advertisement

CALIFORNIA STATE UNIVERSITY, BAKERSFIELD Lee Webb Math Field Day 2010 Team Medley, Junior-Senior Level Each correct answer is worth ten points. Justification of answers is required for full credit. Partial credit may be given. An unanswered question is given zero points. You have 50 minutes to complete the exam. When the exam is over, give only one set of answers per team to the proctor. Two different answers to the same problem will invalidate both. Elegance of solutions may affect score and may be used to break ties. All calculators, cell phones, music players, and other electronic devices should be put away in backpacks, purses, pockets, etc. Leaving early or otherwise disrupting other contestants may be cause for disqualification. 1. Suppose ABCD is a square. Rays AB, BC , CD, DA divide the plane into five regions (the square itself and four unbounded regions outside the square). With 5 colors, how many ways can the regions be painted so that no adjacent regions have the same color (note: not all colors must be used in each of the colorings). 2. Suppose sin cos 3. Find all integer values of x such that x 2 7 x 4 is a perfect square. 4. Find three odd integers, a, b, c such that a b c and 5. Suppose a, b, c are three distinct real numbers such that 7 . Find sin cos and sin 3 cos3 . 3 1 1 1 1 . a b c 3 a b c . What bc ca ab are the possible values for these fractions? 6. Two circles are situated so that the center of each circle is on the other circle. What is the ratio of the area common to both circles to the total area covered by both circles? 7. Let S={1,2,3,4,5,6,7,8,9,10} and that you must choose subsets A,B,C,D,E such that A B C D E . How many possible choices are there for A,B,C,D,E (note some subsets may be empty)? You may leave your answer as a product of smaller numbers. 8. Suppose ABC is an equilateral triangle and that P is a point in the same plane such that PAB, PBC, and PCA are all isosceles triangles. How many such points P are there?