New York State Mathematics Association of Two-Year Colleges Math League Contest ~ Spring 08 Directions: You have one hour to take this test. Scrap paper is allowed. The use of calculators is NOT permitted, as well as computers, books, math tables, and notes of any kind. You are not epected to answer all the questions. However, do not spend too much time on any one problem. Four points are awarded for each correct answer, one point is deducted for each incorrect answer, and no points are awarded/deducted for blank responses. There is no partial credit. Unless otherwise indicated, answers must given in eact form, i.e. in terms of fractions, radicals,, etc. NOTE: NOTA = None Of These Answers.. The Heaviside function, a.k.a. the unit step function, Ht (), is often 0, t 0 defined as Ht (). Which of the following epresses the, t 0 pulse function, pt (), whose graph is shown? A) p( t) H( t a) H( t b) B) p( t) H( t b) H( t a) D) p( t) H( t a) H( t b) C) p( t) H( t a) H( t b). 5%? A) 0.5% B) 5% C) 50% D) NOTA 3. Three identical cubical tanks with edges of length are shown. The spheres in a given tank are the same size and packed wall-to-wall. Thus, the sphere in Tank A has a diameter of length, the eight spheres in Tank B have diameters of length, and the sity-four spheres in Tank C have diameters of length. If the tanks are filled to the top with water, then which tank would contain the most water? A) A B) B C) C D) All three tanks would contain the same amount of water..? A) B) C) i D) i 5. It takes Vivian 30 seconds to walk down an escalator when it is not operating, and seconds to walk down when it is operating. How many seconds should it take her to ride down the escalator while just standing still on it? Assume she and the escalator move at constant rates.
6. When trying to weigh a newborn baby, an accurate reading could not be made due to the baby moving too much. Thus, when I held the baby and stepped on a scale, the nurse read our combined weight as 99 pounds. When the nurse held the baby, I read their combined weight as 5 pounds. When I held the nurse, the baby read our weight as 30 pounds. What is the combined weight of the three of us? 7. The accompanying diagram shows two circles, that are tangent to each other, inscribed in a square with sides of length 5. The smaller circle has a radius of. What is the radius of the larger circle? A) 5 5 B) 3 5 C) 9 5 D) 5 5 8. When the power returns after a power outage, an analog electric clock (i.e. a traditional clock with hour and minute hands) continues measuring time from the moment the power returns. However, a digital clock returns to :00 and measures the amount of time that elapses since. When a man leaves his house at 7:30 AM, his watch, analog clock, and digital clock all agree. When he returns home later that day, his watch reads 5:38 PM, the analog clock shows :50, and the digital clock shows :3. At what time did the power go out? 9. A high school student, after having been absent, learns upon returning to school that she has quizzes in three of her classes net week. What is the probability the quizzes will all be on the same day? Assume that each of the five school days are equally likely. 0. What value for solves the equation log8 log ( )? Epress the answer in simplest form, i.e. not in logarithmic form.. How many solutions does the equation cos( ) sin( ) A) B) C) 3 D) more than 3 have for (0, ]? 00 08 08. Letting A, B 000 5, and C, arranged from smallest to largest we get A) A, B, C B) A, C, B C) B, A, C D) C, B, A 3. A square has three line segments drawn as shown. A segment of length begins at the lower left corner, a perpendicular segment of length is drawn from the end of the previous segment, then a segment of length 3 is drawn perpendicular to the segment of length so it ends at the upper right corner. What is the length of an edge of the square? Note: Diagram not drawn to scale. A) 0 B) 3 36 C) 0 3 D)
. Letting i, ( i) 009 A) i B) 08? 009 i C) 009 009 i D) 009 009 i 5. If and y are real numbers that satisfy the equation A) B) C) D) y y 5, then what is y? 6. A coin with a inch diameter is randomly dropped onto a rectangular table. The table is covered with a grid of lines that are inches apart, thus forming inch by inch squares. Neglecting the width of the lines, what is the probability the coin lands in a square without touching any lines? 7. Two trains are travelling in the same direction along parallel tracks. Train A is 00 meters long and travelling at 80 km/hr. Train B is 300 meters in length and 6 km ahead of Train A. Fifteen minutes later, Train A is 6 km ahead of Train B. Assuming the speed of both trains is constant, distances are measured from the front of the trailing train to the back of the leading train, and the distance between the tracks is negligible, what is the speed of Train B (in km/hr)? 8. Which of the following graphs best represents the graph of sin( y) cos( )? A) B) C) D) 9. The closed curve in the diagram is made up entirely of circular arcs, with centers shown, each with a radius of. What is the area enclosed by the curve? 0. Three students are interested in attending a school play. However, they adhere to the following conditions: If Anna goes to see the play, then so does Bob. If Bob goes to see the play, then so does Cathy. If Bob does not go to see the play, then which of the following must be true? I. Anna does not go see the play. II. Cathy does not go see the play. III. Cathy goes to see the play. A) I only B) II only C) III only D) I and II only
New York State Mathematics Association of Two-Year Colleges Math League Contest ~ Spring 08 ~ Solutions. The graphs of H( t a) and H( t b) appear as shown. Hence, it is clear that the pulse function is H( t a) H( t b).. 5% 0.5 50% Answer: C 3. Since the volume of a sphere is proportional to the cube of the radius, the spheres in Tank B (with radii of the radius of the sphere in Tank A) have a volume that is 3 8 the volume of the sphere in Tank A. Thus, eight of them have the same volume as the sphere in Tank A. Similarly, the spheres in Tank C (with radii of the radius of the sphere in Tank A) have a volume that is 3 6 the volume of the sphere in Tank A. Thus, sity-four of them have the same volume as the sphere in Tank A. Answer: D. i i i ( ) Note: y y only for y, 0 5. Let v be Vivian's speed while walking down the escalator, and e be the escalator's speed for a stationary object to go down. Thus, 30v and ( ve), using distance rate time with the distance being escalator ride. Solving the first equation gives v 30 becomes e 30. The second equation, or e Hence, it should take 0 seconds for Vivian to ride 30 0. down the escalator while just standing still on it. Answer: 0 6. Let b represent the weight of the baby, m represent my weight, and n be the weight of the nurse. The three weighings give the following equations: bm 99, bn 5, and mn 30. Since the question asks for the combined weight, there is no need to solve these equations and determine the individual weights. Simply adding the three equations gives: b m n 63, then dividing by gives the desired result b m n 37. Answer: 37
7. From the diagram, we can see that the diagonal of the square, with measure 5, is composed of the four segments:,, r, and r. Hence, r r 5. Solving for r gives: r 9 5. Answer: C 8. The digital clock reading :3, at 5:38 PM, tells us that the power had been back on for hours 3 minutes. Thus, the power resumed at 5:38 :3 or 3:06 PM. The analog clock reading :50 at 5:38 PM, tells us the power had been out for 5:38 :50 = 8 minutes. Thus, the power went out 8 minutes before 3:06 PM, i.e. :8 PM. Answer: :8 PM 9. Let's call the three classes in which she has quizzes Class I, Class II, and Class III. Suppose the quiz for Class I is set (for any one of the 5 schools days), then the probability the quiz for Class II will be that same day is 5. Now the quiz for Class III must be set, and it also has a probability of being 5 on the same day. Hence, the probability the quizzes for Class II and III fall on the same day as the one for Class I is 5 5 5. Answer: 0.0 5 log ( ) log ( ) 0. Converting to base logarithms:. Which then gives: log log (8) log () ( ) log ( ) 3 log ( ) log ( ) 0 log ( ) log ( ) 0 log ( ) 0 or log ( ) 0. 3 3 3 8 8 3 3 The first equation gives, the second equation yields log 3 ( ). 3 83 Answer:. cos( ) sin( ) The equation is certainly satisfied when. It will also be satisfied when 5 cos( ) sin( ), which occurs when and. Hence, there eactly three solutions for (0, ]. Answer: C. Since all three values are positive, raising each to the same positive power preserves their 00 000 000 00 5 3, numerical ordering. Raising them each to the 000 power: 000 000 000 000 000 000 5 5 5, and 08 08 000 08 08 08 08. Since 000 08 is slightly less than, 000 08 08 is slightly less than 08 08. But, we know 5 05, which means 08 is slightly smaller than 5. Hence, the correct ordering, from smallest to largest is: 00, 08 08, 000 5. r r r Answer: B
3. Drawing the diagonal from the lower left corner to the upper right corner produces two similar right triangles we can use to help solve the problem. The hypotenuse of the smaller triangle is larger triangle is 9 ( ) 3. gives, and the hypotenuse of the 9 ( ). By similar triangle proportions, we have Squaring both sides of this equation and simplifying, 0 or ( )( ) 0. Discarding the negative root, gives us. Thus, the hypotenuses are 5 and 3 5, which makes the length of the diagonal 3 5 5 5. If the length of the sides of the square are s, then the diagonal has length s. Therefore, s 5 or s 5 0. 009 009 08 009 009 009 009 ( i) ( i) i i i i ( ) ( i), with. 5. 009 009 ( ) 009 008 5 i i i i i () i i i. Thus, ( i) i. and 5 08 009 y y 5 y y 5, now completing the square for both variables gives: y y 5 y y 5 ( y ) 5 ( y ) 5. The last equation becomes ( y ) 0, whose only real solution is and y. Hence, y. 6. In order for a coin, with a radius of ½ inch, to land without touching an edge of a square, its center must land further than ½ inch from each edge. Thus, the center must land within the inch by inch square illustrated in the diagram by the dashed square. The probability the center of a coin lands within this by square is the ratio of the areas of the by square and the by square grid. Thus, the probability is to. Answer: ¼ 7. Train A travelled 80 0 km, while Train B travelled km, where is its speed. Also, the initial 6 km difference is measured from the front of Train A to the back of Train B, whereas after the ¼ hour, the 6 km difference is measured from the front of Train B to the back of Train A. Thus, in addition to the 6 km + 6 km Train A made up, it also had to traverse its own length, 0. km, and the length of Train B, 0.3 km. This gives the equation 0 6 6 0. 0.3 or 7.5, which yields 30 km/hr. Answer: 30
8. Since cos( ) sin, we can rewrite the equation sin( y) cos( ) as y sin( ) sin. Also, they are periodic functions with a period of. Hence, the last equation can be written as sin( y m) sin n, where m and n are any integers. Thus, or y n m, which reduces to y m n y k, with k n m is an integer, whose graph is an infinite family of lines with slopes of offset by multiples of. Additionally, since cos( ) cos( ), sin( y) cos( ) can be written as sin( y) cos( ). Then using the same reasoning as before, we get y k, which is an infinite family of parallel lines with slopes of offset by multiples of. Hence, choice A is the most suitable graph. 9. Connecting the si centers forms a regular heagon, which is comprised of si equilateral triangles with sides of length as shown. Then we can see that three sectors outside the heagon are each ⅔ of a circle with radius. While the three empty regions between the heagon and the curve are sectors that are each ⅓ of a circle with radius. Thus, the area of the closed curve is A A A3, where A = the area of the heagon, A = the area of the three outside sectors, and A 3 = the area of the three empty regions. A 6 3, i.e. 6 times the area of one of the equilateral triangles, 3 A 3, and 3 3 A 3. Therefore, the area of the closed curve is 6 3 6 3. Answer: 6 3 0. We need to know when the conditional statement if p then q, symbolized as p q, is true and when it is false. According to logical convention, the conditional statement is false ONLY when p is true and q is false (i.e. when True False ). All other cases (three of them) are true. Hence, if Bob does not go see the play, then the first condition can only be true if Anna also does not go see the play, giving False False True. However, the second condition becomes False Cathy sees the play, which is true regardless of whether Cathy sees the play or not. Hence, only statement I must be true.