Preliminary Round Question Booklet

Transcription

1 First Annual Pi Day Mathematics Cometition Question Booklet 016

2 goes on and on, and e is just as cursed. Iwonder,howdoes begin When its digits are reversed? -MartinGardner

3 Pi Day Mathematics Cometition Rules of the Cometition This test consists of 60 multile choice questions. Each question is followed by answers marked A, B, C, D, and E. Only one of these is correct. Mark your answer to each roblem on the rovided answer sheet. For each question, blacken the circle corresonding to at most once answer choice. Comletely erase errors and any stray marks. Only answers roerly marked on the answer sheet will be graded. SCORING: You will receive one oint for each correct answer. You receive 0 oints for incorrect answers and for questions left unanswered. Contestants may not consult textbooks, notes, other eole (aart from teammates), electronic devices (including calculators, mobile hones, etc.), or any other resources during the test. Figures are not necessarily drawn to scale. Before beginning the test, lease make sure to write the name of your school, your team s name (if you have one) and the names of all members of the team on the answer sheet. You have hours to comlete the questions. At the end of the hours, each team should submit one answer sheet.

4 CMU-Q Pi Day Cometition In general, the wholesale rice of an item deends on the order size. The following grah shows this relation for a articular item. If c a =4,thenwhatisc b? Sale Price ($) c b a 0 0 Quantity (a) 6 (b) 8 (c) 1 (d) 14 (e) 16. If x <x,thenwhatisthelargestintegervalueofx +7?. If (a) 6 (b) 7 (c) 8 (d) 9 (e) x x +0.0 =,thenxequals 4 (a) 100 (b) 10 (c) 10 (d) 11.8 (e) The sum to infinity of the terms of an infinite geometric rogression is 9. The sum of the first two terms is 8. The first term of the rogression is (a) (b) 4 (c) (d) 6 (e) 7. A woman has 100,000 QR to invest. If she invests 0,000 QR at % and 40,000 QR at 6 %, at what rate is the remainder invested for the total interest to be 6,000 QR? (a) 6. % (b) 6.8 % (c) 7.0 % (d) 7. % (e) 7. % 6. The vertices of a arallelogram are located at oints A =(, 1), B =(, ), C =(, ), and D =(a, b). If BD is one of the diagonals of the arallelogram, find the length of this diagonal. (a) 1 (b) (c) (d) 4 (e) 1

5 CMU-Q Pi Day Cometition A number which when divided by 10 leaves a remainder of 9, when divided by 9 leaves aremainderof8,whendividedby8leavesaremainderof7,etc.,downtowhere,when divided by, it leaves a remainder of 1 is: (a) 719 (b) 149 (c) 19 (d) 19 (e) none 8. If a + b 4 + c 6 =0,thenwhichofthefollowingisthevalueofa +b +c? (a) 8 (b) 1 (c) 0 (d) -1 (e) Refer to the figure given below. Let ABCD be a rectangle with AD = 1 cm and AE = EB =cm. If FE = x cm, then find x. D C 1 F x A E B (a) (b) (c) (d) (e) If we are given that log 0 8=.6941, which of the following are we not able to calculate without the use of a scientific calculator? (a) log 0 (b) log 4 10 (c) log 0 (/) (d) log 9 (e) log In a meeting, articiants are o ered tea, milk, or co ee. 7 of the articiants do not drink tea, 6 of the articiants do not drink milk, and of the articiants do not drink co ee. Moreover, 4 of the articiants drink neither tea nor milk, of the articiants drink neither tea nor co ee, and of the articiants drink neither milk nor co ee. One erson drinks none of the three beverages and no one drinks all three at the same time. How many ersons are at the meeting? (a) 10 (b) 11 (c) 1 (d) 1 (e) 14

6 CMU-Q Pi Day Cometition In a street, house numbers are successive odd numbers on one side and successive even numbers on the other side. If the numbers are increasing from left to right, and A, B, C, D are the house numbers of the indicated houses with A B =1,whatisC D? Odd A C Left Right B D Even (a) 9 (b) 11 (c) 1 (d) 1 (e) Let f(x) = x + u x +1 What is u? x 9 and (f f)(x) =,where denotes comosition of functions. x (a) - (b) - (c) -1 (d) 0 (e) If each number in a set of twenty numbers is increased by 100, the arithmetic mean (average) of the numbers (a) remains the same (b) increases by 0 (c) increases by 100 (d) increases by 10 (e) increases by 1. When a clock shows :1, the angle between the hour hand and the minute hand is: (a) 1 (b) 0 (c). (d) (e) The equation x 4 +4x +0x +7=0has (a) no negative real roots (b) no ositive real roots (c) no real roots (d) 1 ositive and 1 negative real roots (e) ositive and negative real roots

7 CMU-Q Pi Day Cometition Mr. A owns a house worth 100,000 QR. He sells it to Mr. B at a 10% rofit based on the worth of the house. Mr. B sells the house back to Mr. A at a 10% loss. Then, (a) A makes 10,000 QR on the deal (b) A loses 10,000 QR on the deal (c) A makes 11,000 QR on the deal (d) A loses 9,000 QR on the deal (e) A comes out even 18. A flower sho has some number of roses for Valentine s day. If roses are groued in fives, then there are extra roses. If roses are groued in sevens, then there are extra roses. What is the minimum ossible number of roses that the flower sho may have? (a) 1 (b) 17 (c) 7 (d) 7 (e) Let f(x) =mx + n. Assume that f 1 () = 4 and f 1 () =, where f 1 (x) istheinverse of f(x). What is the roduct of m and n? (a) -7 (b) -6 (c) - (d) (e) 6 0. The equation q x + x + x +...=issatisfiedwhenx is (a) 1 (b) (c) (d) (e) 4 1. The roduct log b a log a b is equal to: (a) 1 (b) a (c) b (d) ab (e) log ab. Given 1 oints in a lane no three of which are collinear, the number of lines they determine is (a) (b) 66 (c) 84 (d) 100 (e) 144 4

8 CMU-Q Pi Day Cometition 016. A oint is selected at random inside an equilateral triangle. From this oint, erendiculars are drawn to each side of the triangle. The sum of these erendiculars is (a) half the erimeter of the triangle (b) equal to the length of a side of the triangle (c) equal to the altitude of the triangle (d) greatest when the oint is the median (e) least when the oint is the median 4. The sum of the digit number ABC and the two digit number AB is 9. What is the sum A + B + C? (a) 7 (b) 9 (c) 11 (d) 1 (e) 19. A bag contains 10 white marbles and 10 black marbles. A erson randomly selects a marble from the bag, laces it on a table, and continues doing this until the bag is emty. What is the robability that the rd marble taken out is white? (a) 1 4 (b) 1 (c) 1 (d) (e) 4 6. If x + x + x x =,thenx is (a) 1 (b) 4 (c) 1 (d) (e) 7. If x 016 isdividedbyx + 1, the remainder is (a) - (b) -1 (c) 0 (d) 1 (e)

9 CMU-Q Pi Day Cometition An isosceles triangle is a triangle with (at least) two equal sides. Let ABC be an isosceles right triangle. If BD = AC =cm, OA = OC and OD = x cm, then x equals D x C O A B (a) (b) (c) 4 (d) 4 (e) 9. Let x =( ) 4, y = (4),andz =( 1 ). Which of the following is the correct ordering of x, y, andz? (a) z<x<y (b) z<y<x (c) y<x<z (d) x<y<z (e) x<z<y 0. If x +1isafactoroff(x) =x x + x + ax, thena equals (a) 1 (b) 1 (c) 1 (d) 1 (e) 1 1. It is given that one root of the quadratic equation x + ax + b =0isandonerootof the equation x + cx + d =0is value of a c.. If the other roots of the equations are equal, find the (a) 8 (b) 6 (c) (d) (e) 1. Find the sum of the smallest and largest integer values of the exression (a) (b) 6 (c) 7 (d) 8 (e) 9 0 x +6 + x +8.. The di erence of the roots of x 10x is (a) 47 (b) 10 (c) 47 (d) 10 (e) 10 6

10 CMU-Q Pi Day Cometition The digit 1 is used 689 times in the age numbers of a book. How many ages does the book have? (a) 104 (b) 114 (c) 114 (d) 14 (e) 14. What is the arithmetic mean of 6 and 6+? (a) 6 (b) 1 (c) (d) 6 (e) A farmer has 6 di erent size containers:, 9, 1,, and 4 liters. He has filled some of these containers with sunflower oil and some others with olive oil. He has one emty container and the total amount of the sunflower oil in the containers is 4 times the total amount of olive oil in the containers. What is the volume of the emty container? (a) (b) 9 (c) 1 (d) 1 (e) 7. Refer to the figure given below. If R 1 + R =6cmandR R 1 = k cm, find the area between these two concentric circles. R R 1 (a) k cm (b) 4 k cm (c) 6 k cm (d) 8 k cm (e) 9 k cm 8. Find all real numbers for which (x )(x +4) < 0. (x 4) (a) (, ) [ (, ) (c) ( 1, ) [ (, 1) (e) [ (b) (, 0) [ (, ) (d) (, ), ] 9. The bottom, side, and front areas of a rectangular box are known. The roduct of these areas equals: (a) the volume of the box (b) the square root of the volume (c) twice the volume (d) the square of the volume (e) the cube of the volume 7

11 CMU-Q Pi Day Cometition Fadhel looked at the wall clock when he started reading his newsaer. When he checked the clock again after he finished reading, he noticed that the hour and minute hands have switched ositions. For how long did he read the newsaer? (a) 40 min (b) 0 1 min (c) 4 1 min (d) 1 min (e) 60 min 41. Maha just got a book from the library. She realizes that if she starts reading it today and if every day she reads five ages more than she read the revious day, she could read the entire book in 6 days. If she reads 1/ of the book in three days using this rule, how many ages are there in the book? (a) 16 (b) 19 (c) 1 (d) 14 (e) 1 4. In a geometric rogression whose terms are all ositive, any term is equal to the sum of the next two terms. Then the common ratio is: (a) 1 (b) +1 (c) (d) + (e) 4. Refer to the figure given below. Note that BD =4cm, FC =10cm, EF = FT, and DF = x cm. If the line segments AB and TE are arallel, then x is A T B D x 4 F 10 C E (a) 4 (b) 6 (c) 8 (d) 10 (e) Let a, b, and c be ositive real numbers. If a + b c inequalities is absolutely correct < a c +1, then which of the following (a) c<b (b) b<c (c) a<b (d) b<a (e) a<c 8

12 CMU-Q Pi Day Cometition Let log 10 =a and log 10 =b. Find the value of log 10 7 in terms of a and b. (a) b a (b) a b (c) a b (d) a +b (e) a +b 46. If the radius of a circle is increased 100%, the area of the circle is increased (a) 0% (b) 100% (c) 00% (d) 00% (e) 400% 47. The sum to infinity of is 44 (a) 1 (b) 4 (c) Which of the following is the simlest exression equal to a (a) 1 + a (b) 1 a (c) a (d) a (e) a 1 (d) 4 1 (e) A six-sided die is loaded in a way that each even face is three times as likely as each odd face. All even faces are equally likely and all odd faces are equally likely. What is the robability that the outcome of a die roll will be a face showing or less? (a) 1 (b) 1 (c) 14 (d) 1 (e) Let x =sin8, y =tan17, z =cos60,andw =cot7. What is the correct sign of the given trigonometric quantities x, y, z, andw? (a) +,, +, (b),,, + (c) +,,, + (d),,, (e) +,,, 1. Two equal circles in the same lane cannot have the following number of common tangent lines: (a) 1 (b) (c) (d) 4 (e) all the above are ossible values for the number of common tangents. 9

13 CMU-Q Pi Day Cometition 016. Refer to the figure given below. Let O be the center of the given circle and let AT be a line segment tangent to the circle at oint T. If AT =cmandm(oat )=4,find the length of the circular arc BT in centimeters. A 4 T B O (a) (b) (c) 4 (d) 4 (e) 6 sincos7+ cossin7. Let (0, 90 )andsin =. What is in degrees? 4cos84cos6 (a) 1 (b) 1 (c) 18 (d) 0 (e) Ages of two sisters are roortional to 6 and 8 now. After 6 years, their ages will be roortional to 4 and resectively. What is the age of the older sister now? (a) 6 (b) 4 (c) 0 (d) 18 (e) 16. Let A = {a, b, c, d, e}. Find the number of -element subsets that contain d. (a) 4 (b) (c) 6 (d) 7 (e) 8 6. Alice and Bob take turns fliing a fair coin. The first erson to get heads wins. If the game starts with Alice, what is the robability that Bob wins the game? (a) 1 4 (b) 1 (c) 1 (d) (e) 4 7. Which of the following numbers is a rational number? (a) +1 (b) 1 (c) 1 (d) 1+ (e) 10

14 CMU-Q Pi Day Cometition What is the smallest number divisible by 10 and such that the factorial of this number is divisible by 10 1? (a) 40 (b) 0 (c) 60 (d) 80 (e) Refer to the figure given below and find the area of the triangle EBD. Note that m(bac) =90 (right angle), AB =7cm, EC =4cm,and BD = DC. A 7 E 4 B D C (a) (b) 4 (c) 7 (d) 9 (e) Two classes of high school students take the same test. Once class of 0 students had an average score of 8%; the other class of 0 students had an average score of 70%. What is the average score for all students in both classes? (a) 76% (b) 77% (c) 78% (d) 79% (e) 80% 11

15 First Annual Pi Day Mathematics Cometition Question Booklet 016