Ciphering MU ALPHA THETA STATE 2008 ROUND

Transcription

1 Ciphering MU ALPHA THETA STATE 2008 ROUND SCHOOL NAME ID CODE Circle one of the following Mu Alpha Theta Euclidean

2 Round 1 What is the distance between the points (1, -6) and (5, -3)? Simplify: ) 5 2) 3,149 A regular hexagon has a perimeter of 36 centimeters. Find the number of centimeters in the sum of the lengths of all diagonals of the hexagon. (Express your answer in simplest radical form) 3) What is the probability of being dealt a five card poker hand that is a royal flush using a standard 52-card deck? (A royal flush is an ace, king, queen, jack and 10 all of the same suit) 4) 1 649,740

3 Round 2 1 The sum of the solutions to a quadratic equation is, and the product of the 2 3 solutions is. What is the quadratic equation? Answer in y = ax 2 + bx + c form, 4 where a, b and c are the relatively prime integers and a > 0. 1) y = 4x 2 + 2x + 3 How many positive three-digit numbers have exactly two digits that are the same? Solve for x in the following problem: log 2 (x 1) + log 2 (x + 1) = 3 2) 243 3) 3 When General Han counts the soldiers in his army, he uses the following method. He orders them to line up in rows of 11, then rows of 13 and lastly in rows of 17. One morning when counting who is present using this method he found there were 3 extra soldiers when lining up in rows of 11, 4 extra in rows of 13 and 9 extra when lining up in rows of 17. If he has exactly 1,000 soldiers in his army, how many were absent the morning he counted them? 4) 73

4 Round 3 1) What is 10% of 20% of 30% of 50% of 2,000? 1) 6 What is the measure of the space diagonal of a rectangular prism with sides measuring 2cm, 4cm and 6cm? Answer in centimeters and in simplest radical form. Find n such that n = 15! 2) ) 168,168 How many ordered triples of positive prime numbers (a, b, c) exist such that a < b < c and a + b + c = 26? 4) 3

5 Round 4 Find A + B + C A = The number of diagonals in a regular decagon B = The number ways you can choose three cheerleaders from a group of ten cheerleaders C = The number of positive integral factors of 100 1) 164 How many different 5 digit zip codes exist if the first three digits are each greater than 5, the fourth digit is a prime number, and the last digit is odd? Evaluate ) 1,280 3) 6 A circle is tangent to two sides of a square and its diagonal. Given that the length of a side of the square is 4 units, how many units are in the radius of the circle? Express your answer in simplest radical form. 4) 4-2 2

6 Round 5 List in order using A, B, and C from largest to smallest. A= 5 33 B = C = 2 77 What is the sum of the positive integral factors of 100? 1) C,A,B 2) 217 How many non-congruent rectangles with sides of integer length do not have more than 25 square units in their areas? 3) 46 What is the probability of flipping five fair coins and obtaining exactly 3 heads? Express your answer as a common fraction. 4) 16 5

7 Round 6 What is the equation of the line perpendicular to x + y = 4, that passes through the point (1, 1)? Express your answer in slope/intercept form. State all solutions to the following equation. 1) y = x x - 10 x + 21 = 0 2) 49,and 9 The sum of consecutive even integers m = What is the value of m? 3) 100 How many distinct right triangles can you form with all sides being integers from 1-15? 4) 4

8 Round 7 Solve for x and express your answer as a reduced common fraction. 3 x 2 x 2 3 = ) 25 What is the smallest positive prime number that can be written as the sum of three distinct prime numbers? 2) 19 Solve for x and express your answer as a reduced common fraction 1 2x + 3 = ) 2 How many ordered pairs (x, y) of positive integers satisfy the following equation? x + y + xy = 63 4) 5

9 Round 8 What is the sum of the following arithmetic series? ) 4,592 What is the sum of the volume of a sphere with radius 6cm and the volume of a circular cylinder with radius 4cm and height 5cm? Express your answer in terms of π and cubic centimeters. 2) 368π Point P is 9 units from the center of a circle with radius 15. How many different chords of the circle contain point P and have integer lengths? Simplify the following: answer as a common fraction 3) ) 61

10 Round 9 Simplify the following expression to a single fraction with positive exponents. x x 2 1 y y 2 1 1) y + x xy Find A + B + C A= 45 2 B = C = ) 4,846 Solve for x in the following equation: log 2 (log 3 (log 2 (x - 1))) = 1 3) 513 The increasing sequence of positive integers a 1, a 2, a 3, has the property that a 2 + n = a n + a n + 1 where n > 1 If a 7 = 120 then a 8 =? 4) 194

11 Round 10 In a certain cross-country meet between two teams of five runners each, a runner who finishes in the n th position contributes n points to his team s score. The team with the lower score wins. If there are no ties among the runners, how many different winning scores are possible? 1) 13 How many different arrangements using the letters of SACRAMENTO are there? How many positive prime numbers are less than 50? 2) 1,814,400 What is the prime factorization of 2008? 3) 15 4)