Imagine Magazine - Johns Hopkins - March/April 2011 - (Page 19)

From the 2010 hMMT algebra Test Suppose that there exist nonzero complex numbers a, b, c, and d such that k is a root of both the equations ax3 + bx2 + cx + d = 0 and bx3 + cx2 + dx + a = 0. Find all possible values of k (including complex values). From the 2010 MATHCOUNTS National Competition Sprint Round 1. Four standard, six-sided dice are to be rolled. If the product of their values turns out to be an even number, what is the probability that their sum is odd? Express your answer as a common fraction. 2. Charlize accidentally omitted two consecutive integers when adding the elements of the arithmetic sequence, {1,2,3,…, n}. If the sum she obtained is 241, what is the smallest possible value of n? High School Mathematical Contest in Modeling (HiMCM) During a designated 36-hour time period, teams of up to four high school students complete a real-world problem and write up their results in a real-world format, such as a letter to the editor of a science magazine or to the president’s energy advisor. Certificates and recognition awards are granted. See website for prior years’ problems. www.himcm.org Mathematical Olympiads for Elementary and Middle Schools (MOEMS) teams of up to 35 students compete in five monthly (November– March) contests in two divisions: grades 4–6 (Division E) and 6–8 (Division M). Each contest consists of five non-routine problems that require careful mathematical thinking. Certificates, medals, or trophies are awarded to all participants. www.moems.org International Mathematics Olympiad (IMO) this world championship mathematics competition for high school students age 19 and under is held each July in a different country (2011: the Netherlands; 2012: Argentina). Each nation can send up to six students, who take two tests consisting of three problems each. Medals are awarded. See website for prior years’ problems. www.imo-o cial.org From the Division M contests 1. two consecutive positive integers are each less than 100. One integer is divisible by 17 and the other integer is divisible by 21. Find the greater of the two integers. 2. A single piece of wire is folded to form a square with area 225 sq. cm. the wire is then straightened, cut into pieces of equal length, and those pieces fastened together to form the frame of a cube. If no wire is wasted, find the volume enclosed by the cube, in cubic centimeters. Mandelbrot Competition this contest is offered at the national level for more advanced students and the regional level for less experienced problem solvers. the Mandelbrot Competition consists of five rounds spread throughout the year. Each 40-minute round consists of seven short-answer questions covering a variety of non-calculus topics. Mandelbrot team Play, held in three rounds during the winter months, is designed for students who enjoy working on more open-ended problems or who are preparing for events such as ArML or the USAMO. www.mandelbrot.org Moody’s Mega Math (M3) Challenge From Round one of the 2009 Mandelbrot Competition 1. Let f(x) be a function satisfying the equation f(f(x)) = 3x + 1. If we are told that f (0) = 223, then what must be the value of f(4)? 2. How many positive integers from 1 to 2009 have the property that, when tripled, they give a result having all even digits? For example, 1534 is one such number since 3 (1534) = 4602. teams of students in grades 11 and 12 have 14 hours to solve an openended, Internet-based, applied math modeling problem focused on a real-world issue. Scholarships totaling $100,000 are awarded. See website for previous Challenge questions. http://m3challenge.siam.org USA Mathematical Talent Search (USAMTS) Math League Contests In the American Mathematics League contests, students in grades 4–12 take timed exams containing questions from different areas of mathematics. Schools compete in statewide or multistate league competitions. www.mathleague.com/contests.htm A contest for high school students, the USAMtS consists of four, five-problem rounds. Students can start any time; participation in previous rounds is not required. the problems are published on the USAMtS website, and students have four weeks to prepare their solutions using any books, calculators, or computers they wish. Participants who have scored 71 or more after three rounds will be invited to take the AIME (see above). www.usamts.org From the USa Mathematical Talent Search 2009-10 1. Alice has three daughters, each of whom has two daughters; each of Alice’s six grand-daughters has one daughter. How many sets of women from the family of 16 can be chosen such that no woman and her daughter are both in the set? (Include the empty set as a possible set.) 2. Let a and b be positive integers such that all but 2009 positive integers are expressible in the form ma + nb, where m and n are nonnegative integers. If 1776 is one of the numbers that is not expressible, find a + b. From the 2009–2010 8th Grade Math League Contest 1. An equilateral triangle and a square have the same perimeter. If the length of a side of the triangle is 8, what is the area of the square? (A) 16 (B) 24 (C) 36 (D) 64 2. 13.25:1 = ?:8 (A) 53 (B) 106 (C) 122 (D) 150 MATHCOUNTS www.cty.jhu.edu/imagine Solutions: AMC 8: E. AMC 10: C. AMC 12: D. AIME: 8. ARML: 990; 84; 24. CML: 6; 2, 5, 6. HMMT: 20503; 1, -1, i, -i. Mandelbrot: 2011; 127. Math League: C; B. MATHCOUNTS: 8/15; 23. MOEMS: 85; 125cc. USAMTS: 2926; 133. imagine Middle school students compete at the school, chapter, state, and national level. Winners of the approximately 500 chapter meets progress to state contests held in March; the top four scorers and top coach in each state earn an all-expenses-paid trip to Washington, DC, for the national finals in May. www.mathcounts.org 19 http://www.himcm.org http://www.moems.org http://www.imo-official.org http://www.mandelbrot.org http://m3challenge.siam.org http://www.usamts.org http://www.mathleague.com/contests.htm http://www.mathcounts.org http://www.cty.jhu.edu/imagine

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

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