola are 1,2 and 7, 2. The vertex of this parabola lies in the first quadrant. Determine the coordinates of this vertex. Express your answer as the ordered pair x, y. 9. Let x x 2 3. Determine all value(s) for solution(s) for x. Express your answer(s) as an integer or as a common or improper fraction reduced to lowest terms. 10. Determine the exact value of x so that x ABCDEFG is a convex polygon. Determine the number of distinct triangles that can be formed using three of the vertices of ABCDEFG. 12. The circle x y 25 is intersected by the line y x 1 at points A and B. Determine 2 2 the exact length of AB. 13. Let i 1. Let k and w represent positive integers with k w. Determine the value of w so that ( k wi)(2 9 i) 159 7i. 14. Determine all ordered pair(s) of real, x x x y so that 32 16y and 3 27 y.

11 WRITTEN AREA COMPETITION ALGEBRA II ICTM REGIONAL 2017 DIVISION AA PAGE 3 of Let ABC be equilateral with perimeter 24. Determine the exact largest area of a rectangle with lower base on the x-axis and upper vertices on the triangle. y-axis A (x,y) B C x-axis 16. Let x x x x x x P x x. Determine a polynomial in decreasing degree of x. P x. Express your answer as 17. A sequence of similar trapezoids is such that the lengths of the legs form a geometric sequence. The length of a leg of the first trapezoid is 2 and the length of the corresponding leg of the fifth such trapezoid is 288. The ratio of the area of the tenth such trapezoid to the eleventh such trapezoid is k : w where k and w are relatively prime positive integers. Determine this ratio k : w. Express your answer as this ratio k : w. 18. Let f (x) be a quadratic polynomial with non-zero roots p and q, not necessarily distinct. Let g(x) be a cubic polynomial with non-zero roots p, q and r, not necessarily distinct. The leading coefficient of each polynomial is 1 and the constant terms are the same. If f (1) 52, find g(1). 19. When polynomial 10. When P x is divided by P x is divided by x 20, the quotient is Q x and the remainder is Q x is divided by x 17, the remainder is 24. Determine the remainder when x Determine the sum of all possible value(s) for x so that 6 x 2 x. Express your answer as an integer or as a common or improper fraction reduced to lowest terms.

12 2017 RAA Name ANSWERS Algebra II Correct X 2 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. 5 reduced common fraction.) ("triangles" optional.) reduced improper fraction.) reduced improper fraction.) , ordered pair.) reduced improper fraction only.) , x 1: OR exact answer.) ordered pair only.) exact answer.) monomial.) ratio in this form.) reduced common fraction.)

13 WRITTEN AREA COMPETITION PRECALCULUS ICTM REGIONAL 2017 DIVISION AA PAGE 1 OF 3 1. Let 2 k sin 3 and 5 w cot 6. Determine the exact product kw. 2. Arcsin x represents the inverse sine function. Determine the exact value of 7 Arcsin cos 6. f x 2x 5x 9. Then 3. Let 2 of k w p. 2 kx wx p. Determine the value h f x h f x lim h 0 4. A particular triangle has sides of length 20 and 17 and an included angle between those two sides of measure 52. Determine the area of this triangle. Express your answer as a decimal rounded to the nearest thousandth. 5. Let k 18 n 17 n0 20. Determine the exact value of k. B 6. ABC is a right triangle with right angle at C. AB 11, BD 7, and DA 6. Determine the length of BC. Express your answer as a decimal rounded to the nearest thousandth C D 6 A Let matrices A 5 4 and B determinant of the product of matrices A and B.). Determine the value of det A B (the

14 WRITTEN AREA COMPETITION PRECALCULUS ICTM REGIONAL 2017 DIVISION AA PAGE 2 OF 3 8. A Super Ball is launched 100 feet into the air from ground level, measured vertically to the bottom of the ball. It falls back to ground level and then rebounds 3 4 of its height on the previous bounce. The ball continues this pattern until it stops. Determine the total distance, in feet, the ball traveled as measured from ground level vertically to the bottom of the ball The equation ( y 2) 5( x 3) for the standard coordinate system is rewritten in terms of a coordinate system ( x ', y ') for which the new origin is (5, 2) with axes parallel to the original xy- axes. The resulting equation can be written in the form: ( y ') k( x ' w) where k and w are positive integers. Determine the value of k w Assume the following is NOT a degenerate form of a conic. Identify 2 2 6x 5xy 2y x 2y 3 0 as a circle, ellipse, hyperbola, or parabola. Express your answer by writing the whole word for this conic. 11. i 1. Complex z 1 i 3. One square root of z is z a bi with a and b exact real numbers and in simplified radical form and with a 0. Determine the ordered pair a, b. Express your answer as the ordered pair a, b with exact entries. 12. Twins Jordan and Michael have the following growth curves, with x measured in years and x x y in inches: y e and y e. Determine the sum of all the year(s) after birth at which their heights were equal. Express your answer as a decimal rounded to the nearest thousandth. 13. The rectangular coordinates 2, 2 3 can be written in polar form as, r where 0 2 in radian measure. Determine these polar coordinates. Express your answer as the ordered pair r,. 14. In ABC, BAC 74 and AC 100. BC has integral length and is shorter than AC. Determine the sum of all possible integral lengths for BC.

15 WRITTEN AREA COMPETITION PRECALCULUS ICTM REGIONAL 2017 DIVISION AA PAGE 3 OF Today is Aune s fifteenth birthday. Her parents measure her height ( for the last time! ) to be 62.5 inches. On her fourteenth birthday her height was 61.0 inches. They noted her height in inches has been increasing geometrically for the past five years. Determine Aune's height, in inches, on her tenth birthday. Report your answer as a decimal rounded to the nearest quarter of an inch. 16. Let f x 3 4cos x the graph of f x and w represent the y-intercept of the graph of f k w. with x measured in radians. Let k represent the amplitude of x. Determine the sum 17. In ABC, D lies on AC such that BD bisects ABC. AB 18, BC 12, and DC 7. Determine the degree measure, in degrees, for the smallest angle of ABC. Express your answer as a decimal rounded to the nearest thousandth Let sin where 0 2 is measured in radians. The sum of all possible value(s) 2 for is k. Determine the value of k. Express your answer as an integer or as a common or improper fraction reduced to lowest terms. 2sin 2x cos x 4 cos x sin x 1 is solved over 0 x 2, the sum of the solutions 19. When 2 is k. Determine the exact value of k. Express your answer as an integer or as a common or improper fraction reduced to lowest terms. 20. The equation of the directrix of a parabola with vertex 3,2 that passes through 9, 16 and opens down is either x k or y w. Determine the equation of this directrix. Express your answer as the equation x k or y w.

16 2017 RAA Name ANSWERS Pre-Calculus Correct X 2 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. 3 OR 3 1 OR 1.5 OR π OR π π OR 3 3 (Must be this exact answer.) ordered pair with exact values.) decimal.) , π OR 3 4, π answer needed) 6 2, (Either exact coordinate pair accepted; only one.) exact decimal.) exact decimal.) ("feet" or "ft." optional.) ellipse (Must be the whole word, caps optional.) y = 4 decimal, "inches" or "in." optional.) decimal, "degrees" or " " opt.) reduced improper fraction.) equation.)

17 FROSH-SOPH EIGHT PERSON TEAM COMPETITION ICTM REGIONAL 2017 DIVISION AA PAGE 1 of 3 NO CALCULATORS Determine the smaller value for x such that 3x 5x 9 x 4x 24. Express your answer as an integer or as a common or improper fraction reduced to lowest terms. 2. A rectangular pool measures 24 feet by 30 feet. This pool is surrounded by a deck that has the same width on each side of the pool and has a rectangular outer edge. The combined area of the pool and deck is 1080 square feet. Determine the width, in feet, of the deck. Express your answer as an integer or as a common or improper fraction reduced to lowest terms. A D 3. AE is the perpendicular bisector of BD. AE 4 6 and AD 4 2. Determine the length of BD. B C E 4. A regular polygon has sides of integral length. The ratio of the numeric area of this polygon to the numeric length of the apothem of this polygon is 15. Let n represent the number of sides of this polygon. Determine the sum of all possible values for n. 5. In HAM, T lies on HM so that AT AM. MA 9, AT 12, k w p and HAM 120. HT in simplified radical form with q k, w, and p integers with q 0 also an integer. Determine the sum k w p q. H T A M 6. Let f 2x 1 8x 7. Then, as a simplified expression, 2 the sum k w p. f x kx wx p. Determine 7. (All ages are in integral years.) Aune has 3 nephews named Bill, Craig, and David. Currently, Aune is twice Bill's age, Aune is also six years older than twice Craig's age, and Bill is seven years younger than twice Craig's age.. Also using current ages, if one year was added to David's age, he would be as old as half of Aune's age minus Craig's age. Determine the sum of the ages of Aune's three nephews. NO CALCULATORS NO CALCULATORS NO CALCULATORS

18 FROSH-SOPH EIGHT PERSON TEAM COMPETITION ICTM REGIONAL 2017 DIVISION AA PAGE 2 of 3 NO CALCULATORS 8. A right square pyramid has a numeric lateral surface area of 544 and a total numeric surface area of 800. Determine the numeric volume of this figure. A 9. In ABC, AB BC, BD is an altitude to AC, and DE BC. EC 3 and DE 4. Determine the length of AB. Express your answer as an integer or as a common or improper fraction reduced to lowest terms. D B E C 10. The sum of two numbers is 14 and the sum of their reciprocals is common multiple of these two numbers. 7. Determine the least 24 D A 11. Quadrilateral ABCD is circumscribed about the circle. AD 12 and BC 16. Determine the perimeter of quadrilateral ABCD. B 12. Let 3 5 x 31. Determine the reciprocal of the product of the solutions for x in this x 2x 8 equation. C 13. Let 1 a 3. Determine the value of a 1 10 a 10 a. 14. Determine the value(s) for x such that 3x 4 x. List all possible value(s), separating with commas if necessary. NO CALCULATORS NO CALCULATORS NO CALCULATORS

19 FROSH-SOPH EIGHT PERSON TEAM COMPETITION ICTM REGIONAL 2017 DIVISION AA PAGE 3 of 3 NO CALCULATORS 15. For a lab experiment, a chemistry teacher needs 50 gallons of a 3% salt solution. He has 5% and 2% solutions he can mix to make the 50 gallons of the desired solution. Determine the number of gallons of 2% solution he must use. Express your answer as an integer or as a common or improper fraction reduced to lowest terms. 16. Let x, y, and z be positive integers such that x y z. Determine all ordered triple(s) of the form x, y, z such that x y z 1 xyz. List all possible answers as ordered triples, 2 separating with comma(s) if necessary. B 17. Using the grid at the right, determine the number of distinct paths that may be taken to move from point B to point A moving only left and down. 18. Let 8 x 27. Then k 32 x. Determine the value of k. A x x 19. Let 2 and x 0. Then, a solution for x in this equation can be written as x 1 x 1 k w p x where k, w, p, and q are integers, q 0. Determine the sum q k w p q. 20. In the Debate Regional Qualifier, 60 teams compete in head-to head (one on one) competition. There are six rounds and each team either wins or loses each round. Thus, the best record a team can have is six wins and zero losses and the worst record a team can have is zero wins and six losses. In order to advance to the Regional tournament, a team must have at least four wins (and thus two losses) or better. Assume teams are randomly matched in the qualifier. Determine the maximum number of teams that could qualify for the regional tournament. NO CALCULATORS NO CALCULATORS NO CALCULATORS

20 2017 RAA 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 ("feet" or "ft." optional.) OR exact answer or exact decimal ) OR answers needed) (Either answer accepted; only one of these reduced improper fraction.) integer only) ( 1,3,8 ), ( 1,4,5 ) reduced improper fraction, "gal." or "gallons" optional.) (Must have both ordered triples, in either order.) ("paths" or "ways" optional.) ("teams" optional.)

21 JUNIOR-SENIOR EIGHT PERSON TEAM COMPETITION ICTM REGIONAL 2017 DIVISION AA PAGE 1 OF 3 NO CALCULATORS 1. Let (, ) k w represent the norm of the vector represented by, (4,8) (5,32). k w. Determine the value of 2. Three standard fair, cubical dice are thrown. Determine the probability that the product of the three face-up numbers is even or prime. Express your answer as a common fraction reduced to lowest terms. 3. A line passes through 2,3 and 0, k and is perpendicular to the line whose equation is x 6y 8. Determine the value of k. Express your answer as an integer or as a common or improper fraction reduced to lowest terms. 4. Let k w p q when written in simplified radical form. Determine the sum k w p q Let x x Determine the real solution(s) for x. 6. (YES OR NO): Let 6 f x x x x x Is 9 a zero of this function. Report your answer by writing the whole word "yes" or "no" as appropriate. 7. Determine the number of distinct positive integers n that leave a remainder of 17 when 2017 is divided by n. NO CALCULATORS NO CALCULATORS NO CALCULATORS

22 JUNIOR-SENIOR EIGHT PERSON TEAM COMPETITION ICTM REGIONAL 2017 DIVISION AA PAGE 2 OF 3 NO CALCULATORS 2 x 3x 1 8. The graph of y has one or more asymptotes whose equations are of the forms x 1 x a, y b, and/or y kx w. Determine the sum of all values of a,b, k, and w that exist for this graph. 9. The sum of all values for, measured in radians with 0 3, that satisfy the equation 2 tan 3 1 can be expressed as k. Determine the exact value of k. Express your answer as an integer or as a common or improper fraction reduced to lowest terms i 1. z 3 i. Then z k wi answer as the ordered pair k, w with exact entries.. Determine the ordered pair, k w. Express your 11. Three vectors in standard position have endpoints 1,2,3, 3, 2,1, and 2,1, 6. Determine the exact volume of the parallelepiped formed by these three position vectors as sides of the parallelepiped k Determine the value of k. x 2cost 13. The exact area enclosed by the graph of can be expressed as k. Determine the y 4sin t exact value of k. Express your answer as an integer or as a common or improper fraction reduced to lowest terms. 14. Let k, w, and p each represent positive integers such that k 1, w 2, and p 2. Each of k cases contains w boxes, and each box contains p items. If w p and the total number of items in the k cases is 56, determine the smallest possible value of w p. NO CALCULATORS NO CALCULATORS NO CALCULATORS

23 JUNIOR-SENIOR EIGHT PERSON TEAM COMPETITION ICTM REGIONAL 2017 DIVISION AA PAGE 3 OF 3 NO CALCULATORS 15. Determine the exact sum of all of the six limits below that exist. Ignore those that do not exist. Report your answer as an integer or as a common or improper fraction reduced to lowest terms. 2 4 x 4x 12 sin x 3x 2x 8 lim lim x2 x 2 0 x lim 4 x x 2 x cos x tan x cos x lim sin x lim 3 lim x x x x /2 sin x 16. Determine the non-zero value(s) for x such that arctan 5x arctan x arctan 2x arctan 3x. (Note: arctan x is the inverse tangent relation of x.) 17. Let k log 5log 6log 7log 8log Determine the exact value for k. Express your answer as an integer or as a common or improper fraction reduced to lowest terms Let k. Determine the exact value of k. Express your answer as an integer or 2 n1 n 3n as a common or improper fraction reduced to lowest terms. 19. Ye Olde Ice Cream Shoppe offers 18 distinct flavors of ice cream. Their special is a bowl with three different flavor scoops of ice cream. Determine the number of different specials they could serve without repeating a bowl. 20. The solutions for 3 9 x x 3 3 can be written as are positive integers. Determine the sum k w p. k w where k, w, and p p NO CALCULATORS NO CALCULATORS NO CALCULATORS

24 2017 RAA School ANSWERS Jr/Sr 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 reduced common fraction.) real number only.) NO whole word, capitalization optional.) (Must have both answers in this form, any order, may 11 11, use "and" or "or", or exact simp. equiv.) OR ,16 ordered pair.) reduced common fraction. ("Bowls" or "specials" optional.)

25 CALCULATING TEAM COMPETITION ICTM REGIONAL 2017 DIVISION AA PAGE 1 of 3 Exact answers are required unless otherwise specified in the question. Answers must be simplified and in the specific form if so stated. Except where noted, angles are in radians. No units of measurement are required. 1. Determine the x-intercept of the line passing through 20,17 and perpendicular to the line 2.38x 15.7 y Express your answer as a decimal rounded to the nearest thousandth. 2. Determine the least solution for x such that x x a decimal rounded to the nearest thousandth Express your answer as the initial input of 2017.) Express your answer as a decimal rounded to four significant digits Let f x x. Determine the value of f (2017 iterations of f x with 4. According to the June 14, 2016, Chicago Tribune, the owner of a house in Chicago, which is valued by the tax assessor at $225,000 will pay a property tax of $3,633 using a tax rate of C per cent. The owner of a house in a far west suburb, which is valued by the tax assessor at $450,000, will pay a property tax of $11,850 using a tax rate of S per cent. Let S kc. Determine the value of k. Express your answer as a decimal rounded to the nearest thousandth. 5. Let e be the base of the natural logarithms. Consider the arithmetic sequence whose first term is 24, second term is 24 2e, third term is 24 4e and so forth. Determine the positive integer that represents the term number of the term that is closest to A circle with center at point O has circumference log The circle with center P is internally tangent to Circle O, passes through point O, and has circumference log x 12. Determine the value of x. Express your answer as a decimal rounded to four significant digits Let k Determine the value of k. Express your answer as a decimal rounded to the nearest thousandth.

26 CALCULATING TEAM COMPETITION ICTM REGIONAL 2017 DIVISION AA PAGE 2 of 3 8. Point A lies on the graph of 20x 17y 42 0 and point C lies on the graph of 10x 24y These lines intersect at point B to form acute ABC. Determine the degree measure of this ABC. Express your answer as a decimal rounded to the nearest thousandth. 9. In the diagram, circles with centers at A, B and C are mutually tangent to each other. AB 5, AC 7, and BC 4. Determine the numeric area of the shaded region, the region enclosed by the triangle but not interior to any circle. Express your answer as a decimal rounded to four significant digits. A B C 10. StarPeet Coffee wants to blend coffee that sells for $8.49 per pound with coffee that sells for $15.25 per pound to obtain a blend that sells for $11.75 per pound. For every 10 pounds of the $8.49 coffee, StarPeet must add k pounds and w ounces (rounded to the nearest ounce) of the $15.25 coffee. Determine the ordered pair k, w. 11. A circular dart board (target) consists of a circle circumscribing an equilateral triangle with another circle inscribed in the triangle. The diameter of the dart board has length 8 3. A randomly thrown dart lands on the target. Determine the probability the dart landed interior to the triangle but exterior to the circle inscribed in the triangle. Express your answer as a decimal rounded to four significant digits. 12. Let k 1. Determine the value of k. Express your answer as a decimal rounded to the nearest thousandth. 13. At noon, the hour and minute hands of a standard analog 12-hour clock keeping correct time are both vertical. In k minutes and w seconds (rounded to the nearest second) after noon, the hour and minute hand will form a right angle for the second time. Determine the ordered pair k, w. 14. A circle with radius 20 is inscribed in a regular pentagon. Determine the area of this pentagon. Express your answer as a decimal rounded to the nearest thousandth.

27 CALCULATING TEAM COMPETITION ICTM REGIONAL 2017 DIVISION AA PAGE 3 of It is known that Tiger will play golf on a Tuesday 95% of the time. It is known that Phil will play golf on a Tuesday 90% of the time. It is known that Rory will play golf on a Tuesday 85% of the time. It is known that Jordan will play golf on a Tuesday 80% of the time. Determine the probability that at least three of the four will play golf next Tuesday. Express your answer as an exact decimal. 16. Determine the sum of all values for x such that the graph of y e x 2 intersects the 2 2 x y ellipse 1. Express your answer as a decimal rounded to the nearest thousandth The Pigeon Hole Computer program can quickly assign 11 distinct student name labels, one each, to 11 distinctly numbered cubbyholes, formatting one such assignment set per page, but the attached printer can only print one page every 3 seconds. The printer process began the moment March 1, 2016 began and prints continuously. Determine the number of years from the start of March 1, 2016 to the completion of this printing project. Express your answer as a decimal rounded to the nearest thousandth. Assume the printer is an "infinite" printer and ignore issues of toner or ink, paper, and repair. 18. A right circular cone with the radius of the base of length 20 has a numeric lateral area that is at most Determine the largest possible length of a lateral edge of this cone. Express your answer as a decimal rounded to the nearest thousandth. 19. Aune has 3 sticks, one with a length of 8, a second with a length less than 11, and a third with a length less than 12. Determine the probability the 3 sticks can form an obtuse triangle with the longest side having a length of 8. Express your answer as a decimal rounded to four significant digits. 20. On July 1, 2016, the UFC franchise was sold for 4 billion dollars. It was purchased on July 1, 1986 for 2 million dollars. Using these numbers as exact, if the 2 million dollars had grown to 4 billion dollars in a bank savings account paying an annual percentage rate of k %, compounded monthly, determine the value of k. Express your answer as the value of k only and as a decimal rounded to the nearest thousandth.

28 2017 RAA School ANSWERS Calculator Team (Use full school name no abbreviations) Correct X 5 pts. ea. = Note: All answers must be written legibly. Exact answers are required unless otherwise specified in the question. Answers must be simplified and in the specific form if so stated. Except where noted, angles are in radians. No units of measurement are required. (Must be decimal.) this decimal.) OR.613 (Must be this decimal.) decimal.) decimal.) ("376th term" or "term" optional.) OR , OR decimal, trailing zero necessary.) ordered pair.) decimal, comma usage optional.) exact decimal.) decimal.) decimal.) decimal, " " or "degrees" optional.) OR.6696 decimal.) ,5 ordered pair.) OR.1384 decimal.) decimal, "yr.s". or "years" optional.) decimal.) decimal.) decimal with no percent sign used.)

29 FROSH-SOPH 2 PERSON COMPETITION ICTM 2017 REGIONAL DIVISION AA PAGE 1 OF 2 1. If f ( x) = 3x + 2 and g ( x) = 2x 3, determine the value of f ( g ( 4) ) g ( f ( 4) ). 2. A square has a perimeter of k units, an area of 2k square units and a diagonal of d units. Let m be the distance between the x and y intercepts of the line 3x 3y = 12. Determine the value ( d m) Let n be the smallest possible integer such that 2 3 n is less than 11 largest possible integer such that 6 5 m 5 is greater than Let m be the 5. Determine the sum ( n m) A bag of coins has 2 pennies, 3 nickels and a dime. Two coins are selected at random from this bag. What is the probability that the value of the coins is at least 10 cents? Express your answer as a common fraction reduced to lowest terms. 5. The larger of two concentric circles has numeric area P and the smaller of the circles has numeric area Q. A chord of the larger circle has length 12 and is tangent to the smaller circle. Let A represent the smallest integral value of x such that 2x + 1 < 9. The expression ( P Q + A) = kπ + w. Determine the value of ( k w) A kite has consecutive side lengths of 25 and 29. A diagonal of this kite has length 40 and is bisected by the other diagonal whose length is k. The angles of a triangle have degree k + x. measures of ( x + ), ( x ) and ( x ). Determine the sum ( ) 2x + 3y = 9 7. The system of equations has an infinite number of solutions. The system of ax + by = 27 2x y = 8 equations has no solutions. Let w represents the sum of all values that k 4x 2y = k a + b + w. CANNOT have. Determine the sum ( ) 8. Triangle DEF is inscribed in rectangle ABCD as shown. The percent of the area of rectangle ABCD that is inside triangle DEF is k %. Determine the value of k. Express your answer as a decimal rounded to the nearest tenth. Do not use the percent sign in your answer. D 14 A 5 E 19 C 6 F B

30 FROSH-SOPH 2 PERSON COMPETITION ICTM 2017 REGIONAL DIVISION AA PAGE 2 OF 2 9. Let m represent the slope of a line that is perpendicular to the line through ( 7,4) and ( 13, 6). Two regular hexagons have perimeters that are in the ratio 4 : 25. If one side of the smaller of the two hexagons has length 100, the perimeter of the larger of the two m + P. hexagons is P. Determine the value ( ) 10. Determine the sum of all possible values of the y-coordinates of the first quadrant lattice points (points with integral values of x and y ) on the graph of 3x + 7 y = 120.

31 ICTM Math Contest Freshman Sophomore 2 Person Team Division AA

32 FROSH-SOPH 2 PERSON COMPETITION LARGE PRINT QUESTION 1 ICTM 2017 REGIONAL DIVISION AA NO CALCULATORS ALLOWED 1. If f ( x) = 3x + 2 and g ( x) 2x 3 =, determine the value of ( ( 4) ) ( 4) ( ) f g g f.

33 FROSH-SOPH 2 PERSON COMPETITION LARGE PRINT QUESTION 2 ICTM 2017 REGIONAL DIVISION AA NO CALCULATORS ALLOWED 2. A square has a perimeter of k units, an area of 2k square units and a diagonal of d units. Let m be the distance between the x and y intercepts of the line 3x 3y = 12. Determine the value ( d + m).

34 FROSH-SOPH 2 PERSON COMPETITION LARGE PRINT QUESTION 3 ICTM 2017 REGIONAL DIVISION AA NO CALCULATORS ALLOWED 3. Let n be the smallest possible integer such that 2 3n 11 is less than 3. 5 Let m be the largest possible integer such that 6 5m 7 is greater than 5. 9 Determine the sum ( n + m).

35 FROSH-SOPH 2 PERSON COMPETITION LARGE PRINT QUESTION 4 ICTM 2017 REGIONAL DIVISION AA NO CALCULATORS ALLOWED 4.A bag of coins has 2 pennies, 3 nickels and a dime. Two coins are selected at random from this bag. What is the probability that the value of the coins is at least 10 cents? Express your answer as a common fraction reduced to lowest terms.

36 FROSH-SOPH 2 PERSON COMPETITION LARGE PRINT QUESTION 5 ICTM 2017 REGIONAL DIVISION AA NO CALCULATORS ALLOWED 5.The larger of two concentric circles has numeric area P and the smaller of the circles has numeric area Q. A chord of the larger circle has length 12 and is tangent to the smaller circle. Let A represent the smallest integral value of x such that 2x + 1 < 9. The expression ( ) P Q + A = kπ + w. Determine the value of ( k + w).

37 FROSH-SOPH 2 PERSON COMPETITION LARGE PRINT QUESTION 6 ICTM 2017 REGIONAL DIVISION AA CALCULATORS ALLOWED 6.A kite has consecutive side lengths of 25 and 29. A diagonal of this kite has length 40 and is bisected by the other diagonal whose length is k. The angles of a triangle have degree measures of ( 4x + 11), ( 7x 3) and ( 10x 17). Determine the sum ( k + x).

38 FROSH-SOPH 2 PERSON COMPETITION LARGE PRINT QUESTION 7 ICTM 2017 REGIONAL DIVISION AA CALCULATORS ALLOWED 7.The system of equations 2x + 3y = 9 ax + by = 27 has an infinite number of solutions. The system of equations 2x y = 8 4x 2y = k has no solutions. Let w represents the sum of all values that k CANNOT have. Determine the sum ( a b w) + +.

39 FROSH-SOPH 2 PERSON COMPETITION LARGE PRINT QUESTION 8 ICTM 2017 REGIONAL DIVISION AA CALCULATORS ALLOWED 8.Triangle DEF is inscribed D 14 C 6 F in rectangle ABCD as A 5 E 19 B shown. The percent of the area of rectangle ABCD that is inside triangle DEF is k %. Determine the value of k. Express your answer as a decimal rounded to the nearest tenth. Do not use the percent sign in your answer.

40 FROSH-SOPH 2 PERSON COMPETITION LARGE PRINT QUESTION 9 ICTM 2017 REGIONAL DIVISION AA CALCULATORS ALLOWED 9.Let m represent the slope of a line that is perpendicular to the line through ( 7,4) and ( 13, 6). Two regular hexagons have perimeters that are in the ratio 4 : 25. If one side of the smaller of the two hexagons has length 100, the perimeter of the larger of the two hexagons is P. Determine the value ( m + P).

41 FROSH-SOPH 2 PERSON COMPETITION LARGE PRINT QUESTION 10 ICTM 2017 REGIONAL DIVISION AA CALCULATORS ALLOWED 10. Determine the sum of all possible values of the y-coordinates of the first quadrant lattice points (points with integral values of x and y) on the graph of 3x + 7 y = 120.

42 FROSH-SOPH 2 PERSON COMPETITION EXTRA LARGE PRINT QUESTION 11 ICTM 2017 REGIONAL DIVISION AA CALCULATORS ALLOWED 11. A circle has area 2017 square units. A cube has edge length the same length as a radius of the circle. Determine the length of a diagonal of the cube. Express your answer as a decimal rounded to four significant digits.

43 FROSH-SOPH 2 PERSON COMPETITION EXTRA LARGE PRINT QUESTION 12 ICTM 2017 REGIONAL DIVISION AA CALCULATORS ALLOWED 12. Four positive integers have a sum of 50. Determine the largest possible product of these four integers.

44 2017 RAA School ANSWERS Fr/So 2 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 exact answer.) reduced common fraction.) decimal, no % in the answer.) TOTAL SCORE: Score (to be filled in by proctor) (*enter in box above) Extra Questions: ANS ANS ANS decimal.) * Scoring rules: Correct in 1 st minute 6 points Correct in 2 nd minute 4 points Correct in 3 rd minute 3 points PLUS: 2 point bonus for being first In round with correct answer

45 JUNIOR-SENIOR 2 PERSON COMPETITION ICTM 2015 REGIONAL DIVISION AA PAGE 1 OF 1 i =. Let ( 4 3 ) ( 3 2 ) Determine the sum ( k + w) i + i = k + wi when expanded and simplified completely Let k be such that two of the roots of x + 20x 17x + k = 0 are additive inverses of each other. Let w be the numerical area of a convex parallelogram whose diagonals have lengths 20 and 17 and form a 30 angle at their point of intersection. Determine the value of k + w. ( ) 3. From a right isosceles triangle, an equilateral triangle, and a triangle with side lengths of 15, 36, and 39, one angle is selected at random. Find the probability that the tangent of this angle is defined and is greater than one. Express your answer as a common fraction reduced to lowest terms. 4. Let k be the numerical coefficient of the 2 3 x y term when ( 2x y 2) 6 completely simplified. Let w be the slope of the line determined by k Determine the value of the quotient w. + is expanded and x = 4t + 7. y = 12t Determine the product of the roots for the equation ( x 1) ( 2x 1) ( x 2) + + = +. n 6. Let A = A. Then A = k w when written in simplified radical form. Let p be the number of distinct positive integral factors of Determine the sum ( k + w + n + p). 7. A pulley with diameter 9 inches is turning at 1200 revolutions per minute and is connected with a belt to a smaller pulley with diameter of 6 inches. Let k be the number of revolutions per minute for the smaller pulley. Let w be the sum of the real solution(s) for the polynomial formed by 0 5 n 5 3 = x n= 0 x. Determine the sum ( k w ) Let k be a positive integer. When written as a positive integer, k! ends in exactly 56 trailing zeros. Determine the sum of all possible distinct values of k. (Trailing zeros are the zeros that follow the last non-zero digit of a number reading left to right.) 9. Let x + sin x lim x 0 k = = +. Determine the sum ( k + w). x Express your answer as an integer or as a common or improper fraction reduced to lowest terms.. Let 2 w lim ( 4x 7x 2x) x 10. The area of the region located between the graphs of x + y = 12 and x + y = k is 160. Determine the sum of all possible distinct values of k.

46 ICTM Math Contest Junior Senior 2 Person Team Division AA

47 JUNIOR-SENIOR 2 PERSON COMPETITION LARGE PRINT QUESTION 1 ICTM 2017 REGIONAL DIVISION AA NO CALCULATORS ALLOWED 1. i = 1. Let 2 3 ( 4 3 ) ( 3 2 ) i + i = k + wi when expanded and simplified completely. Determine the sum ( k + w).

48 JUNIOR-SENIOR 2 PERSON COMPETITION LARGE PRINT QUESTION 2 ICTM 2017 REGIONAL DIVISION AA NO CALCULATORS ALLOWED 2.Let k be such that two of the roots of 3 2 x + 20x 17x + k = 0 are additive inverses of each other. Let w be the numerical area of a convex parallelogram whose diagonals have lengths 20 and 17 and form a 30 angle at their point of intersection. Determine the value of ( k + w).

49 JUNIOR-SENIOR 2 PERSON COMPETITION LARGE PRINT QUESTION 3 ICTM 2017 REGIONAL DIVISION AA NO CALCULATORS ALLOWED 3.From a right isosceles triangle, an equilateral triangle, and a triangle with side lengths of 15, 36, and 39, one angle is selected at random. Find the probability that the tangent of this angle is defined and is greater than one. Express your answer as a common fraction reduced to lowest terms.

50 JUNIOR-SENIOR 2 PERSON COMPETITION LARGE PRINT QUESTION 4 ICTM 2017 REGIONAL DIVISION AA NO CALCULATORS ALLOWED 4.Let k be the numerical coefficient of the 2 3 x y term 2x y + 2 is when ( ) 6 expanded and completely simplified. Let w be the slope of the line determined by x = 4t + 7. Determine the y = 12t 5 value of the quotient k w.

51 JUNIOR-SENIOR 2 PERSON COMPETITION LARGE PRINT QUESTION 5 ICTM 2017 REGIONAL DIVISION AA NO CALCULATORS ALLOWED 5. Determine the product of the roots for the equation ( x 1) ( 2x 1) ( x 2) + + = +.

52 JUNIOR-SENIOR 2 PERSON COMPETITION LARGE PRINT QUESTION 6 ICTM 2017 REGIONAL DIVISION AA CALCULATORS ALLOWED 6. Let A = A. Then A = k n w when written in simplified radical form. Let p be the number of distinct positive integral factors of Determine the sum ( k + w + n + p).

53 JUNIOR-SENIOR 2 PERSON COMPETITION LARGE PRINT QUESTION 7 ICTM 2017 REGIONAL DIVISION AA CALCULATORS ALLOWED 7.A pulley with diameter 9 inches is turning at 1200 revolutions per minute and is connected with a belt to a smaller pulley with diameter of 6 inches. Let k be the number of revolutions per minute for the smaller pulley. Let w be the sum of the real solution(s) for the polynomial formed by 0 5 n 3 x 5. x n= 0 = Determine the sum ( k w) +.

54 JUNIOR-SENIOR 2 PERSON COMPETITION LARGE PRINT QUESTION 8 ICTM 2017 REGIONAL DIVISION AA CALCULATORS ALLOWED 8. Let k be a positive integer. When written as a positive integer, k! ends in exactly 56 trailing zeros. Determine the sum of all possible distinct values of k. (Trailing zeros are the zeros that follow the last non-zero digit of a number reading left to right.)

55 JUNIOR-SENIOR 2 PERSON COMPETITION LARGE PRINT QUESTION 9 ICTM 2017 REGIONAL DIVISION AA CALCULATORS ALLOWED 9. Let Let k x + sin x = lim x 0 x. w = lim 4x + 7x 2x. x Determine the sum ( k + w). Express your answer as an integer or as a common or improper fraction reduced to lowest terms. 2

56 JUNIOR-SENIOR 2 PERSON COMPETITION LARGE PRINT QUESTION 10 ICTM 2017 REGIONAL DIVISION AA CALCULATORS ALLOWED 10. The area of the region located between the graphs of x + y = 12 and x + y = k is 160. Determine the sum of all possible distinct values of k.

57 JUNIOR-SENIOR 2 PERSON COMPETITION EXTRA LARGE PRINT QUESTION 11 ICTM 2017 REGIONAL DIVISION AA CALCULATORS ALLOWED 11. Let k = lima when t t 6 1 = a = 1 and a a 0 t t 1 Let w = Determine the product ( kw ).

58 JUNIOR-SENIOR 2 PERSON COMPETITION EXTRA LARGE PRINT QUESTION 12 ICTM 2017 REGIONAL DIVISION AA CALCULATORS ALLOWED 12. The vertices of a hyperbola are ( 0,4) and ( ) 6,4. The asymptotes of this hyperbola 5 have equations y = ± ( x 3). 3 In simplified radical form, the coordinates of the foci are then ( ) k ± w p,4 with w > 0, p positive integer. Determine the sum ( k + w + p).

59 2017 RAA School ANSWERS Jr/Sr 2 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 reduced common fraction.) reduced improper fraction.) + OR 4( ) OR 4( ) TOTAL SCORE: Score (to be filled in by proctor) (*enter in box above) Extra Questions: OR 21 OR * Scoring rules: Correct in 1 st minute 6 points Correct in 2 nd minute 4 points Correct in 3 rd minute 3 points PLUS: 2 point bonus for being first In round with correct answer