By Richard Rusczyk, Sandor Lehoczky

The artwork of challenge fixing, quantity 1, is the vintage challenge fixing textbook utilized by many profitable MATHCOUNTS courses, and feature been a massive development block for college students who, just like the authors, played good adequate at the American arithmetic Contest sequence to qualify for the mathematics Olympiad summer season application which trains scholars for the USA overseas Math Olympiad workforce. quantity 1 is suitable for college kids simply starting in math contests. MATHCOUNTS and beginner highschool scholars rather have came upon it necessary. even if the artwork of challenge fixing is familiar through scholars getting ready for arithmetic competitions, the e-book isn't just a suite of methods. The emphasis on studying and figuring out tools instead of memorizing formulation allows scholars to unravel huge periods of difficulties past these provided within the ebook. talking of difficulties, the paintings of challenge fixing, quantity 1, comprises over 500 examples and routines culled from such contests as MATHCOUNTS, the Mandelbrot pageant, the AMC checks, and ARML. complete ideas (not simply answers!) can be found for the entire difficulties within the answer handbook.

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**Example text**

For example, if Jim is 20 years old, Jane is 10, and Sam is 5, we can say that the ratio of their ages is 20 :10 : 5, or 4 : 2 : 1 . ) Clearly, we can't use fractions in a convenient way to express the relationship among more than two quantities. EXAMPLE 4-1 A ten foot pole casts an eight foot shadow. How long is a pole which casts a twelve foot shadow? Solution: Since the height of an object is directly proportional to the length of its shadow, we can write pole _ 10 shadow 8 < 28 > the ART o f PROBLEM SOLVING < 29 We can simply substitute 12 in where 'shadow' is and get our answer: pole _ 10 ~12 ~ ~8 ' Hence, the pole is 15 feet long.

How long will it take 2 people to paint 6 houses? CHAPTER 4. PROPORTIONS 30 > Solution: Is the number of houses directly proportional or inversely proportional to the number of days and the number of people? If we have more houses, we will need more time, so the number of houses is directly proportional to the number of days. Similarly, if we have more houses and keep time constant, we need more people, so the number of houses is directly proportional to the number of people as well. Direct proportion means that the quotient is constant, so both days and people are on the opposite side of the fraction from houses.

Since Jim's mother's house is 40 miles away, it takes him one hour to get there and two to get back, so the whole trip takes 3 hours. The round trip is eighty miles, so his average speed is 80/3 = 26 1 mph. All of these "moving" problems, whether they involve driving, walking, rowing a boat, or flying a plane, can be solved using the basic formula rate x time = distance. This simply means that the rate (speed) you travel times the time you travel is the distance you travel. If you understand this you can do all of these problems.