State Math Contest (Senior)

Transcription

1 Name: Student I: State Math ontest (Senior) Instructions: o not turn this page until your proctor tells you. nter your name, grade, and school information following the instructions given by your proctor. alculators are not allowed on this exam. This is a multiple choice test with 40 questions. ach question is followed by answers marked a), b), c), d), and e). Only one answer is correct. Mark your answer to each problem on the bubble sheet nswer Form with a # pencil. rase errors and stray marks. Only answers properly marked on the bubble sheet will be graded. Scoring: You will receive 6 points for each correct answer,.5 points for each problem left unanswered, and 0 points for each incorrect answer. You will have hours and 0 minutes to finish the test.

2 . Which of the following could not be the lengths of the sides of a triangle? a) 5, 7, 8 b), 4, 5 c) 4, 7, 0 d),, e), 6, 0 Three numbers represent the lengths of sides of a triangle if and only if they satisfy the triangle inequality. Notice that + 6 < 0 implies doesn t satisfy the triangle inequality.. bumble bee is traveling back and forth between the front end of two trains moving towards each other. If the trains start 90 miles away from each other and one train is going 0 miles per hour while the other train is going 0 miles an hour and the bumble bee is traveling 00 miles per hour, how many miles does the bumble bee travel before being smashed by the two trains colliding? a) 450 miles b) 90 miles c) 00 miles d) 80 miles e) 70 miles orrect answer: c The trains start 90 miles apart and collectively are going 0 mile per hour. It will take hours for the trains to collide. Since the bubble bee is traveling 00 miles per hour and travels for hours, the bee travels 00 miles.. Four people s car keys are accidentally dropped into a pond. When the keys are fished out, they look alike except for the notches, so the four keys are returned in random order to the four owners. What is the probability that none of the owners will be able to get into his or her own car later? a) /4 b) /8 c) 5/ d) / e) 5/8 orrect answer: b Keys 4 have 4 permutations. Possible totally wrong orders: Which of the following expressions indicate the shaded region. ( is the complement of relative to P.) P

3 I. II. III. P ( ) IV. P ( ) V. P ( ) VI. P ( ) VII. VIII. a) only II. b) only V. c) only VIII. d) II., V., and VII. e) III., V., and VIII. 5. rectangle is partitioned into 4 subrectangles as shown below. If the subrectangles have the indicated areas, find the area of the unknown rectangle. 60? a) 7 b) c) 70 d) 48 e) 64 b s in the figure to the right, cb = 60, bd = 00, and ad = 80. Then ac = ad c cb 60 = 80 = 80 d bd 00 = 48. a? c 80 d 6. In the expansion of (x + y) 0, what is the coefficient of x y 8? a) b) 45 c) 9 d) 568 e) choose is 45 and multiplying by gives If cos(θ) + sin(θ) = 0 5 and tan(θ) = a b, then a b + b a = a) 5 b) 0 c) 0 d) 0 e)

4 orrect answer: b b and cos(θ) = a + b a + b = sin(θ) + cos(θ) =. Squaring and simplifying we ob- a + b If tan(θ) = a a, then sin(θ) = b 0 Thus 5 tain a + b = 0ab. Thus a b + b a = 0. a + b. 8. Two straight roads intersect at right angles. car and bicycle meet at the intersection each traveling on a different road. The car is going 0 miles an hour and the bicycle is traveling at one fifth the speed of the car. fter 0 minutes how far apart are they? a) 0 miles b) 0 miles c) 04 miles d) miles e) 8 miles orrect answer: c fter 0 minutes the car is 0 miles and the bicycle is miles. The hypotenuse is x = Find the sum of all the even integers from 546 to 854 inclusive: a) 06,400 b) 09,00 c) 07,800 d) 07,00 e) 08,500 ( ) + ( ) + ( ) + + ( ) + ( ) Since = 5 and we have only even numbers we have 76+ = 77 times 400 are added together. (77 400) = Find the shortest path which starts at the origin and visits all five of the following points and returns to the origin: {(0,0), (,0.5), (,), (,0), (0,)}. a) (0,0),(,0.5),(,),(,0),(0,),(0,0) b) (0,0),(,0),(,0.5),(,),(0,),(0,0) c) (0,0),(,0.5),(,0),(,),(0,),(0,0) d) (0,0),(0,),(,0.5),(,0),(,),(0,0) e) (0,0),(0,),(,0),(,0.5),(,),(0,0)

5 orrect answer: c y inspection the path (c) is shorter than the path (d) and the path (a) is shorter than the path (e). Let x be the distance from (0,0) to (,0.5). Then the distance from (,0.5) to (,0) and the distance (,0.5) to (,) are also both equal to x. Notice that x >. Let y be the (,0) to (0,) and z be the distance from (,) to (0,). Then z < y. The length of the path (a) is x + y + 4. The length of the path (b) is x + z + 5. The length of the path (c) is x + z + 4. Thus the path (c) is the shortest.. ladder 0 feet long leans against a building. If the bottom of the ladder slides away from the building horizontally at a rate of 4 ft/sec, how fast is the ladder sliding down the house when the top of the ladder is 6 feet from the ground. a) ft/sec b) 4 ft/sec c) ft/sec d) 4 in/sec e) 5 ft/sec orrect answer: a When the top of the latter is 6 feet from the ground the distance from the bottom of the latter to the building is ft. 0 = + 6 Letting length of the latter be z, the distance from the bottom of the latter to the building be x, and the distance from the top of the latter to the ground be y find an equation for rates of change. x( x/sec) + y( y/sec) = z( z/sec) Since the length of the latter does not change we have Plugging in for x, y and x/sec, x( x/sec) + y( y/sec) = 0. y/sec = f t/sec Since the question is asking how fast the latter is sliding down the answer is ft/sec.. Let z be a complex number of magnitude such that z is not or. Let i =, and let c and d be real numbers with c + di =. Which of the following is a possible value for c? + z a) b) c) d) e)

6 orrect answer: a Let z = a + bi. Then + z = + z + z + z ( + a) bi =. Thus c = ( + a).. The number of real solutions to x = sin(x) is which of the following? 00 a) b) 56 c) 6 d) 64 e) 65 orrect answer: c On the interval [0,π], sin increases 0 to and then decreases back to 0. On the interval [π,π], sin decreases from 0 to - and then increases back to 0. This repeats because sin is periodic with period π. The graph of y = x/00 is a straight line. This line intersects the graph of y = sin x twice on the interval [0,π], [π,π],... as long as the right endpoint is less than 00. Notice that π < 00 and π > 00. So for x 0, there are solutions. Similarly for x 0, there are solutions. The solution x = 0 was counted twice so there are 6 solutions. 4. onvert 60. from base 7 to a decimal (base 0). What is the digit in the tens position? a) 0 b) c) d) 5 e) 7 In decimal the number is Let n be greatest integer that will divide the three integers 5, 90 and 4589 and leave the same remainder. Which is the sum of the digits of n? a) 7 b) 9 c) d) 7 e) 4 The differences of the numbers are 078, 686, and 9. n must divide each of these numbers. Factoring them gives ()(7)(7)(), ()7, and 7. Thus n = young man has to walk from his home to his work. (His mom picks him up so he does not walk home.) How many days can he walk to work in a different way by walking a total of three blocks north and seven blocks east in any order? (No cutting diagonal through the blocks). WORK HOM

7 a) b) 0 c) 80 d) 0 e) 00 The young man needs to walk a total of 0 blocks. of the blocks need to be north and the remaining east. 0 = 0 7. Find the largest integer n such that n has exactly 4 positive divisors and n divides 00!. a) 7659 b) 86 c) 59 d) 979 e) 00 To have exactly 4 positive divisors a number needs to be the product of distinct prime numbers or a prime cubed. Since 00! = (00)(99)(98)...(5)(4)()()() if you choose the largest two prime which are less than 00 you get (97)(89) = 86. The largest prime whose cube divides 00! is and = onsider the checkerboard below, with some of the squares removed as signified by the s. How many ways are there to place 5 circles on the remaining squares (those without s) so that no two circles are on the same row or column? a) 44 b) 0 c) 40 d) 96 e) 45

8 orrect answer: a First let us label the columns and rows. 4 5 Now make a large table of combinations which do not put a circle on an lizabeth is a contestant on the Price is Right and her game is PLINKO. She has one token to drop from the top. On the game board below lizabeth chooses slot to drop her token into. t each divider there is an equal chance of going left and right. What is the probability lizabeth will get the prize of $00?

9 $00 a) 5 b) 6 c) 8 d) 0 e) Thus the probability is $00 ( ) = The line H is perpendicular to line and H is on the line. The angle and both measure π 6. If H is length 4, what is the length of? iagram is may not be drawn to scale.

10 H a) 8 b) 4 c) 6 d) 0 e) 4 orrect answer: a The triangle H is congruent to the triangle H. The angle H is complementary to the angle H since H is a right triangle. ( π Thus H is a 6, π, π ) right triangle and H has length 4. So the length of is 8.. Let f (x) be an odd function defined on R satisfying (a) f (x + ) = f (x), for all real numbers x; (b) f (x) = x when 0 x. Find the value of f (0 ). a) 0 6 b) c) 0 d) e) 0 6 orrect answer: b Since f (x + ) = f (x), we have f (x + 4) = f (x + ) = f (x). Hence f is a periodic function with period 4. Then using that f (x) is an odd function and f (x) = x when 0 x, we have f (0 ) = f (0 6) = f ( 0 + 6) = f ( 0 + 8) = ( 0 + 8) = 0 + 6, since 0 < <.. How many four-digit integers either leave a remainder of when divided by 7, or a remainder of 4 when divided by 5, but not both? a) 570 b) 57 c) 575 d) 084 e) 086

11 orrect answer: b Let be the number of four-digit integers that leave a remainder of when divided by 7. Let be the number of four-digit integers that leave a remainder of 4 when divided by 5. Let be the number of four-digit integers that leave a remainder of when divided by 7 and a remainder of 4 when divided by 5. Our goal is to find ( ) + ( ). Find Looking for integers x such that 000 7x + < 0000 Subtract and divid by x < Since x is an integer. 4 x 48 = = 86. Find Similar calculations with y = 800. Find Similar calculations with z 000 5y + 4 < y z + 9 < z 85 = 57 ( ) + ( ) = 57. Let a n and b n be sequences of numbers satisfying a =, b =, a n+ = b n, b n+ = a n b n for all natural number n. ompute b 07 + b 06. a) 05 b) 07 c) 04 d) 06 e) 5 06

12 Since a n+ = b n, b n+ = a n+ b n+, we have b n+ + b n+ = (b n+ + b n ) = ( ) (b n + b n ) = = ( ) n (b + b ), which implies b 07 + b 06 = ( ) 05 ( 8) = Find all real numbers a such that f (x) = x + a x is an increasing function on the interval [0, ). a) [, ) b) [, 0] c) (, 0] d) [0,] e) [,] orrect answer: b For 0 x, f (x) = x ax + a. Thus f (x) is increasing on the interval [0,] if and only if a 0. On the interval [, ), f (x) = x + ax a. Hence, f (x) = x +ax a is increasing on [, ) if and only if a. Hence [,0] is the set of real numbers such that the function is increasing. 5. Find the area of triangle if = = 50 in and = 60 in. a) 000 square inches b) 500 square inches c) 000 square inches d) 400 square inches e) 00 square inches Since the triangle is isosceles, the altitude to the side bisect. Hence, using the Pythagorean Theorem on the right triangle, we have = 40 in. Therefore, the area is ()()/ = 60 40/ square inches = 00 square inches. 6. group of airplanes is based on a small island. The tank of each plane holds just enough fuel to take it halfway around the world. ny desired amount of fuel can be transferred from the tank of one plane to the tank of another while the planes are in flight. The only source of fuel is on the island. It is assumed that there is no time lost in refueling either in the air or on the ground. The planes have the same constant speed and rate of fuel consumption. What is the smallest number of planes that will ensure the flight of one plane around the world on a great circle and have all the planes return safely to their island base? a) b) 4 c) 6 d) 7 e) 9

13 orrect answer: a Planes,, and take off together. fter going 8 of the distance, transfers 4 tank to and to and returns. Planes and continue 4 another 8 of the way, then transfers 4 of a tank to and returns. flies 4 the way around and is met by who transfers 4 of a tank to. turns around and follows to the base. t 8 the way from the base they are met by who transfers 4 to the base safely. of a tank to each and they all return 7. Find the units digit of 07. a) 5 b) 7 c) 9 d) e) The units digit of 07 is the same as that of 07. Since 07 = , we have So the units digit is. 07 = = = ( 4 ) 504 = ( ) ( + x x + x ) 8. Let f (x) = ln. What is f x + x in terms of f (x)? a) f (x) b) f (x) e c) f (x) + f (x) d) f (x) + f (x) e) f (x) ( x + x ) ( + x+x ) ( +x f + x = ln + x + x + x ) = ln x+x + x x x = ln +x ( ( + x) ( x) ) = f (x) 9. my likes either red or green clothes, but not both. She likes either turtleneck or V-neck sweaters, but not both. my never wears a sweater that is both the color and the type she likes, nor does she wear one that is neither the color nor the type she likes. my wears a red turtleneck. If you want to buy a sweater for my that she will wear, should you buy a red V-neck, a green V-neck, or a green turtleneck? a) red V-neck b) green V-neck c) green turtle neck d) my won t wear any of those sweaters e) Not enough information is given to answer the question

14 orrect answer: b She likes exactly two of the four options, {red, green, V-neck, turtleneck}. She also likes exactly one of the options {red, turtleneck}. Thus she must like exactly one of the options {green, V-neck}. 0. The parabolas y = x and y = x + 4x 5 have two common tangent lines. The one with smaller slope intersects the x-axis somewhere between x = 0 and x = 5. Where? a) t x = 0 b) t x = 5 c) t x = 4 d) t x = e) t x = orrect answer: c The common tangent line is the line with slope through the points (/,/4) and (/,5/4). (The other common tangent has slope and intersects x-axis at 4.). Let 50 = a + b + c +, where a, b, c, d and e are integers. Find c and e. d + e a) c = and e = b) c = and e = c) c = and e = d) c = and e = 4 e) c = and e = 5 orrect answer: a (a, b, c, d, e) = (,,,5,) by calculation.. Find the greatest common divisor of 4400 and a) b) c) 5 d) 7 e) 5 orrect answer: c For n = 4400 and m = , we have n = 0000m+5 So a common divisor of n and m must divide 5. oth n and m is divisible by 5 since their last digit is 0 or 5. The number n is not divisible by since the sum of the digits of n is not divisible by. Thus the greatest common divisor is 5. onsider the sequence x 0 =, x =, x = 5, x = 4, x 4 = 8,... The pattern is x n = x n x n n >. Find the limit of these numbers. for

15 a) b).5 c).9 d) e).4 Numbers are x n = n n. 4. Find the area of overlap between the closed disc x + y and the parabolic region y x. a) π + b).5. c) π 6 + d). e) π + qual to one-fourth of circle of radius plus twice the area between y = x and y = x. 5. How many pairs of integers (x, y) satisfy the following equation. x + 6x + 8x = y + 9y +. a) 0 b) c) d) e) None of the above orrect answer: a Notice that x + 6x + 8x = x(x + 6x + 8) = x(x + 4)(x + ) which must be divisible by three because either x, x +, x + 4 is divisible by. However, y + 9y + has a remainder of when divided by. Thus there 0 pairs of integers satisfying the equation. 6. Find the shortest distance from the point (6,/) to the parabola y = x in the plane. a) 7 7 units b) 4 units c) 5 units d) 0 units e) 0 units orrect answer: a Line from point (6,/) to (x, x ) must have slope. Thus the equation x x = 6 with solution x =. istance from (,4) to (6,/) is the answer =

16 a) - b) c) 0 d) e) Let = + 5 and = 5. Then we wish to solve for. Notice that = + 5 ( 5) = 0 and = ( + 5)( 5) =. ( ) = + = ( ) = 0 9( ) ( ) + 9( ) 0 = ( ( ) )( ( ) + ( ) + 0 ) = 0. Since is real, =. 8. Let S be a subset of the set {,6,6,0,57,,,465,9,856,77,74,4865,97,59454} so that the elements in S add up 557. Find the two smallest elements of S. a) and 77 b) and 856 c) and d) 6 and 9 e) 6 and 0 ach term in the sequence is greater than the sum of all the terms before it (superincreasing sequence). So, if 557 is a sum, the largest term in the sum must be 97. The difference between 557 and 97 is 584. So the next largest term must be 77. The difference between 584 and 77 is 5, so the next largest term must be 856. The difference between 5 and 856 is 69, so the next largest term must be. The difference between 69 and is 6 so the last two terms must be 6 and Suppose real numbers x, y, z satisfy the equation x y + z + ompute the value of x y + z + y z + x + y z + x + z x + y. z x + y =. a) b) - c) d) 0 e) -

17 We first note that x + y + z 0. Otherwise, Thus, we have ( x y + z + x y + z + y z + x + Simplifying it, we have y z + x + z x + y z x + y =. ) (x + y + z) = (x + y + z). x y + z + x + y z + x + y + z + z = (x + y + z), x + y which yields x y + z + y z + x + z x + y = 0. ( 40. Let f (x) = 4sin x sin x + sin x ) cos x. What is the minimal period of f (x)? a) π b) π/ c) π/ d) π e) π orrect answer: c Using the triangle identities, we have ( f (x) = 4sin x sin x + sin x cos x ) ( = 4sin x sin x + sin x cos x ) = 4sin x sin x + = sin x(4sin x ) + = sin x((sin x ) + sin x) + = sin x( cos x + sin x) + = sin x( cos x (cos x sin x)) + = sinxcos x sin xcosx + = sinx +. The answer is π/. N OF M