Math is Cool Championships

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1 Individual Contest Express all answers as reduced fractions unless stated otherwise. Leave answers in terms of π where applicable. Do not round any answers unless stated otherwise. Record all answers on the colored cover sheet. 1 Evaluate: Evaluate: 4 9 / Evaluate: Evaluate: Evaluate: Evaluate: and express your answer as a mixed number What is the slope of the line 3x-5y=17? My favorite number is tripled, the result is then reduced by 34, and then that result is doubled. The final result is 46. What is my favorite number? What are the coordinates, in (x, y) form, of the y-intercept of the line 3x-2y=7? What is the length, in centimeters, of the hypotenuse of a right triangle with legs measuring 8 cm and 20 cm? [Give an exact simplified answer.] When two fair six-sided dice are rolled, what is the probability of rolling a sum of five or six? If 1327=x mod 17, and 0<x<16, what is the value of x? A pentagonal pyramid has exactly one of its base vertices (and no other vertices) cut off by a plane cut. How many total vertices are there on the two resulting solids? What is the surface area of a right rectangular prism with edge lengths of 7, 9, and 3? What is the volume, in cubic centimeters, of a right circular cone with a base radius measuring 3 cm and a height of 8 cm?

2 16 How many positive two-digit integers consist of two different digits? 17 Express the base nine number 678 9in base ten. 18 What is the area, in square centimeters, of a rectangle with a perimeter measuring 68 cm and a diagonal measuring 26 cm? I have 22 coins, each either a nickel or a quarter, with a total value of \$2.50. How 19 many nickels are there? 20 If the probability of event A is 3 4, the probability of event B is 2, and the 3 probability of both events occurring is 3, what is the probability that at least 5 one of the events occurs? What is the largest integer satisfying 47-3x <88? What is the sum of the twelve smallest positive multiples of twelve? A bag contains the six letters of the word BANANA. If two letters are drawn from the bag, what is the probability that they are the same as one another? Evaluate: If f(x) = 4x x2, evaluate f(f(5)). How many positive four-digit numbers are divisible by three but not two? In the diagram AD=10, CD=2, CE=3 and BE=5, what is the ratio of the area of the quadrilateral ADEB to the area of triangle ABC. Express your answer as m:n where m and n are relatively prime integers. D C E A B 28 What is the sum of an infinite geometric series with a second term of 4 3 and a common ratio of 34? 29 If a 1 = 1, a 2 = 2, and a n = 3a n 2 + a n 1 evaluate a 6.

3 Challenge Questions 30 What is the maximum value of the function 2 f ( x ) = 3x 2x? 31 What is the fewest number of triangles that can be formed by five lines in a plane if each line must intersect at least two others and no more than two may be parallel? 32 Evaluate: log 4 (128) If (2x+1)=7 mod 13 and x is an integer between 0 and 100 inclusive, what is the 33 sum of all possible values of x? A set contains the integers from 1 to 1001 inclusive. What is the highest number 34 of elements in a subset such that no two elements from the subset add to a multiple of 5? 35 A nine term arithmetic sequence a 1,a 2,...,a 8,a 9 satisfies a 5 + a 7 = 17and a 4 + a 6 = 17. What is the sum of the terms of the sequence? If A = and B = , what is the sum of the elements of AB? 2 A circle is tangent to two perpendicular lines. There is a point on the circle that is 8 units from one line and 9 units from the other. What is the radius of the circle? What is the length of the major axis of the ellipse with equation 4x 2 24x + y 2 + 2y = 11? A triangle has two sides measuring 2 cm and 2 cm with an included angle of 135. How long, in cm, is the third side? When four cards are drawn from a standard 52-card deck, what is the probability of getting at most three different ranks (e.g. one King, one seven, and two fours)?

5 Team Contest Express all answers as reduced fractions unless stated otherwise. Leave answers in terms of π where applicable. Do not round any answers unless stated otherwise. Record all answers on the colored cover sheet. 1 What is the remainder when is divided by 17? 2 How many prime numbers between 100 and 300 have digits that sum to 15? 3 The line y = 4 x + 2 intersects the parabola y = 2x x 19 at (a, b) and (c, d). Evaluate a + b + c + d and express as a common fraction. 4 How many lattice points (points with integer coordinates) lie on the graph of 2 2 x 4x + y + 2y = 80? Evaluate: If points (0, 0, 0), (0, 1, 0), and (0, 0, 1) are three of eight vertices of a cube, how many possible locations could be the other vertices of the cube? 7 How many right circular cones with base radii of 1 and heights of 1 would it take to have the same total volume as a sphere with a radius of 6? 8 Two real numbers, x and y, are randomly chosen between 0 and ten. What is the 2 2 probability that x + y > 10 and x + y < 100? 9 What is the sum of the ten smallest perfect positive squares that are not perfect cubes? 10 If f(x) = 2x 3 3x + 2 and g(x) = f(f(x)), evaluate g( 1).

6 Pressure Round Contest 1 Find the minimum value of f(x) = 3x 2 2x Let n be the total number of positive integer factors of How many positive integer factors does n have? 3 When all the 5-digit numbers that can be made by arranging the digits 2, 9, 7, 8, and 4 (each used exactly once) are listed in increasing order, what will be the 50th number on the list? 4 Triangle ABC is inscribed in a circle with radius 4 (not drawn to scale). If AB = CB and AC = 8, find the shaded area. A C B Find 1 1 k. (The symbol Π indicates a product.) k =2

7 Mental Math Contest Express all answers as reduced fractions in terms of radicals and π where applicable, unless stated otherwise. PERSON What is the sum of the first 10 positive odd integers? What is the value of x, if 6 x plus 11 equals 8? -1/2 1.3 What is the measure of each exterior angle in a regular dodecagon, in 30 [ ] degrees? 1.4 What is the probability of getting exactly two heads when I flip a fair coin 4 times? 3/8 PERSON What is the least common multiple of 4, 6 and 10? What is the sum of the solutions to 2 x squared minus 5 x plus 10 equals 5/2 0? 2.3 What is the area of an isosceles triangle with side lengths of 10, 10 and 48 [un 2 ] 12? 2.4 Given that I get at least one head on 5 flips of a fair coin, what is the probability that all 5 are heads? 1/31 PERSON What is the smallest three-digit prime number? If Y equals 3 and Z equals 5, what is Y cubed times Z squared divided by 75 9? 3.3 If the probability of event A is two-thirds, what is the probability of A 0 intersected with the complement of A? 3.4 What is the perimeter of a rhombus with diagonals of 10 and 24? 52 [un] PERSON What is 25 squared minus 24 squared? In a two-person race between John and Tom in which there are no ties 2 : 3 allowed, the probability that John wins is three-fifths, what are the odds of Tom winning? [Express answer in the form of a colon b.] 4.3 What is the minimum value of the function: f of x equals x squared plus 6 0 x plus 9? 4.4 What is the total surface area of a cone with a base diameter of 8 and a height of 3? 36π [un 2 ]

8 COLLEGE KNOWLEDGE BOWL ROUND #1 # Problem Answer 1 A 72-degree arc of a circle has a length of 12, what is the area of the circle? 2 The second and third terms of an infinite geometric sequence are 12 and 4, what is the sum of the infinite sequence? 3 Luke is dealt a pair of Jacks in a poker game. Three additional cards are turned over. What is the probability that these three cards contain exactly one Jack? 4 What is the maximum number of points that a circle could intersect with rectangle? 5 If Y plus Z equals 10 and Y times Z equals 20, what is Y squared plus Z squared? 6 What is the smallest integer greater than 25 that has a remainder of 3 when divided by 5 and a remainder of 7 when divided by 11? 7 What is the smallest base that the base 10 number 321 will only have two digits? 7.1 Extra Problem Only if Needed 900 / π [un 2 ] / [int] [base] 18 8 The houses on my block are numbered 103, 107, 111, continuing until 139. What is the sum of the house numbers? 1210

9 COLLEGE KNOWLEDGE BOWL ROUND #2 # Problem Answer 1 Anakin was adding the positive integers up to 40, but accidentally skips one, getting total of 799. What number did he skip? 2 What is the smallest positive integer, n, for which n squared plus n plus 1 is not prime? 3 What is the difference in area between a square of side length 6 and the circle that is inscribed in it? 4 What is the smallest solution to x cubed plus 3x squared minus 4x minus 12 equals 0? 5 An urn contains 4 red and 5 blue marbles. If I draw out two marbles without replacement, what is the probability that both are the same color? 6 What is the area of an ellipse with a major axis of 12 and a minor axis of 1? 7 How many subsets with three elements can be made from a set containing: Hydrogen, Helium, Nitrogen, Oxygen, Silver, Gold and Uranium? 7.1 Extra Problem Only if Needed π [un 2 ] -3 4 / 9 3π 35 [subsets] 8 What is the positive difference between the mean and median of the numbers: 23, 42, 55, 34, and 12? Express answer as a decimal. 0.8

10 COLLEGE KNOWLEDGE BOWL ROUND #3 # Problem Answer 1 What is the sum of the positive integers less than with exactly three positive integer factors? 2 A parabola with an axis of symmetry parallel to the y- (-4,0) axis and vertex at the point 1 comma 8 crosses the axis at the point 6 comma 0. At what other point does it cross the x axis? 3 ObiWan knows that the gravitational force between two 16x [newtons] bodies varies inversely as the square of the distance between them. At 400 thousand feet, the force on his ship is x newtons toward the planet. What will be the force, in newtons, at 100 thousand feet in terms of x? 4 Expressed in base 10, what is the largest 4 digit number 624 in base 5? 5 What is the average speed, in miles per hour, of a car 600/11 [mph] that travels at 50 miles per hour to Spokane and 60 miles per hour back? 6 An equilateral triangle has the same area of a [un] triangle with a hypotenuse of 4 times the square root of 2. What is the side length of the equilateral triangle? 7 What is the probability of getting 1, 2 or 3 heads with four flips of a fair coin? 7/8 7.1 Extra Problem Only if Needed 8 Using pennies, nickels, dimes and quarters, how many ways can you make 37 cents? 24 [ways]

11 School Name Team # Proctor Name Room # Final Score: KEY First Score STUDENT NAME Individual Contest Score Sheet DO NOT WRITE IN SHADED REGIONS Answer 1 or 0 1 or 0 Answer 1 or 0 1 or / / , , [num] / : /9 9 (0, -7/2) [cm] 30 9/ / [triangles] / [vert] [un2 ] [elements] 15 24π [cm3 ] / [int] [ 10] [un] [cm2 ] [un] [nickels] [cm] 20 49/

12 School Name Team # Proctor Name Room # STUDENT NAME Final Score: KEY First Score (out of 18) Individual Multiple Choice Contest Score Sheet Correct responses are worth 2 points, incorrect responses are worth -1 point and no response is 0 points. DO NOT WRITE IN SHADED REGIONS Answer -1, 0 or 2-1, 0 or 2 1 B 2 A 3 A 4 D 5 C 6 A 7 C 8 D 9 B 10

13 School Name Team # Proctor Name Room # STUDENT NAME Final Score: KEY First Score (out of 20) Team Contest Score Sheet DO NOT WRITE IN SHADED REGIONS [primes] Answer 2 or 0 2 or [points] 5 194, [locations] [cones] 8 π (or (π-2)/4)

14 School Name Team # Proctor Name Room # Final Score: KEY First Score STUDENT NAME Pressure Round Answers Answer 1 23/3 2 4 [factors] π 8 [un 2 ]

Math is Cool Masters

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