MEET 1 - OCTOBER 2006

Transcription

1 ROU D Arithmetic - Open MEET - OCTOBER CALCULATORS ARE OT ALLOWED O THIS ROU D.. The repeating decimal can be expressed as a rational fraction p q. When reduced to lowest terms, what is the square root of the sum of its numerator and denominator? How many two-digit primes have exactly one digit that is a 7 or a 9?. When is written in the positive integer base x, it is represented as 45, and when it is written in base (x + ) it is represented 5 How would be written in base (x )?

2 ROU D Algebra Word Problems. mph. $ CALCULATORS ARE OT ALLOWED O THIS ROU D.. Al runs three miles at the rate of 5mph. He returns over the same route by ambulance. The entire trip took 4 minutes. What was the average speed of the ambulance in mph? On an 8 foot seesaw, the fulcrum (balance point) is located at the 0 mark. Alice is seated on the 7 mark and Beth is seated on the mark. If Alice weighs 56 lbs, Beth weighs 08 lbs and Mary weighs 40 lbs, at what mark should Mary be seated so that the seesaw will balance?. The ratio of nickels to dimes to quarters is ::. If the ratio of nickels to dimes to quarters for the same number of coins were ::, the value of the coins would be reduced by $ Originally, in dollars and cents, by how much did the value of the quarters exceed the value of the nickels?

3 ROU D Algebra Exponents and Radicals. :. (, ) CALCULATORS ARE OT ALLOWED O THIS ROU D.. If x 9, evaluate x ( x ) / x / / as a simplified ratio of integers. Simplify the following expression: 6 6. Given that a b+ b a a b b a a b b a a b+ b a ab (a > 0, b > 0 and a b) solve for a explicitly in terms of b, i.e. determine the ordered pair of constants (m, n) for which a mb + n.

4 ROU D 4 Algebra Factoring.. CALCULATORS ARE OT ALLOWED O THIS ROU D.. Factor completely: x 5 y(y + 0) Factor completely: 7 x x 5 x If x (y z) + y (z x) + z (x y) is factored completely over the integers, determine a nonzero sum of the factors? (There are three possible answers.)

5 ROU D 5 Trig: Angular and Linear Velocity, Right Triangles.. CALCULATORS ARE OT ALLOWED O THIS ROU D.. In an isosceles triangle, the legs are each inches long and the base is 4 inches long. Determine the exact value of the cosine of the complement of a base angle. A point on a circle of radius 6 cm is moving around the circle at π/ radians per second. Determine the linear velocity of the midpoint of a radius of this circle (in cm/sec).. A ship sailing due east sights a buoy 0 north of east. Continuing due east, four miles later, it sights the same buoy 60 north of east. If the easterly course is maintained, how close will the ship come to the buoy? If necessary, express your answer in simplified radical form. (For these small distances the curvature of the earth may be ignored.)

6 TEAM ROU D pts.. pts. 4 pts.. SAT APPROVED CALCULATORS ARE ALLOWED O THIS ROU D.. Farmer MacDonald brought chickens and 5 turkeys to market. He sold some of each of them to the local butcher for a total of $84. If he received $5 for each chicken and $6 for each turkey, determine the maximum number of animals that were not sold? Dick s stamp collection consists of books. One-eighth of his stamps are in the first book, several sevenths are in the second book and there are 006 stamps in the third book. Find all possible solutions for the number of stamps in Dick s entire collection?. Two natural numbers are simultaneously chosen at random from a box containing the natural numbers,, 4,, 9, 0. In how many ways can the two numbers be selected from the box so that their product has exactly four factors? The order in which the two numbers are selected is irrelevant.

7 Answer Sheet Round (6) Round. 6 Round 4. (x + y + 5)(x y 5) ( x + 4)( x + 5)( x 5). (x z) or x z (y x) or y x (z y) or z y Round 5 7. $5.60 Round. 4 0 π 4. : 4 6. (, 4) or 6. Team Round. 7 ( pts) 6,608 or,6 ( pts). 48 (4 pts)

8 Detailed Solutions for GBML Meet OCTOBER 006 ROU D. Let n Then 00n Subtracting 57n 99 or n 99/57 Since this fraction is already reduced, the sum of the numerator and denominator is 56, giving a square root of 6. The required prime must be of one of the following 4 forms: 7_ 7, 7 7_ 7, 7, 47, 67 _9 9, 9, 59, 89 9_ none Total: 0. Instead of trying n > by trial and error, we choose to proceed algebraically. 4x + 5x + (x + ) + 5(x + ) + x +7x + 5 x x 0 x 6x 6 (x 8)(x + ) 0 x 8 45 (8) 4(64) + 5(8) + 99 (0) 5 (6) Read the remainders UP from the bottom Quotient Rem ROU D. Let r the rate of the ambulance (in mph). Time Distance/Rate /5 + /r 4/60 (hours) 6r r 5r 80 r 6 B08lbs M40lbs A56lbs 0 7 To be in balance (or equilibrium), the clockwise torque (or tendency to turn) must equal the counterclockwise torque. Thus, Beth creates 9(08) 97 foot-pounds of CCW torque and Alice creates 7(56) 09 foot-pounds of CW torque. Mary must be seated to the left of the fulcrum, say at a distance of x feet and 40x x. Therefore, Mary must be seated at the 7 marker.. Originally, N : D : Q x : x : x ( with a value in of 5x + 0x + 75x 00x) The changed ratio, with the same number of coins, implies N : D : Q x : x : x (with a new value in of 5x + 0x + 50x 75x) Thus, 00x 75x + 00 x 8 and the original 8 quarters were worth 40 and the original 4 quarters were worth $6 producing a difference of $5.60

9 ROU D. 9 (9 ) / 9 / / (7) 4 4 : Simplifying the left hand side of the equation we have ( a b+ b a) ( a b b a) a b b a a b + ab ab+ b a a b + ab ab b a 4 ab ab ab( a b b a) 4ab a b b a ab(a b) a b 4 a b + 4 (m, n) (, 4) ab ROU D 4. x 5 y(y + 0) x (y + 0y + 5) x (y + 5) As the difference of perfect squares, this factors to (x + y + 5)(x y 5). 7 x x 5 x 00 x + 4 x 5 x 00 x ( x + 4) 5( x + 4) ( x + 4)( x 5) ( x + 4) ( x + 5) ( x 5). x (y z) + y (z x) + z (x y) x y x z + y z y x + z (x y) (x y y x) (x z y z) + z (x y) xy(x y) z(x y ) + z (x y) (x y)[xy z(x + y) + z ] (x y)[xy zx zy + z ] (x y)[y(x z) - z(x z)] (x y)(x z)(y z) Finally, the sum of the three factors is x z or (x z). Other possible factorizations are: ( -x + y)(-x + z)(y z) sum y x or (y x) ( x y)(-x + z)(-y + z) sum z y or ((z y) (-x + y)(x z)(-y + z) sum 0 (rejected)

10 ROU D 5. By the Pythagorean theorem BM Since θ is the complement of a base angle, applying SOHCAHTOA, we have cos(θ ) 4 0 θ B The point rotates 5 each second, completing one revolution in 4 seconds. The circumference of the circle π(6) (d)4 d π/ cm 7 M C Thus, the point on the circumference travels π/ cm /sec, i.e. its linear velocity is π/ cm/sec. The midpoint of a radius travels only half as far in the same time period, i.e. its linear velocity is π/4 cm/sec. A B. Consider the diagram at the right. We must find BR. 4x x (4 + ) + x x x x + 8x - 8 x x - x 6 0 Using the quadratic formula x ± + 4 ± P x Q y R x

11 TEAM ROU D. Let C and T respectively denote the number of chickens and turkeys sold. (9/4)C + 6T 84 9C + 4T 6 C + 8T C ( 8T)/ Separating the numerator into parts that must be divisible by and parts that are not necessarily divisible by, C ( 9T + + T)/ C 7 T + +T If T, 5, 8, or 4, the last term is an integer. The corresponding values of C are:, 4, 6, 8 and 0. Thus, the possible transactions were (C, T) (8, ) or (6, 8), leaving either 7 (or ) unsold. Let x denote the number of stamps in Dick s collection. Let y denote the number of sevenths in book (y > ) x xy Then x 7x + 8xy + 56(006) 56x x-8xy x Clearly, the numerator is only divisible by the primes, 7, 7 and y (y, 49 8y) (, ), (, 5), (4, 7), (5, 9), (6, ) y 4 x 6,608 and y 6 x,6. Since 4 can be factored only as or 4 (implying exponents used in the prime factorization of the products we seek must be either or 0), the integers with exactly 4 positive factors must either be the product of two distinct primes or a perfect cube of a prime. The 0 primes in the box are:,, 5, 7,,, 7, 9,, and 9 Thus, there are 0 C 45 possible pair products, plus the perfect cubes 8, 7 and 5 48