Monthly Archives: October 2015

House of Cards

A 4-story house of cards requires cards (as shown). How many cards are required to build a 98-story house of cards? Source: NCTM Mathematics Teacher, December 2005 SOLUTION If we number the rows from the top down as rows , … Continue reading

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Find All Integers

Find all integers such that is also an integer. Source: NCTM Mathematics Teacher, December 2005 SOLUTION The expression is an integer if is a divisor of . The divisors of are: Eight integers make the expression also an integer. Answer: 8

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Right Triangle

Triangle is right-angled at . is an altitude of the triangle. and . What is the value of ? Source: NCTM Mathematics Teacher, December 2005 SOLUTION Theorem: If the altitude is drawn to the hypotenuse of a right triangle, then … Continue reading

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Sum of Terms

In the sequence of numbers , each term (starting with the third) is equal to the term preceding it minus the term preceding that one. What is the sum of the first terms of this sequence? Source: NCTM Mathematics Teacher, … Continue reading

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Integral Factors

What is the total number of positive integral factors of ? Source: NCTM Mathematics Teacher, November 2006 SOLUTION Integral factors are made by multiplying the powers of and . We can choose from¬† powers of from to , powers of … Continue reading

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Planting Flowers

Four triangular gardening plots form a square as shown. Each plot will contain one kind of flower, and flowers in plots that share an edge will be different. How many different ways can the garden be planted if the flowers … Continue reading

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Integral Points

A point is called integral if both and are integers. How many points on the graph of are integral points? Source: NCTM Mathematics Teacher, November 2006 SOLUTION Divide by , Divide by Divide by , no solution , duplicate solution … Continue reading

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Mean and Median

What is the number of distinct real numbers that have the property that the median of the five numbers is equal to their mean? Source: NCTM Mathematics Teacher, November 2006 SOLUTION Suppose is the smallest number. We order the five … Continue reading

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Painting Train Cars

Each car of a five-car train must be painted a solid color. The only color choices are red, blue, and yellow. If each of these colors must be used for at least one car, in how many ways can this … Continue reading

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