FIFTY-SIXTH ANNUAL MICHIGAN MATHEMATICS PRIZE COMPETITION

Transcription

1 FIFTY-SIXTH ANNUAL MICHIGAN MATHEMATICS PRIZE COMPETITION sponsored by The Mihigan Setion of the Mathematial Assoiation of Ameria Part I Tuesday Otober 2, 202 INSTRUCTIONS (to be read aloud to the students by the supervisor or protor). Your answer sheet will be graded by mahine. Carefully read and follow the instrutions printed on the answer sheet. Chek to ensure that your six-digit ode number has been reorded orretly. Do not make alulations on the answer sheet. Fill in irles ompletely and darkly. 2. Do as many problems as you an in the 00 minutes allowed. When the protor asks you to stop, please quit working immediately and turn in your answer sheet. 3. Consider the problems and responses arefully. You may work out ideas on srath paper before seleting a response. 4. You may be unfamiliar with some of the topis overed in this examination. You may skip over these and return to them later if you have time. Your sore on the test will be the number of orret answers. You are advised to guess an answer in those ases where you annot determine an answer.. For eah of the questions, five different possible responses are provided. In some ases the fifth alternative is (E) none of the others. If you believe none of the first four alternatives is orret, hoose response (E). 6. Any sientifi or graphing alulator is permitted on Part I. (Unaeptable mahines inlude omputers, PDAs, poket organizers, ell phones, and similar devies. All problems will be solvable with no more tehnology than a sientifi alulator. The Exam Committee makes every effort to struture the test to minimize the advantage of a more powerful alulator.) No other devies are permitted. 7. No one is permitted to explain to you the meaning of any question. Do not ask anyone to violate the rules of the ompetition. If you have questions onerning the instrutions, ask them now. 8. You may now open the test booklet and begin.

2 . Consider five boxes with the dimensions listed below. Whih one has the largest surfae area? A: B: 8 8 C: 4 6 D: E: All have the same surfae area 2. log 8 (log 2 6) = A: 2 3 B: 3 2 C: 2 D: 3. In the xy-plane, the set of points that are equidistant from the point (0, ) and the line y = 0 forms A: a parabola B: a hyperbola C: a irle D: an ellipse 4. What is the equation of the straight line through the point (2, 3) that is parallel to 2x+3y =? A: 2x 3y = 3 B: 2x + 3y = C: 3x + 2y = D: 2x + 3y = 0. The larger square in the figure below has side 7. The orners of the smaller square touh the sides of the larger square, as shown. What is the area of the irle insribed in the smaller square? A: 2π B: 49π/4 C: 2π D: 2π/4 6. The faes of two die are numbered in a non-standard manner. One has one, two 2s, and three 3s. The other has three 4s, two s, and one 6. The die are arefully balaned so that eah fae is equally likely to land on top. When the pair of die are rolled what is the probability that the sum of the two die is 7? A: 7/8 B: /8 C: /3 D: /4 7. For integer values of n with 0 n 202, the minimum value of (202 n)!(n!) 2 ours when n is equal to A: 006 B: 67 B: 670 D: 4 E: 44

3 8. If y = ax bisets the area of the triangle in the xy-plane bounded by x + y = 0, 3x y = 0, and the x-axis, what is a? A: B: 3 C: 4 D: The arithmeti mean of 8 and 8 exeeds the geometri mean of these numbers by A: 3 B: C: D: 2 0. From a pile of pennies, nikels, dimes, and quarters I grab a handful of oins. The ombined value of the fifteen oins is $.0, and there are 3 more quarters than nikels. How many dimes do I have? A: B: 2 C: 3 D: There is no solution to this problem.. Suppose that f(a + b) 2f(a) + f(a b) = 2f(b ) for all integers a and b. Suppose also that f() = 2. Then f( ) + f(0) = A: 2 B: 4 C: 2 D: 4 2. We have two similar solids. The volume of the larger one is 6 2 times the volume of the smaller one. The sum of their surfae areas is 26 square inhes. What is the surfae area of the smaller solid? A: 6 square inhes B: 24 square inhes C: 48 square inhes D: 8 square inhes 3. For what values of a does x + a + x + 2 = have an infinite number of solutions? A: 7 and 3 B: 7 only C: 3 and 7 D: 7 only E: No suh value of a exists. 4. A snail is at the bottom of a well, feet deep. On day, it starts limbing up the side. That night it rests and slips down one foot. It repeats this proess, limbing up 3 feet eah day and sliding down one foot eah night until it reahes the top of the well. On what day number with the snail reah the top? A: 39 B: 7 C: 8 C:. Two asteroids in spae hurtling diretly toward eah other. Initially, they are 202 kilometers apart. Asteroid A travels at a rate of 720 kilometers per hour. Asteroid B is travels at a rate of 280 kilometers per hour. How many kilometers apart are the two asteroids one minute before they rash? A: B: 60 ( )60 D: 202 C: ( + i ) 4,444, = 2 A: B: C: D: 2 2,222, ,222,222

4 7. One ar goes around a trak in 30 seonds, and a seond ar goes around the same trak in 0 seonds. How long will it take the faster ar to gain one lap? A: 7 seonds B: 0 seonds C: 300 seonds D: 00 seonds 8. The fration is expanded in base b where b is an integer with b 2. The expansions need 4 not terminate. If only the digits 0 and are required, then the possible values of b are A: 2, 4 B: 2, 4, C: 3, 4, D: 3, 4, 6, 7 E: There are no possible values for b. 9. The polynomial x 3 + ax 2 + bx + has three distint real roots. Two of them are and 2. Whih of the following statements must be true? A: = 2a + 6 B: = 2a 6 C: a + b + = D: a and are either both positive or both negative 20. How many integers satisfy the inequalities x(x 6)(x 8) 0 and A: B: 2 C: 3 D: 4 x x 2 3x + 2 0? 2. Suppose a = 2, a 2 = and a n+2 + a n+ + a n = 7 for n. 3 Find a k. k= A: 2 B: 29 C: 262 D: The numbers, 2, 3, 4,, 6, 7, 8, 9 are arranged in the 9 small squares in the figure below so that eah square ontains a different number. The numbers obey the inequalities indiated on the boundaries between adjaent squares. What number must loated in the square marked X? < > < > X > < < > < > < < A: 9 B: 8 C: 7 D: When rolling three fair, ubial die, the probability of getting doubles (exatly two die with the same value) is A: 2 B: 36 C: 36 D: 8 E: 6

5 24. A pizza parlor offers a basi pizza for $2.9. Additional toppings ost $.00 eah. Customers are required to pay sales tax of 6% as well as a % servie fee. [The basis for the servie fee is the food harge only, not inluding the sales tax.] A group of friends wants to order a pizza, but they have only $20 between them to pay for everything. How many toppings an they afford to put on their pizza? A: 6 B: C: 4 D: 3 2. Consider a right triangle whose legs have lengths a and b, and whose hypotenuse has length. Of all the points on the hypotenuse, the one losest to the opposite vertex lies what distane from that vertex? a b A: ab B: 2ab C: a + b D: ab 26. Mr. Sott, his sister, his son, and his daughter are tennis players. The people mentioned in the following fats refer to these four people.. The best player s twin and the worst player are of opposite sex. 2. The best player and the worst player are the same age. Whih one of the four is the worst player? A: Son B: Daughter C: Sister D: Mr. Sott E. The given information is ontraditory. 27. Two ards are drawn at random from a standard dek of 2. What is the probability that at least one of the drawn ards is an ae? A: B: 2 3 C: 2 26 D: A square of side length is partitioned into four trapezoids and a entral square of equal area as indiated in the diagram. The length of the lines onneting the orners of the outer square and the entral square is A: 2 ( ) B: C: D: 2 3 E: If tan θ =, then the possible values of sin θ are 0 A: and B: 0 only C: only 0 D: and 0 E: and 0

6 30. A planar figure is said to be tiled by objets if it is exatly overed by them with no overlaps exept at orners or along edges. You are given a set of one hundred 2 2 square tiles and one hundred 3 3 square tiles. Whih of the following sized retangles annot be tiled using squares from the set? You are allowed to use both 2 2 squares and 3 3 squares in your tiling. You are also allowed to use only 2 2 squares or only 3 3 squares in your tiling. A: 6 B: 7 8 C: 8 8 D: The letters a, b,, d, e, f, g represent the numbers, 2, 3, 4,, 6, 7 in a one to one fashion, though not neessarily in that order. Suppose that a + b + + d + e = 6. Suppose also that there are integers x, y suh that gxy = 4. Then f = A: 7 B: 6 C: D: annot be determined from the given information. 32. There are 30 students in a math lass. There are two more females than males, and there are twie as many right-handed students as left-handed students. If there are left-handed males, then how many left-handed females are in the lass? A: 9 B: 7 C: 6 D: 33. A square of flexible, strethable material has its left and right sides glued together mathing the arrows labeled a in the diagram and has its top and bottom sides glued together mathing the arrows labeled b in the diagram. The result an be deformed into the shape of b a a A: a torus (surfae of a doughnut) B: a double torus (surfae of a doughnut with two holes) C: a sphere D: a ylinder 34. The piture below shows a series of blak squares inside a square of side length. The largest blak square has side equal to half the side of the original square. Eah sueeding blak square has side equal to half the side of preeding one. What is the sum of the areas of the blak squares? b A: 3 B: 2 3 C: 2 D:

7 3. Two real numbers are suh that the ratio of their differene, their sum, and their produt is : 9 : 00. The produt of the two numbers is A: 200 B: 20 C: 000 D: 00 E: Let S be the set of the following nine points in the plane: { (, ), (, 2), (, 3), (2, ), (2, 2), (2, 3), (3, ), (3, 2), (3, 3) }. How many distint lines pass through two or more points in S? A: 20 B: 28 C: 36 D: The produt of two onjugate omplex numbers is A: always non-real B: always purely imaginary C: always a negative real number D: always real 38. Suppose that a, b,, d are non-zero real numbers and that the equation ax 3 + bx 2 + x + d = 0 has three real roots. Then the sum of the reiproals of these roots is A: d B: b a C: d D: b d E: a b 39. The streets of the town of Blaise are laid out as indiated in the digram. How many ways are there to travel from the southwest orner A to the northeast orner B staying on the streets and traveling only north and east? B A: 6 B: 34 C: 42 D: 70 E: 0 A 40. Suppose that in the triangle ABC, os A = 2, os B = 4, and the length of segment AB is 2. What this the length of segment BC? A: 2 2 B: 7 C: D: There is not enough information to determine the length of BC.

8 The Mihigan Mathematis Prize Competition is an ativity of the Mihigan Setion of the Mathematial Assoiation of Ameria. DIRECTOR Stephanie Edwards Hope College OFFICERS OF THE MICHIGAN SECTION Chair Dan Isaksen Wayne State University Past Chair Mike Bolt Calvin College Vie Chairs Steve Blair Eastern Mihigan University Franes Lihtman Delta College EXAMINATION COMMITTEE Chair Sid Graham Central Mihigan University Robert Messer Albion College Hugh Montgomery University of Mihigan Daniel Frohardt Wayne State University Seretary-Treasurer Mark Bollman Albion College Governor John Fink Kalamazoo College ACKNOWLEDGMENTS The following individuals, orporations, and professional organizations have ontributed generously to this ompetition: Hope College The Mihigan Counil of Teahers of Mathematis The Mihigan Assoiation of Seondary Shool Prinipals has plaed this ompetition on the Approved List of Mihigan Contests and Ativities.