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WRITTEN AREA COMPETITION ICTM REGIONAL 2016 DIVISION AA ALGEBRA I PAGE 1 OF 3 1. Let x 5 and y 3 . Determine the value of 2 x3 5 y 2 . 2. Let x and y be integers such that 17 x 20 and 7 y 9 . Determine the smallest possible value of the product xy . 3. A line through points k , 4 and 6, 9 is perpendicular to the line with an equation 3 x 2 y 5 0 . Determine the value of k . 4. The degree measures of the angles of a triangle are in the ratio 8:11:13. One of these angles is selected at random. Determine the probability that the degree measure of this angle is an integer. Express your answer as a common fraction reduced to lowest terms. 5. Determine the least integer solution for the inequality 9 2 x 15 . 6. The product of two positive numbers is 768, and their sum is 7 times their positive difference. Determine the larger of the two numbers. 7. For all x and y , f x y f x f y 1 . f 1 2 . Determine the value of f 3 . 8. Ten students in a class took a test. Two of the students scored a 79. Two others scored 83. Four of the rest had scores that averaged 92. The class average for these ten students was an 88. Determine the average score of the remaining two students. WRITTEN AREA COMPETITION ICTM REGIONAL 2016 DIVISION AA ALGEBRA I PAGE 2 OF 3 9. The total cost of prom this year was $17,850. The cost was to be shared equally by those who planned to attend. Regrettably, at the last minute, 10 people could not attend, raising the cost for each person by $4.25. Determine the number of students who actually attended prom. 10. A palindromic number is a positive integer that reads the same backwards and forwards, such as 64746. Craig is challenged to produce the largest palindromic number possible, subject to the requirements that no digit is 0, at least four different digits are used, and the sum of all of the digits in the number is 25. Craig is successful with his offering of N. Determine the sum of the first five digits of N. (The first five digits when reading the digits of N from left to right.) 11. For some value of k , 13kx 12 x 9 1 has exactly one real solution for x . Determine the ordered pair k , x . Express your answer as the ordered pair k , x . 2 12. Let k be the largest integer which leaves the same remainder when each of the three numbers 163, 305, and 518 are divided by k . Determine the value of k . 13. Dave notices his odometer in his car reads 686,686 miles, which is a palindromic number (reads the same forwards and backwards.) If Dave could drive at an uninterrupted 55 miles per hour, determine the number of hours Dave would need to drive until the odometer reached the first palindromic mileage reading greater than seven hundred thousand miles. Express your answer as an exact decimal. 14. When a woman drives directly from home to work at an average speed of 26.25 miles per hour, she arrives 2 minutes late. When she drives the same route at an average speed of 35 miles per hour, she arrives 2 minutes early. Determine the number of miles in the length of this route. WRITTEN AREA COMPETITION ICTM REGIONAL 2016 DIVISION AA 15. For all valid replacements of a and b , a b ALGEBRA I PAGE 3 OF 3 3a b . Determine the value of 4 6 9 . b 16. The base 10 number 218 is represented by 431 in base b . (That is 218 431b ). Then 253b k where k is a base 10 integer. Determine the value of k . 17. Jack began mowing a field for 30 minutes. Jill joined him and together they finished the field in another hour. Jill can mow the same field by herself in 2 hours and 30 minutes. Determine the number of minutes additionally that Jack would have had to mow the field if he had mowed alone. Express your answer to the nearest whole minute. 18. Determine the first (leading) digit in the sum 11 22 33 4 4 55 999999 10001000 . 1 x 19. When simplified, 1 x 1 1 1 1 1 1 kx w . Determine the ordered pair k , w . Express your answer as the ordered pair k , w . 20. In the town of Chambana, Illinois, the total population is 120,840 people. Exactly one-third of the population have exactly two-letter initials, at least seven-fifteenths of the remainder of the population have exactly three-letter initials, and the rest of the population have exactly four-letter initials. Assuming the normal English alphabet, determine the minimum number of people in Chambana, Illinois that have the same exact initials. 2016 RAA Algebra I Correct X 2 pts. ea. = Name School ANSWERS (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 3 specified in the question. No units of measurement are required. 0, or 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 295 11. 144 13.5 OR 1 3 27 1 OR 13 2 2 (Must be this reduced common fraction.) 2 94 9 14. 7 17. 18. ("students" optional.) 19. 20. 4 (Must be this decimal.) ("miles" optional.) 5 136 120 ("minutes" optional.) 1 OR ONE 2, 1 (Must be this ordered pair.) 63 ("people" optional.) 0.0.75 also accepted 71 242.2 16. 8 1 3 OR 4, OR 4,1 2 2 13. 15. 32 200 12. 4,1.5 60 also accepted as correct WRITTEN AREA COMPETITION ICTM REGIONAL 2016 DIVISION AA GEOMETRY PAGE 1 OF 3 1. Two of the faces of a solid cube block were painted orange, and the rest of the faces were painted blue. Determine the number of faces of this cube that were painted blue. 2. Point A 2,1 is reflected over the line y x to point A ' . Point B y-axis has coordinates 8, 4 . Determine the area of A ' BA . Express your answer as an exact decimal. B(8,4) A(-2,1) x-axis 3. One of the angles of an isosceles trapezoid has a degree measure of 42. Determine the largest possible sum of the degree measures of two of the other angles of the trapezoid. 4. In the diagram (not drawn to scale), points A , B , and D lie on circle O , CD is tangent at point D , and A , B and C are collinear. AB is a diameter, CD 12 , and CB 4 . Determine the area of COD . B C O D 5. A is an acute angle such that A 5 x 10 . Then k x w is the restriction on x . Determine the ordered pair k , w . Express your answer as this ordered pair. 6. A right rectangular box is 60 centimeters wide, 75 cm. long, and 45 cm. high. Determine the length, in centimeters, of the longest stick that can be placed inside this box. Express your answer as a decimal rounded to 4 significant digits. 7. A triangle has sides with lengths of 5, 6, and 8. The area of this triangle can be expressed as k w in simplified and reduced radical form where k , w , and f are positive integers. f Determine the value of k w f . A WRITTEN AREA COMPETITION ICTM REGIONAL 2016 DIVISION AA GEOMETRY PAGE 2 OF 3 A 8. AB FC DE . AB 11 and FC 3 . Determine the length of DE . Express your answer as a common or improper fraction reduced to lowest terms. B F C E 9. Three congruent circles are externally tangent to each other and each has radius 6. The common external tangents, two circles at a time, intersect to form a triangle. Determine the exact area of this triangle. D 10. Consider the statement "You can go outside after you eat your supper." Which statement(s) below is/are logically equivalent to the given statement? Express your answer as the capital letter(s) corresponding to your choice(s). A) If you eat your supper, then you can go outside. B) If you don't eat your supper, then you can't go outside. C) If you can't go outside, then you didn't eat your supper. D) If you can go outside, then you ate your supper. 11. Two intersecting spheres have diameters 30 and 40 and the centers of the spheres are 25 units apart. The area of the circle formed by the intersection of the spheres has numeric area k . Determine the exact value of k . 12. In the diagram, ACB 90 , AD 5 , DC 10 , CB 20 , and EB 17 . Determine the area of ADE . A D E C B 2 1 1 13. Three points 3 , 2 , 5, 2 , and k , 2 are collinear. Determine the value of k . 5 2 2 Express your answer as a common or improper fraction reduced to lowest terms. 14. In the right triangle shown, x y z 2 k w p in simplified and reduced radical form with k , w , and p positive integers. Determine the sum k w p . z x y 2 10 WRITTEN AREA COMPETITION ICTM REGIONAL 2016 DIVISION AA GEOMETRY PAGE 3 OF 3 15. Quadrilateral ABCD is inscribed in Circle O as shown, (but not drawn to scale.) AB 2 , BC 4 , CD 5 , and DA 3 . The ratio of the area of FAB to EAD is k : w where k and w are relatively prime positive integers, Determine the ratio k : w . Express your answer as that ratio k : w . D C E O A B 16. To estimate the height of a vertical climbing wall, you place a mirror horizontally on the ground so that its center is 85 feet from the base of the wall and then keep walking backward from the mirror so that the mirror is between you and the wall until you can see the top of the wall centered in the mirror. At this point, you are 6.5 feet from the center of the mirror and your eyes are 5 feet vertically from the ground. Assuming the ground is flat, determine the height of the wall in feet. Express your answer as a decimal rounded to the nearest hundredth. 17. The four vertices of quadrilateral ABCD has coordinates A 0, 0 , B b, a , C k , w and D a, b . Determine the coordinates of point C so that quadrilateral ABCD is a parallelogram. Express your answer as the ordered pair k , w where k and w are expressed in terms of a and b . 18. Two sides of a right triangle have measure 12 and 13. Determine the absolute value of the difference between the largest and smallest possible measures of the third side. Express your answer as a decimal rounded to four significant digits. 19. Kite ABCD has right angles DAB and DCB . EB 8 and AD 3 . Determine the exact numeric area for kite ABCD . D 3 A E C 8 20. In ABC , the altitude from A to BC intersects BC between B and C and the foot of the altitude has coordinates 2,3 . An equation of the perpendicular bisector of BC is 2 x y 4 and an equation of the median from A is 3 x 4 y 16 . Determine the coordinates of A . Express your answer as an ordered pair x, y . B F 2016 RAA Name Geometry Correct X School 2 pts. ea. = ANSWERS (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. 2. 3. 4. 5. 6. 7. 8. 9. 10. 4 ("faces" optional.) (Must be this exact decimal.) 19.5 276 ("degrees" or " " optional.) 96 2, 20 (Must be this ordered pair.) (Must be this exact decimal.) 106.1 406 33 8 (Must be this reduced improper fraction.) 144 3 216 OR 72 2 3 3 OR 216 144 3 OR 72 3 2 3 A, C (Must have both capital letters, either order.) 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 144 16 57 8 (Must be this reduced improper fraction.) 41 4 : 27 65.38 (Must be this exact ratio and in this form.) ("feet" or "ft." optional.) a b, a b OR a b, b a ordered pair.) OR b a, b a OR b a, a b (Must be this 12.69 (Must be this exact decimal.) 18 2 (Must be this exact answer.) 4,7 (Must be this ordered pair.) WRITTEN AREA COMPETITION ICTM REGIONAL 2016 DIVISION AA ALGEBRA II PAGE 1 OF 3 1. Let k 1 be a positive real number. Determine all value(s) of k such that log k 5 log5 k . Express your answer(s) as an integer or common or improper fraction reduced to lowest terms. 2. (Yes or No) Let k represent an irrational number and let w represent an irrational number. Determine if it is possible for the product kw to represent a rational number. Express your answer as the whole word "yes" or "no". 3. Determine the value of n such that n ! 24 . 4. Let i 1 . The roots for x of x 4 7 x3 ax 2 cx 24 0 are p , p 1 , and bi where p is a positive integer and b 0 . Determine the exact value of the sum p b . 2 x 12 . Determine the exact value of f 2016 10 . (That is, 2015 function x2 compositions of f x or f f f 10 with 2016 " f " s .) 5. Let f x 6. A parabola has directrix y 3 and endpoints of the latus rectum are 2, 1 and 6, 1 . An equation for this parabola can be written either as 4 p y k x h or 4 p x h y k . Determine the sum h k p . 2 2 7. Let w be the symbol for the digit twelve in a base system. Let k be a positive integer that is a number base. Let x represent a base 10 number such that x 16k and x3 281wk (where w is the units digit of the number). Determine the value of k . Express your answer in base ten. WRITTEN AREA COMPETITION ICTM REGIONAL 2016 DIVISION AA ALGEBRA II PAGE 2 OF 3 8. $1200 is invested at an Annual Percentage Rate of 6% compounded monthly. Determine the amount of interest earned during the 121st month of this investment. Express your answer in dollars rounded to the nearest cent. 9. Three monkeys randomly toss 24 indistinguishable ping pong balls into three containers labeled A, B, C. Each ball is equally likely to end up in any one of the containers and all balls go into one of the containers. Determine the number of distinct outcomes that are possible. The order in which the balls are placed in the containers does not matter. 10. A ball is dropped from a height of 100 feet and it rebounds to forever. Determine the total number of feet the ball travels. 11. Let 2 x 7 x2 x 6 3 of its height each time 5 1 . Determine all possible value(s) for x such that this equation is true. k p w in simplified and reduced radical form q and k , w , p , and q are positive integers. Determine the value of the sum k w p q . 12. Let x 33 33 33 . Then x 13. f x 2 is quadratic function with roots of 5 and 3 . The graph of f x passes through 2, 45 . Then f x can be written as f x ax 2 bx c where a , b , and c are integer coefficients and. Determine the sum a b c . 14. Determine the value of k such that f x 28 x 123 is its own inverse. 7x k WRITTEN AREA COMPETITION ICTM REGIONAL 2016 DIVISION AA ALGEBRA II PAGE 3 OF 3 15. Joe invests $12000 divided into three funds. Fund A earns 2%, Fund B 3.5% and Fund C 5%, all annual percentage rates and compounded annually. At the end of 1 year, Joe earned $480 total interest on his investments. Joe had D dollars invested in Fund B. The number of dollars invested in Fund A is then w M kD where M is some number of dollars. Determine the product Mk . 16. Let y vary directly as the square root of x and inversely as z 2 . When y is tripled and z quadrupled, x is multiplied by k . Determine the value of k . 17. P x is a polynomial function. When P x is divided by x 3 , the remainder is 1. When P x is divided by x 1 , the remainder is 3. Determine the remainder when P x is divided by x 2 2 x 3 . Express your answer as a polynomial expression in terms of x with integer or common or improper coefficients reduced to lowest terms. 18. An unfair weighted coin results in heads being tossed 60% of the time. The coin is tossed 4 times and the results recorded consecutively as H or T (for heads or tails.) Determine the probability there is exactly one occurrence of two consecutive tosses the same. Express your answer as common fraction reduced to lowest terms. 19. Bob's boat travels at a constant speed in still water. The current in the river has a constant speed also. Bob makes a round trip, driving the boat upstream from the ramp to his cabin and then returning to the ramp. The average speed of this trip is 36% of his still water speed. The speed of the current is k % of his still water speed. Determine the value of k . Do not use the % symbol as part of your answer. 20. A group of 2n mathematicians consisting of exactly n men and of exactly n women are having a dinner meeting. One of the mathematicians, Cindy, notes that there are k distinct arrangements possible if the group were to sit at a round table with no distinguishing marks and if men and women must alternate. Another of the mathematicians, Tom, suggests that the group find the minimum value for n such that there would be more than 100, 000k distinct arrangements possible if the group were to sit at a round table with no distinguishing marks and if men and women did not necessarily alternate. Determine that minimum value of n . 2016 RAA Algebra II Correct X Name 2 pts. ea. = ANSWERS 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 all 4 sol'ns, 1 (Must have both answers, either order, with reduced in any order.) 5 common fraction.) 11. 1. (Must be the whole word, capitalization optional. 5, 2. 3. 4. 5. 6. 7. 8. 9. 10. 4, 3, 2,3 YES 12. 4 3 2 (Must be this exact answer.) OR 2 3 10 17 OR 1710 OR 17ten 325 400 14. 15. 3 10.92 13. (Must be this exact decimal, "dollars" "$" optional.) 16. 17. 18. ("outcomes" optional.) ("feet" or "ft." optional.) 19. 20. 137 36 4 2000 2304 1 5 x 2 2 228 625 80 11 ("dollars or "$" optional.) (Must be this polynomial expression with rational coefficients.) (Must be this reduced common fraction.) (Must be this integer, with no % symbol used.) WRITTEN AREA COMPETITION ICTM REGIONAL 2016 DIVISION AA PRECALCULUS PAGE 1 OF 3 1. Determine the value of the indicated vector inner (dot) product 5,3 6,8 . 5 x3 3x 2 7 x 3 . x 0 2x 3 2. Determine the value of lim kw p 1 with in quadrant I. Then the exact value for cos cot in 2 q simplified and reduced radical form with k , w , p , and q integers. Determine the sum 3. Let tan k w p q . 4. Determine the value of the sum (2 5 2 k 1) . 4 10 5. A sine curve has a maximum value of 3 at x , and a minimum value of 3 at x 3 3 with exactly one zero between these two values for x . Determine the standard equation y A sin B x C D for this curve with x measured in radians and where A 0 is the least possible value. Express your answer as the sum A B C D written as a common or improper fraction reduced to lowest terms. 3 6. Let f x x3 , g x 2 x x , and h x sin 2 x . Determine the value of 2 f g h . Express your answer as a common or improper fraction reduced to lowest 6 terms. WRITTEN AREA COMPETITION ICTM REGIONAL 2016 DIVISION AA PRECALCULUS PAGE 2 OF 3 7. Let f x x a x b x c with 0 a b c . Determine the solution for f x 0 . 2 Express your answer(s) in interval notation. (For example, the interval k , w represents k x w .) 8. In a geometric sequence, the last term is 1458, the common ratio is 3 , and the sum of the terms is 1094. Determine the second term of this geometric sequence. 9. Let a , b , c , and d be positive real numbers. Let a b c d 10 , let c d , and let k w a 2 b 2 c 2 d 2 30. The largest possible value of c can be expressed as in f simplified and reduced radical form where k , w , and f are positive integers. Determine the value of k w f . 10. Determine the numeric area of a triangle with sides 20 and 16 and the included angle between those two sides having measure 150 . 11. In ABC , 2sin A 3cos B 4 and 3sin B 2 cos A 3 . Determine the degree measure of C . 12. Points 1,3, 1 , 2,1,5 , and 3, 2, 6 lie on a plane. A standard equation for that plane can be written as Ax By Cz D 0 where A , B , C , and D are relatively prime integers and A 0 . Determine the sum A B C D . 13. A parabola has the equation of 2 x 2 12 x 3 y . The equation of the directrix of this parabola can be expressed in the form x k or y w . Determine the value of k or w , whichever is appropriate, as an exact decimal. Express your answer as the equation x k or y w. WRITTEN AREA COMPETITION ICTM REGIONAL 2016 DIVISION AA 14. PRECALCULUS PAGE 3 OF 3 3 5 is one zero of a fourth-degree polynomial with integer coefficients and leading coefficient 1 when written in decreasing degree of x . Determine the constant term for this polynomial. 15. Determine the number of distinct arrangements of the letters in the word MATHEMATICS. 16. The roots for x in x3 ax 2 bx 105 0 are each two more than the roots for x in x3 dx 2 ex f 0 . a , b , d , e , and f are constant real numbers. Determine the value of the sum 4d 2e f . 17. For all real values of such that cos( ) sin( ) 0 , Determine the value of 7 k 28w . cos3 ( ) sin 3 ( ) k w sin 2 . cos( ) sin( ) 1 24tn for n 1 be a sequence. Determine the value of t2016 . 24 3tn Express your answer as a common or improper fraction reduced to lowest terms. 18. Let t1 77 and tn 1 19. Determine the number of points of intersection for the curves y 1 x is measured in radians. 20. In the diagram with coordinates as shown, AC 51 and BC 55 . The ratio of the area of PAB to the area of PAC to the area of PBC is 1:2:3. Determine the ordered pair that represents point P. Express your answer as an ordered pair k , w with each member of the ordered pair expressed as a common or improper fraction reduced to lowest terms. x sin x and y where 4 x C(x,y) A(0,0) P B(26,0) 2016 RAA Pre-Calculus Correct X 2 pts. ea. = Name ANSWERS 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. ("degrees" or " " optional.) 6 1. 1 2. 22 3. 4. 5. 6. 7. 8. 9. 10. 23 6 (Or any value of the form 97 64 c, a 6 12 80 90 12. 52 13. 56 23 4n 6 11. where n is an integer) (Must be this improper fraction.) (Must be only this ordered pair used as interval notation.) 14. 15. 16. 17. 18. 19. 20. y 6.375 (Must be this equation in y using this exact decimal value.) 4 4989600 ("arrangements" and comma usage optional.) 97 21 1847 207 2 739 110 , 78 13 (Must be this reduced improper fraction.) ("points of intersection" or "points" optional.) (Must be this ordered pair with reduced improper fraction entries.) FROSH-SOPH EIGHT PERSON TEAM COMPETITION ICTM REGIONAL 2016 DIVISION AA PAGE 1 of 3 NO CALCULATORS 1. Determine the number of distinct arrangements using all the letters of the word ILLINOIS exactly once. 2. The diagonals of a rhombus are length 6 and 10. Determine the perimeter of this rhombus. A 3. AD and BC are common internal tangent segments to Circles E and F as shown. EB 3 , GC 3 , and CF 2 . Determine the sum of the lengths of these two common internal tangents. E 3 3 C 2 G B F D 4. A farmer's rectangular field is to be bounded on one side by a river and fenced in on the other three sides. The farmer only has 200 meters of fencing. Determine, in square meters, the maximum area of the field he can contain using the river and fence. . A , D , E , and B are 5. A is the center of arc BC collinear as are A , F , and C . DEF FEC CEB . AFD EFC . FD AB . Determine the degree measure of ABC . Express your answer as an integer or exact decimal. D E A F 6. A solid cube whose sides have length 8 has a vertex cut off, leaving an equilateral triangle trace with sides of length 1. The surface area of the now 7-faced solid can be written in the kw p form when simplified and reduced radical form with k , w , p , and q integers. q Determine the value of the sum k w p q . 7. The sum of two numbers is 10. The product of these same two numbers is 7. Determine the sum of the reciprocals of these two numbers. Express your answer as a common or improper fraction reduced to lowest terms. NO CALCULATORS NO CALCULATORS NO CALCULATORS B C FROSH-SOPH EIGHT PERSON TEAM COMPETITION ICTM REGIONAL 2016 DIVISION AA PAGE 2 of 3 NO CALCULATORS x 8. The figure shows a square and two similar, non-square rectangles, where y x . The ratio of the area of the larger non-square rectangle to x k w that of the smaller non-square rectangle may be written as in p simplified and reduced radical form where k , w , and p are positive x x x integers. Determine the sum k w p . 9. A right triangle has legs of 12 and 35. The altitude to the hypotenuse divides the hypotenuse into two parts k and w . Determine the geometric mean of k and w . Express your answer as an improper fraction reduced to lowest terms. y y 35 12 k w 10. Seven good friends eat at the same restaurant. All of them are eating there today but only one of them eats there every day. The second eats there every 2 days, the third every 3 days, the 4th every 4 days, the fifth every 5 days, the 6th every 6 days, and the 7th every 7 days. Determine the number of days from today before they will all be eating there again on the same day. 11. Determine the value of k such that k 17 1 1 8 8 . 12. Determine the base 3 representation for the base 4 number 1334 . M 13. In right EBM with right angle at B , points G and O trisect MB while points L and U trisect EB . ML 8 and OE 6 . Determine the length of ME . G O E U L B 14. A circle is circumscribed about a square and at the same time the circle is inscribed in another square. Determine the probability that a point selected in the interior of the larger square is also a point in the interior of the smaller square. Express your answer as a common fraction reduced to lowest terms. NO CALCULATORS NO CALCULATORS NO CALCULATORS FROSH-SOPH EIGHT PERSON TEAM COMPETITION ICTM REGIONAL 2016 DIVISION AA NO CALCULATORS PAGE 3 of 3 A B 10 x 19 A B Cx D 2 . Then 2 . Determine the values of C x 3 x 4 x x 12 x 3 x 4 x x 12 and D . Express your answer as the ordered pair C , D . 15. Let 16. Determine the number of positive even integers of three digits that can be formed using the digits 1, 2, 3, 4, 5, 6, and 7 if no digit is repeated. 17. A rectangular solid prism has sides of length 6, 8, and 24 and is inscribed in a sphere. The surface area of this sphere is k . Determine the value of k . 18. A regular octagon is inscribed in a circle with diameter 8. A square is inscribed in the same circle so that the vertices of the square are also vertices of the octagon. Determine the exact area inside the octagon but outside the square. 19. When the Cousins Club decided to plan their summer barbecue, they found that 14 members could meet in June only, 10 members could meet in July only, and 12 members could meet in August only. Also, 3 members could meet only in July or August, 2 members could meet only in June or August, and 2 members could meet only in June or July. There were 22 cousins who could meet in June. Determine the number of cousins who could meet in July. 20. The linear function f x 6 x 72 has the value of its x-intercept doubled and the value of its y-intercept tripled to create a new linear function g x . Determine the slope of the graph of g x . NO CALCULATORS NO CALCULATORS NO CALCULATORS 2016 RAA Fr/So 8 Person Correct X School ANSWERS (Use full school name – no abbreviations) 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. 1. 3360 2. 4 34 3. 15 4. 5. 6. 7. 8. 5000 82.5 (Must be this exact answer.) ("square meters" or " m 2 " optional.) (Must be this exact decimal, " " or "degrees" optional.) 1541 10 7 (Must be this reduced improper fraction.) 9. 10. 420 12. 13. 14. 15. 16. 17. 4 1011 OR 10113 OR 1011three 3 10 1 2 4, 37 90 676 32 2 32 OR 32 10 420 37 11. (Must be this reduced common fraction.) (Must be this ordered pair.) ("integers" optional.) 2 1 OR 32 1 2 32 32 2 OR 32 1 2 OR 32 2 1 (Must be this reduced improper fraction.) ("days" optional.) 18. 19. 20. 19 9 ("cousins" or "members" optional.) JUNIOR-SENIOR EIGHT PERSON TEAM COMPETITION ICTM REGIONAL 2016 DIVISION AA NO CALCULATORS PAGE 1 OF 3 1. Let f x 2.8 3.5cos 2.3 x . Determine the maximum value of this function. 4 Express your answer as an exact decimal. 2. Let k 0.23 in base 3. Determine the value of k in base 10. 3. There are 76 marbles in a jar, either orange or blue. Ella draws one marble at random and does not replace it in the jar. The marble is blue. The probability the next marble Ella draws 3 will be blue is . Determine the ratio of orange marbles to blue marbles that were in the jar 5 before Ella drew any marbles. Express your answer as a ratio k : w where k and w are relatively prime positive integers. 4. Let f x x 2 2 x and g x 2 x 6 . Determine the largest value for x such that f g x g f x . 13 1 5. Let A log 2 2k . Determine the value of A . 4 k 0 x 3 . Express your answer as an integer or common or x 9 improper fraction reduced to lowest terms. 6. Determine the value of lim x 9 NO CALCULATORS NO CALCULATORS NO CALCULATORS JUNIOR-SENIOR EIGHT PERSON TEAM COMPETITION ICTM REGIONAL 2016 DIVISION AA NO CALCULATORS PAGE 2 OF 3 3 2 5 7. Let A x 4 1 . Determine the average of the values for x such that det A 0 . 1 x 2 Express your answer as a common or improper fraction reduced to lowest terms. 8. Henrik and Will need to create 150 ounces of a 50% acidic solution. They have 80% acidic solution and 30% acidic solution available to combine. Determine the number of ounces of the 30% acidic solution that will be used. 8 12 9. Let k sin Arc tan Arc cos . Determine the value of k . Express your answer as a 15 13 common or improper fraction reduced to lowest terms. (Note: This notation uses the convention arcsin x represents the inverse sine relation while Arc sin x represents the inverse sine function, etc.) 10. Let be in radian measure such that 2 cos 2 k sin 3 0 are 6 and 2 2 2 . The solutions to the equation . Determine the value of k . sin 27 cos 27 sin 27 cos 63 k w 11. Let . Determine the sum k w p q . cos 27 sin 27 sin 63 cos 27 p q kw p 1 3 . Then the largest real solution for x can be written as in reduced 2 x q and simplified radical form with k , w , p , and q integers with q 0 . Determine the value 12. Let x 2 of the sum k w p q . NO CALCULATORS NO CALCULATORS NO CALCULATORS JUNIOR-SENIOR EIGHT PERSON TEAM COMPETITION ICTM REGIONAL 2016 DIVISION AA PAGE 3 OF 3 NO CALCULATORS 13. Determine the sum of all solution(s) for x when 2 x 7 9 4 x 74 . Express your answer as an integer or common or improper fraction reduced to lowest terms. 14. Determine the remainder when 32016 is divided by 11. 15. All angles are measured in radians. Determine the sum 1 1 1 1 1 1 cos 0 sin cos 2 sin 3 cos 4 sin 2015 cos 2016 . 4 4 4 4 4 4 16. Determine the sum of the first 19 terms of an arithmetic series whose tenth term is 41. 17. Determine the numeric coefficient of the 6th term of 3m 2n when expanded and 7 completely simplified in decreasing degree of m . 18. Consider the conic represented by the equation x 2 3 y 2 15 . Determine the exact eccentricity of this conic. Express your answer as a simplified rational expression. 19. Let g x ax11 bx5 cx 3 where a , b , and c are real valued constants. g 2 7 . Determine the value of g 2 . 20. Let x 2 y 2 6 xy and 0 y x . Determine the value of NO CALCULATORS NO CALCULATORS x y . x y NO CALCULATORS 2016 RAA Jr/Sr 8 Person Correct X School ANSWERS (Use full school name – no abbreviations) 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. (Must be this exact decimal.) 6.3 1. 2. 1 3. 15 : 23 4. 6 21 OR 1 6 6. 1 10 7. 9. (Must be this exact ratio.) 21 6 63 5. 8. 11. 10. 171 221 90 OR (Must be this reduced common fraction.) (Must be this reduced common fraction.) ("ounces" or "oz." optional.) 12. 13. 14. 15. 16. 17. 18. 171(Must be this reduced common fraction.) 221 19. 3 20. 2 9 2 3 (Must be this reduced common fraction.) 3 1 779 6048 6 3 13 2 (Must be this exact and simplified rational expression.) CALCULATING TEAM COMPETITION ICTM REGIONAL 2016 DIVISION AA 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) 1. Concave quadrilateral ABCD has vertices A 1.379,5.431 , B 2.543,3.791 , C 4.972, 3.145 and D 0.2473,1.895 . Determine the numerical area of this quadrilateral. 2. At the same time, two ladybugs leave two different points which are 240 meters apart. They fly directly back and forth between the two points without stopping at rates of 10 meters per second and 5 meters per second respectively. Determine the exact number of seconds that will elapse before they meet or pass for the second time. 3. The vertices of ABC are A 2, 4 , B 1, 2 and C 7,1 . Determine the length of the altitude from vertex B to the line containing side AC . 4. The number 2016! ends in k trailing zeros (the zeros to the right of the last non-zero digit in the expansion of 2016! ). Determine the largest possible positive integer n such that n ! ends in k 3 trailing zeros. 5. Ten committees of 8 people each must be chosen from a group of 90 people with no one serving on more than one committee. Determine the number of possible ways these committees can be chosen. Express your answer in scientific notation. 6. Let D lie on AB in right ABC with right angle at C such that CD AB . AD 4 and BD 25 . Let k be the arithmetic mean, w the geometric mean, and p the harmonic mean between AD and DB . Determine the sum k w p . 23 23 y 1! b 7. Let x ! a where a is a positive integer. Determine the value of b x 0 y 0 y ! when a is as small as possible. CALCULATING TEAM COMPETITION ICTM REGIONAL 2016 DIVISION AA PAGE 2 of 3 8. Determine the largest possible degree measure for such that 0 360 and 2 5 sin . 5 . 9. Let f x x x x2 x x x is an integer. 2x . Determine the sum of all real function value(s) f x when 10. Let g x 3 x 2 10 x 1 and f x g 2 x 1 . Determine the value of f 5. 11. Three pumps, A , B , and C , pump at different but constant non-zero rates and are available to pump water into or out of a swimming pool. With all three pumps working together, they would fill the empty pool in 2 hours and 6 minutes. If A and B are pumping water into the empty pool but C is accidently set to pump water out, it would take 20 hours to fill the pool. Determine the number of hours it would take for C to fill the empty pool alone. 1 12. Determine the exact numerical coefficient for the term involving x16 when x 4 2 is x expanded and completely simplified in decreasing degree of x . 10 13. Let f x 4 x 5 3 x 4 72 x3 54 x 2 320 x 240 . Let k be the largest positive zero and w be the smallest positive zero for f x . Compute the value of k w . CALCULATING TEAM COMPETITION ICTM REGIONAL 2016 DIVISION AA PAGE 3 of 3 14. A right circular cylinder has a base radius of x 4 , height of 2 x 3 , and numeric volume 20. Determine the sum of all possible value(s) for x . 15. A regular pentagon PENTA has perimeter 10. An isosceles triangle is created by drawing two segments from vertex P of the pentagon to the midpoints of two of the non-consecutive sides EN and TA of the pentagon and the segment joining those midpoints.. Determine the perimeter of this isosceles triangle. 16. The previous census determined that babies were born at the rate of 103 boys for every 100 girls. A family has three children. At this rate of birth, determine the probability the three children are two of one sex and one of the other. 17. Determine the standard deviation for the set of numbers 8, 9, 6,8,5, 6, 7, 6,9,8,8,5 . 18. Let k 3 2 10 i 1 1 i . Determine the value of k . 19. On an analog clock face, the hour hand is 6 inches long and the minute hand is 7.5 inches long, both attached at the center of the circular clock face. Determine the positive difference in number of inches between the arc length distances the tips of the hands travel between 1:00 pm and 6:00 pm, inclusive. 20. Let k 20 15 20 15 . Determine the value(s) of k . 2016 RAA Calculator Team Correct X School ANSWERS (Use full school name – no abbreviations) 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. 1. 2. 3. 4. 5. 8.559 48 OR 48.00 3.258 3.606 10 (Must be this answer 85 and in this form.) 31.40 7. 82.75 8. 116.6 9. 2.894 143.6 12. 13. 2029 6. 10. ("seconds" or "sec." optional.) 11. 14. 15. (Must be this decimal with trailing zero necessary.) 16. ("degrees" or " " optional.) 4.693 210 (Must be this exact answer.) 2.371 4.716 7.612 0.7498 OR .7498 17. 1.382 18. 5.994 19. 219.9 20. ("hours" or "hrs" optional.) 4.946 ("inches" or "in." optional.) FROSH-SOPH 2 PERSON COMPETITION ICTM 2016 REGIONAL DIVISION AA PAGE 1 OF 2 1. Given 4k 3 k 1 5 k 2 3 and 3 2w 1 13 4 w 5 . Determine the sum 2. Given ABC DEF . If A 3 x 29 , B 3 y 2 , D 5 y 3 and k w . E x 24 , determine the degree measure of F . 3. Two of the five lines 2 x 6 y 11 , 3 x y 9 , x 3 y 15 , 4 x 12 y 7 , and 9 x 3 y 20 are selected at random . Determine the probability that the selected lines are perpendicular. Express your answer as a common fraction reduced to lowest terms. 4. One square has a diagonal of length 20. A second square has a diagonal of length 24. Determine the exact positive difference between the perimeters of these squares. 5. Determine the sum of all integers that are solutions to both x 2 2 x 15 0 and x2 4 6. In the “triskadec” measurement system, one “triskafoot” is equivalent to 13 “triskainches”. Determine the number of cubic “triskafeet” in a rectangular solid with length 40 “triskainches”, width 10 “triskainches” and height 70 “triskainches”. Express your answer as a decimal rounded to the nearest tenth. 7. A circle is inscribed in a right triangle with legs of lengths 9 and 12. A second circle is inscribed in an equilateral triangle with sides of length 6. Determine the exact absolute value of the difference between the areas of the two circles. 8. Rich drove a distance of 100 miles on a trip. On a second trip, he drove a distance of 400 4 miles. His second trip took the number of hours the first trip took. His average speed 3 on the second trip was k times his average speed on the first trip. Determine the value of k. 9. Determine the number of rectangles in a standard 8 8 chess board that are not squares. 10. Determine the largest integer x for which both integers. 32 72 and represent positive x3 x5 FROSH-SOPH 2 PERSON COMPETITION ICTM 2016 REGIONAL DIVISION AA 30 EXTRA QUESTIONS 11-12 PAGE 2 OF 2 11. In a circle with radius 12. Jack and Jill can mow the golf course together in 6 hours and 40 minutes. Jack and Joe can mow the same golf course together in 7 hour and 30 minutes. Jill and Joe can mow 2 the same golf course together in 8 hours and 34 minutes. Determine how long, in 7 hours, it will take for all three together to mow the same golf course. units, let L represent the length of a minor arc of a circle formed by a central angle with measure 60 . In the coordinate plane, let D represent the distance between points 5, 2 and 10,10 . Determine the sum L D . ICTM Math Contest Freshman – Sophomore 2 Person Team Division AA FROSH-SOPH 2 PERSON COMPETITION ICTM 2016 REGIONAL DIVISION AA LARGE PRINT QUESTION 1 NO CALCULATORS ALLOWED 1. Given 4k 3 k 1 5 k 2 3 and 3 2w 1 13 4 w 5. Determine the sum k w . FROSH-SOPH 2 PERSON COMPETITION ICTM 2016 REGIONAL DIVISION AA LARGE PRINT QUESTION 2 NO CALCULATORS ALLOWED 2. Given ABC DEF . If A 3 x 29 , B 3 y 2 , D 5 y 3 and E x 24 , determine the degree measure of F . FROSH-SOPH 2 PERSON COMPETITION ICTM 2016 REGIONAL DIVISION AA LARGE PRINT QUESTION 3 NO CALCULATORS ALLOWED 3. Two of the five lines 2 x 6 y 11, 3 x y 9, x 3 y 15, 4 x 12 y 7, and 9 x 3 y 20 are selected at random . Determine the probability that the selected lines are perpendicular. Express your answer as a common fraction reduced to lowest terms. FROSH-SOPH 2 PERSON COMPETITION ICTM 2016 REGIONAL DIVISION AA LARGE PRINT QUESTION 4 NO CALCULATORS ALLOWED 4. One square has a diagonal of length 20. A second square has a diagonal of length 24. Determine the exact positive difference between the perimeters of these squares. FROSH-SOPH 2 PERSON COMPETITION ICTM 2016 REGIONAL DIVISION AA LARGE PRINT QUESTION 5 NO CALCULATORS ALLOWED 5. Determine the sum of all integers that are solutions 2 to both x 2 x 15 0 and x 2 4 FROSH-SOPH 2 PERSON COMPETITION ICTM 2016 REGIONAL DIVISION AA LARGE PRINT QUESTION 6 CALCULATORS ALLOWED 6. In the “triskadec” measurement system, one “triskafoot” is equivalent to 13 “triskainches”. Determine the number of “cubic triskafeet” in a rectangular solid with length 40 “triskainches”, width 10 “triskainches” and height 70 “triskainches”. Express your answer as a decimal rounded to the nearest tenth. FROSH-SOPH 2 PERSON COMPETITION ICTM 2016 REGIONAL DIVISION AA LARGE PRINT QUESTION 7 CALCULATORS ALLOWED 7. A circle is inscribed in a right triangle with legs of lengths 9 and 12. A second circle is inscribed in an equilateral triangle with sides of length 6. Determine the exact absolute value of the difference between the areas of the two circles. FROSH-SOPH 2 PERSON COMPETITION ICTM 2016 REGIONAL DIVISION AA LARGE PRINT QUESTION 8 CALCULATORS ALLOWED 8. Rich drove a distance of 100 miles on a trip. On a second trip, he drove a distance of 400 miles. His second trip 4 took the number of hours 3 the first trip took. His average speed on the second trip was k times his average speed on the first trip. Determine the value of k . FROSH-SOPH 2 PERSON COMPETITION ICTM 2016 REGIONAL DIVISION AA LARGE PRINT QUESTION 9 CALCULATORS ALLOWED 9. Determine the number of rectangles in a standard 8 8 chess board that are not squares. FROSH-SOPH 2 PERSON COMPETITION ICTM 2016 REGIONAL DIVISION AA LARGE PRINT QUESTION 10 CALCULATORS ALLOWED 10. Determine the largest integer x for which both 32 72 and represent x3 x5 positive integers. FROSH-SOPH 2 PERSON COMPETITION ICTM 2016 REGIONAL DIVISION AA EXTRA LARGE PRINT QUESTION 11 CALCULATORS ALLOWED 11. In a circle with radius 30 units, let L represent the length of a minor arc of a circle formed by a central angle with measure 60. In the coordinate plane, let D represent the distance between points 5, 2 and 10,10 . Determine the sum L D . FROSH-SOPH 2 PERSON COMPETITION ICTM 2016 REGIONAL DIVISION AA EXTRA LARGE PRINT QUESTION 12 CALCULATORS ALLOWED 12. Jack and Jill can mow the golf course together in 6 hours and 40 minutes. Jack and Joe can mow the same golf course together in 7 hour and 30 minutes. Jill and Joe can mow the same golf course together in 8 hours 2 and 34 minutes. Determine 7 how long, in hours, it will take for all three together to mow the same golf course. 2016 RAA Fr/So 2 Person Team School ANSWERS (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 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 3 Score (to be filled in by proctor) 95 ("degrees" or " " optional.) 8 2 (Must be this exact answer.) 2 5 12 (Must be this reduced common fraction.) 12.7 (Must be this decimal, :"triskafeet" optional.) 1092 (Must be this integer, comma usage and "rectangles" optional.) 6 3 (Must be this exact answer.) 13 TOTAL SCORE: (*enter in box above) Extra Questions: 11. 12. 13. 14. 15. 23 5 ("hours" optional.) * Scoring rules: Correct in 1st minute – 6 points Correct in 2nd minute – 4 points Correct in 3rd minute – 3 points PLUS: 2 point bonus for being first In round with correct answer JUNIOR-SENIOR 2 PERSON COMPETITION ICTM 2016 REGIONAL DIVISION AA PAGE 1 OF 2 1. The point P is located on the x axis and the point Q is located on the y axis . Each point is 10 units from the point 6, 8 . Determine the largest possible length of PQ . 2. Let 10log3log 2 2k and 4 w 8w2 2 2 w . Determine the sum k w . 3. In TRI , TR 8 and RI 10 . Determine the value of the expression answer as a common or improper fraction reduced to lowest terms. sin T . Express your sin I 4. On a number line, the distance between 5 and a number x is 1 less than the distance between 2 and x . Determine the value of x . 5. i 1 . Let 1i 2i 2 3i 3 4i 4 a bi where a and b are real numbers. Let 4 3 16 3 3 2 2 3 54 c 3 d in simplified and reduced radical form and where d is a positive integer. Determine the value a 2b 3c 4d . 6. Let P be the point that is the vertex of the parabola y 3x 2 24 x 50 . Let Q be the point of intersection of the lines with equations 4 x 3 y 2 and y 3 x 1 . Determine the length of PQ . 1 7. The n term of 6 sequences are described as follows: tn 5n 2 , tn 8 , tn 3 2n , 2 2n 3 n3 tn , tn , and tn 8 2n . Two of these sequences are selected at random. 7 n2 Determine the probability that both are arithmetic sequences. Express your answer as a common fraction reduced to lowest terms. n th 8. Determine the sum of all integers that are solutions for x to both log 3 7 2 x 2 and x 2 10 x 24 . 3 x 9. Let f x 2 x 1 x 2 3x 14 for x 5 for 5 x 5 . Determine the sum of all x for which f x 4 . for x 5 10. Two standard and fair 6-sided die with faces numbered uniquely with integers 1 to 6 inclusive are tossed. Determine the probability that the sum of the two numbers showing is greater than the product of the same two numbers. Express your answer as a common fraction reduced to lowest terms. JUNIOR-SENIOR 2 PERSON COMPETITION ICTM 2016 REGIONAL DIVISION A EXTRA QUESTIONS 11-12 PAGE 2 OF 2 11. Let 1 2 3 2016 k w when in reduced and simplified radical form. Determine the sum k w . 12. Let cos 2 x cos x with x measured in radians. The greatest solution for x when this equation is solved over the interval 2 , 0 is k . Determine the value of k . Express your answer as a common or improper fraction reduced to lowest terms. ICTM Math Contest Junior – Senior 2 Person Team Division AA JUNIOR-SENIOR 2 PERSON COMPETITION ICTM 2016 REGIONAL DIVISION AA LARGE PRINT QUESTION 1 NO CALCULATORS ALLOWED 1. The point P is located on the x axis and the point Q is located on the y axis. Each point is 10 units from the point 6, 8. Determine the largest possible length of PQ . JUNIOR-SENIOR 2 PERSON COMPETITION ICTM 2016 REGIONAL DIVISION AA 2. Let 10 LARGE PRINT QUESTION 2 NO CALCULATORS ALLOWED log3 log 2 and 2k 4 8 2 w w2 2w . Determine the sum k w . JUNIOR-SENIOR 2 PERSON COMPETITION ICTM 2016 REGIONAL DIVISION AA LARGE PRINT QUESTION 3 NO CALCULATORS ALLOWED 3. In TRI , TR 8 and RI 10. Determine the value of sin T the expression . sin I Express your answer as a common or improper fraction reduced to lowest terms. JUNIOR-SENIOR 2 PERSON COMPETITION ICTM 2016 REGIONAL DIVISION AA LARGE PRINT QUESTION 4 NO CALCULATORS ALLOWED 4. On a number line, the distance between 5 and a number x is 1 less than the distance between 2 and x . Determine the value of this number x . JUNIOR-SENIOR 2 PERSON COMPETITION ICTM 2016 REGIONAL DIVISION AA LARGE PRINT QUESTION 5 NO CALCULATORS ALLOWED 5. i 1. Let 2 3 4 1i 2i 3i 4i a bi where a and b are real numbers. Let 3 3 3 3 4 16 3 2 2 54 c d in simplified and reduced radical form and where d is a positive integer. Determine the value a 2b 3c 4d . JUNIOR-SENIOR 2 PERSON COMPETITION ICTM 2016 REGIONAL DIVISION AA LARGE PRINT QUESTION 6 CALCULATORS ALLOWED 6. Let P be the point that is the vertex of the parabola 2 y 3 x 24 x 50. Let Q be the point of intersection of the lines with equations 4 x 3 y 2 and y 3 x 1. Determine the length of PQ . JUNIOR-SENIOR 2 PERSON COMPETITION ICTM 2016 REGIONAL DIVISION AA LARGE PRINT QUESTION 7 CALCULATORS ALLOWED 7. The n term of 6 sequences are described as follows: th tn 5n 2, t n 3 2 , n 1 tn 8 , 2 2n 3 tn , 7 n3 tn , and tn 8 2n. n2 n Two of these sequences are selected at random. Determine the probability that both are arithmetic sequences. Express your answer as a common fraction reduced to lowest terms. JUNIOR-SENIOR 2 PERSON COMPETITION ICTM 2016 REGIONAL DIVISION AA LARGE PRINT QUESTION 8 CALCULATORS ALLOWED 8. Determine the sum of all integers that are solutions for x to both log 3 7 2 x 2 and 2 x 10 x 24. JUNIOR-SENIOR 2 PERSON COMPETITION ICTM 2016 REGIONAL DIVISION AA 9. Let 3 x f x 2x 1 x 2 3x 14 LARGE PRINT QUESTION 9 CALCULATORS ALLOWED for x 5 for 5 x 5 for x 5 Determine the sum of all x for which f x 4. JUNIOR-SENIOR 2 PERSON COMPETITION ICTM 2016 REGIONAL DIVISION AA LARGE PRINT QUESTION 10 CALCULATORS ALLOWED 10. Two standard and fair 6-sided die with faces numbered uniquely with integers 1 to 6 inclusive are tossed. Determine the probability that the sum of the two numbers showing is greater than the product of the same two numbers. Express your answer as a common fraction reduced to lowest terms. JUNIOR-SENIOR 2 PERSON COMPETITION ICTM 2016 REGIONAL DIVISION A EXTRA LARGE PRINT QUESTION 11 CALCULATORS ALLOWED 11. Let 1 2 3 2016 k w when in reduced and simplified radical form. Determine the sum k w . JUNIOR-SENIOR 2 PERSON COMPETITION ICTM 2016 REGIONAL DIVISION A EXTRA LARGE PRINT QUESTION 12 CALCULATORS ALLOWED 12. Let cos 2 x cos x with x measured in radians. The greatest solution for x when this equation is solved over the interval 2 ,0 is k . Determine the value of k. Express your answer as a common or improper fraction reduced to lowest terms. 2016 RAA Jr/Sr 2 Person Team School Total Score (see below*) = ANSWERS (Use full school name – no abbreviations) 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 Score (to be filled in by proctor) 20 1. 1 2. 5 4 3. 2 4. (Must be this reduced improper fraction.) 33 5. 5 6. 7. 1 5 (Must be this reduced common fraction.) 11 36 (Must be this reduced common fraction.) 65 2 8. 9. 10. TOTAL SCORE: (*enter in box above) Extra Questions: 11. 12. 13. 14. 15. 2 3 14131 OR 2 3 N/A N/A N/A (Must be this reduced common fraction,) * Scoring rules: Correct in 1st minute – 6 points Correct in 2nd minute – 4 points Correct in 3rd minute – 3 points PLUS: 2 point bonus for being first In round with correct answer ORAL COMPETITION ICTM REGIONAL 2016 DIVISION AA 1. Discuss the product (or composite) of two rotations. 2. Six reflections are performed in succession. The order is that of the axes as given here: First: y = x +1 Second: y = x + 3 Third: x = 0 Fourth: x = 5 Fifth: y = x - 4 Sixth: y = x - 6 Discuss the product (or composite) of these six reflections. 3. Let a transformation T be given by the equations x¢ = y + 4, y¢ = x -10. (a) Explain why T is an isometry. (b) Analyze this isometry, giving details. ORAL COMPETITION ICTM REGIONAL 2016 DIVISION AA EXTEMPORANEOUS QUESTIONS Give this sheet to the students at the beginning of the extemporaneous question period. STUDENTS: You will have a maximum of 3 minutes TOTAL to solve and present your solution to these problems. Either or both the presenter and the oral assistant may present the solutions. 1. A reflection has the property that if it is composed with itself, the resulting isometry is the identity. Are there any other isometries that have this property? 2. What is the composite of a glide-reflection with itself? 3. Reflections are sometimes called “the building blocks of the isometries,” much like the relation between the prime numbers and the set of integers. Explain this description. ORAL COMPETITION ICTM REGIONAL 2016 DIVISION AA PAGE 1 OF 2 JUDGES’ SOLUTIONS 1. Discuss the product (or composite) of two rotations. Solution: The product is either a rotation or a translation, depending on whether the sum of the angles is or is not a multiple of 360° . The following table summarizes the possibilities: SUM OF ANGLE MEASURES Multiple of 360° Not a multiple of 360° C E N T E R S Same (A) Identity Rotation about A through sum of given angles Different (First A, then B) Translation with vector 2AB Rotation through sum of given angles. The center is the intersection of the perpendicular bisectors of PP¢ and QQ¢, where P¢ and Q¢ are the images of two chosen points P and Q under the product of the two rotations. 2. Six reflections are performed in succession. The order is that of the axes as given here: First: y = x +1 Second: y = x + 3 Third: x = 0 Fourth: x = 5 Fifth: y = x - 4 Sixth: y = x - 6 Discuss the product (or composite) of these six reflections. Solution: The first two give the translation T1 with vector < -2, 2 >. The third and fourth give the translation T2 with vector < 10, 0 >. The fifth and sixth give the translation T3 with vector < 2, - 2 >. Since translations commute with each other the product T3T2T1 is the same as T3T1T2 , but this is just T2 since T1 and T3 are inverses of each other. The net result is T2. ORAL COMPETITION ICTM REGIONAL 2015 DIVISION AA PAGE 2 OF 2 JUDGES’ SOLUTIONS 3. Let a transformation T be given by the equations x¢ = y + 4, y¢ = x -10. (a) Explain why T is an isometry. (b) Analyze this isometry, giving details. Solution: (a) One answer: The equations fit the form (iii) of Property XV in the reference, with a = 90°, h = 4, k = 10 , so T is an opposite isometry. Second answer: Let P = (r, s) and Q = (u, v) be distinct arbitrary points. Then P¢ = (s + 4, r -10) and Q¢ = (v + 4, u -10). The distance P¢Q¢ = [(s + 4) - (v + 4)]2 +[(r -10) - (u -10)]2 = (s - v)2 + (r - u)2 = PQ, so T preserves distance and is an isometry by definition. (b) Solving for fixed points we get x = y + 4 and y = x -10 . These equations give x = x - 6, so there are no solutions and thus no fixed points and T is neither a rotation nor a reflection. Since the equations are not of the form for a translation, T must be a glide-reflection. Choose two points, such as P(0, 0) and Q(10,-4). Then P¢ = (4,-10) and Q¢ = (0, 0). The midpoints M (2,-5) of PP¢ and N(5,-2) of QQ¢ lie on the axis of the reflection part and the equation of this line is y = x - 7. The glide-vector is MM ¢ = < -1- 2, - 8 - (-5) > = < -3,-3 > . (Or use NN ¢ = < 2 - 5, - 5 - (-2) > = < -3,-3 > .) Note that the glide-vector is parallel to the reflection’s axis, as it should be. ORAL COMPETITION ICTM REGIONAL 2015 DIVISION AA PAGE 1 OF 1 JUDGES’ SOLUTIONS EXTEMPORANEOUS QUESTIONS Give this sheet to the students at the beginning of the extemporaneous question period. STUDENTS: You will have a maximum of 3 minutes TOTAL to solve and present your solution to these problems. Either or both the presenter and the oral assistant may present the solutions. 1. A reflection has the property that if it is composed with itself, the resulting isometry is the identity. Are there any other isometries that have this property? Solution: The identity, of course, but also a half-turn (180 degree rotation). No others. 2. What is the composite of a glide-reflection with itself? Solution: Since the translation part and the reflection part commute with each other, the composite is the translation composed with itself, i.e., the translation whose vector is twice that of the given translation part. 3. Reflections are sometimes called “the building blocks of the isometries,” much like the relation between the prime numbers and the set of integers. Explain this description. Solution: Every isometry is either a reflection or the composite of two or three reflections, so the reflections are, in some sense, primary.