AreTeem / Math Zoom Summer Camp 2014 Assessment Test Page 1 AreTeem / Math Zoom Summer Camp 2014 Assessment Test Cover Sheet Your name (please print) Last First Current Grade School Address Phone # Email (please print legibly) Number of pages (not including this cover sheet) Math Zoom Academy, 4850 Barranca Pkwy, #203, Irvine, CA 92604 (949) 679-8989 http://www.mathzoom.org AreTeem / Math Zoom Summer Camp 2014 Assessment Test Page 2 AreTeem / Math Zoom Summer Camp 2014 Assessment Test • The Assessment Test aims to determine the skill levels of students with little or no experiences in math competitions, for the purpose of placement in the Math Zoom Summer Camp. If you meet one of the following requirements, you are not required to submit the Assessment Test: (a) Qualified for AIME; (b) Scored 19 or above in AMC 8; (c) Entered the MathCounts State level competition as an individual or in a team; (d) Top performer in ARML; (e) Good performance in other math contests comparable to the above—please provide details. • There are 16 problems in this test. Give yourself 5 days to work on them. You don’t have to solve all the problems. Solve as many of the problems as you can. Choose six (6) of your favorite solutions and submit them. You are encouraged to submit more than six if possible. Don’t be discouraged if you can’t solve six of them; simply send whatever you have done after 7 days. • All of these problems can be solved using only elementary techniques. Some are quite easy, and some pretty challenging. Most of the time, you need to find clever ways to combine the elementary techniques in order to find a solution. • Complete solutions are required for all problems. Partial credits are given to well-reasoned progress toward a solution, even though the solution is incorrect or incomplete. Do not simply submit an unsupported answer. You should include all significant steps in your reasoning and computation. • There are a cover sheet and an answer sheet. Print out both sheets, and make several copies of the blank answer sheet. Fill out the top of each answer sheet as you go, and then fill out the cover sheet when you’re finished. Start each problem on a new answer sheet. • All the work you present must be your own. • You must submit your solutions by regular mail, e-mail or fax (714-784-7838) as soon as possible. Make sure that the cover sheet is the first page of your submission, and that it is completely filled out. Solutions are to be mailed to the following address: Math Zoom Summer Camp 4850 Barranca Pkwy, Suite 203 Irvine, CA 92604 If you e-mail your solutions, please send them to mathcamp@mathzoom.org E-mailed solutions may be written and scanned or typed in TEX. They should be sent as an attachment in either .doc or .pdf format. If you write and scan your solutions, insert the scans into a .doc or .pdf file, and send just the one file. Please go to the next page for the problems. Math Zoom Academy, 4850 Barranca Pkwy, #203, Irvine, CA 92604 (949) 679-8989 http://www.mathzoom.org AreTeem / Math Zoom Summer Camp 2014 Assessment Test Page 3 AreTeem / Math Zoom Summer Camp 2014 Assessment Test 1. The digits 2, 4, 5, 8 and 9 are each used once to form the largest possible odd five-digit number. What is this five-digit number? 2. Given numbers a = 2555 , b = 3333 , c = 5222 . Arrange them in increasing order. 3. The average of the five numbers in a list is 130. The average of the first and the last numbers is 100. What is the average of the three numbers in the middle? 4. How many positive integers below 400 have at least one “3” as a digit? 5. Six passengers A,B,C,D,E,F are taking the same train. They came from six different cities: Boston, New York, Los Angeles, Dallas, Seattle, Atlanta. Given that (a) A and the person from Boston are both doctors; E and the person from New York are both teachers; C and the one from Los Angeles are both engineers. (b) A,B,F, and the Dallas person served in the military. The Los Angeles person never did. (c) The person from Seattle is older than A. The person from Atlanta is older than B. F is the youngest among the six. (d) B is traveling together with the person from Boston. C is traveling together with the person from Seattle. Please determine the origin and occupation of each passenger. 6. It takes 60 minutes for Dave to walk from home to school. It takes 20 minutes if he rides a bike instead. One day, he first rides a bike for 12 minutes before the bike breaks. He next walks to school for the remaining distance. How many minutes in total does Dave spend on the road? 7. The Goldbach Conjecture says that each even integer greater than 2 can be written as the sum of two prime numbers. For example, 4 = 2 + 2, 6 = 3 + 3, 12 = 5 + 7, etc. The number 126 can be written as the sum of two prime numbers in many different ways. How many different ways can you come up with? Write as many as possible. 8. The equiangular convex hexagon ABCDEF has AB = 1, BC = 4, CD = 2, and DE = 4. Find [ABCDEF ], where [ABCDEF ] denote the area of the hexagon. Math Zoom Academy, 4850 Barranca Pkwy, #203, Irvine, CA 92604 (949) 679-8989 http://www.mathzoom.org AreTeem / Math Zoom Summer Camp 2014 Assessment Test Page 4 9. The side length of square ABCD is 2. Let E be a point inside square ABCD such that CDE is an equilateral triangle. Find the area of triangle AEC. 10. In quadrilateral ABCD, BD is perpendicular to AD and BC. Given that BD = 8, AB = 17, CD = 10. Find the area of ABCD. 11. The numbers u, v, w, x, y, z satisfy the following system: u+v v+w w+x x+y y+z = = = = = 2 9 37 12 40 What is the value of u + z? 12. A quadratic equation ax2 − 10ax + 5b = 0 has two real solutions. What is the average of the solutions? 13. How many positive factors does the number 3600 have? 14. At 7:18pm, what is the measure (in degrees) of the angle between the hour hand and the minute hand on a standard clock? 15. If |m − 2011| = −(n − 2012)2 , what is (m − n)2013 ? 16. In triangle ABC, let A′ be a point on BC such that BA′ = 2A′ C, B ′ on AC such that CB ′ = 2B ′ A, and C ′ is on AB and AC ′ = 2C ′ B. Connect AA′ , BB ′ , and CC ′ , and an internal triangle P QR is formed, as shown. Find the ratio of areas [P QR]/[ABC]. ([P QR] and [ABC] denote the areas of △P QR and △ABC respectively.) Math Zoom Academy, 4850 Barranca Pkwy, #203, Irvine, CA 92604 (949) 679-8989 http://www.mathzoom.org AreTeem / Math Zoom Summer Camp 2014 Assessment Test Page 5 Your name (please print) Problem Number Page of Write neatly! Write all work inside the box. Do NOT write on the back of the page. Math Zoom Academy, 4850 Barranca Pkwy, #203, Irvine, CA 92604 (949) 679-8989 http://www.mathzoom.org