Imagine Magazine - Johns Hopkins - March/April 2011 - (Page 18)

National MathematicsCompetitions American Mathematics Competitions http://amc.maa.org AMC 8 Students in grade 8 or below and homeschool students age 14 or younger take this 40-minute exam that consists of 25 multiple-choice questions. topics include arithmetic of integers, fractions and decimals, percent and proportion, number theory, informal geometry, perimeter, area, volume, probability and statistics, and logical reasoning. High-scoring students are awarded certificates. the 12 top-scoring students are invited to the Olympiad Awards Ceremony in Washington, DC. the top six scorers on the USAMO form the team that represents the U.S. in the International Mathematics Olympiad, a 10–14 day trip and examination for the most talented students from more than 70 countries (see IMO below). American Regions Math League Competition (ARML) From the 2010 aMC 8 the lengths of the sides of a triangle measured in inches are three consecutive integers. the length of the shortest side is 30% of the perimeter. What is the length of the longest side? (A) 7 (B) 8 (C) 9 (D) 10 (E) 11 t eams of 15 high school students represent their school, city, county, or state in this annual competition. Students compete in five rounds: Individual (students solve ten problems independently), t eam (all 15 members work together on short-answer questions), Power (students cooperate on a proof-oriented question), and relay (students work in teams of three to solve problems using answers from the teammate ahead of them). www.arml.com AMC 10 and AMC 12 From the 2010 aRML Individual: Let x be a real number in the interval [0, 360] such that the four expressions sin x˚, cos x˚, tan x˚, cot x˚ take on exactly three distinct (finite) real values. Compute the sum of all possible values of x. team: Compute the smallest positive integer n such that nn has at least 1,000,000 positive divisors. relay: If A, R, M, and L are positive integers such that A2 + R2 = 20 and M2 + L2 = 10, compute the product A·R·M·L. the AMC 10 is for high school students in grade 10 or below and homeschool students age 16 or younger. the AMC 12 is for advanced high school students in any grade and homeschool students age 18 or younger. Both 75-minute exams consist of 25 multiplechoice questions on non-calculus secondary school mathematics topics. Students who score at least 100 out of the possible 150 points on the AMC 12 and those who score at least 120 out of 150 on the AMC 10 are invited to take the AIME (see below). From the 2010 aMC 10 Shelby drives her scooter at a speed of 30 miles per hour if it is not raining, and 20 miles per hour if it is raining. today she drove in the sun in the morning and in the rain in the evening, for a total of 16 miles in 40 minutes. How many minutes did she drive in the rain? (A) 18 (B) 21 (C) 24 (D) 27 (E) 30 Continental Mathematics League CML oﬀers a series of math meets for students in grades 2–9 and a calculus league for high school students. Participants at each grade level work independently, with a school’s top six scores becoming its team score. All teams receive certificates and medals; national and regional awards go to top individuals and teams. www.continentalmathematicsleague.com From the 2010 aMC 12 For what value of x does log√2 √x + log2 x + log4 (x2) + log8 (x3) + log16 (x4) = 40? (A) 8 (B) 16 (C) 32 (D) 256 (E) 1024 From the 1999 CML the number of square units of the surface area of a cube is greater than the number of cubic units of the volume until the side of a cube reaches a length of __. (Pythagorean Division, Grade 9) the product of three whole numbers is 60. the numbers are all diﬀerent and greater than 1. the sum of the three numbers is 13. What are the numbers? (Euclidean Division, Grade 7) American Invitational Mathematics Examination (AIME) the three-hour AIME is made up of 15 questions, with integer answers from 0 to 999, that can be solved using pre-calculus methods. All participants receive a Certificate of Participation, and the top-scoring participants (based on a weighted average that includes AMC 10 and AMC 12 scores) are invited to take the USAMO or the USAJMO (see below). Harvard-MIT Mathematics Tournament (HMMT) t eams of up to eight high school students register for one of two divisions and then compete in both team and individual events. Individuals choose one of six 120-minute short-answer tests covering such topics as algebra, geometry, calculus, and combinatorics. t eams participate in both a 90-minute test of 14–20 proof-style problems and the Guts round, an 80-minute event consisting of 36 questions. http://web.mit.edu/hmmt/www From the 2010 aIME Find the smallest positive integer n with the property that the polynomial x4 – nx + 63 can be written as a product of two nonconstant polynomials with integer coeﬃcients. USA Mathematical Olympiad (USAMO) and USA Junior Mathematical Olympiad (USAJMO) From the 2010 hMMT General Test John needs to pay 2010 dollars for his dinner. He has an unlimited supply of 2, 5, and 10 dollar notes. In how many ways can he pay? Six weeks after the AIME, approximately 180 qualifying students take the USAMO or the USAJMO, a two-day, nine-hour exam consisting of six essay/proof questions. 18 imagine Mar/Apr 2011 http://amc.maa.org http://www.arml.com http://www.continentalmathematicsleague.com http://web.mit.edu/hmmt/www

Table of Contents for the Digital Edition of Imagine Magazine - Johns Hopkins - March/April 2011

Imagine Magazine - Johns Hopkins - March/April 2011
Contents
Big Picture
In My Own Words
Problem Solving
Orange County Math Circle
Number Theory
Count Me In
National Mathematics Competitions
Math at the Science Fair
Just My Speed
MathPath
When Origami Meets Rocket Science
Selected Opportunities & Resources
Making a Difference
High School Options for Gifted Students
Off the Shelf
Word Wise
Exploring Career Options
One Step Ahead
Planning Ahead for College
Students Review
Mark Your Calendar
Knossos Games

Imagine Magazine - Johns Hopkins - March/April 2011