Monthly Archives: August 2015

Similar Triangles

In triangle , and . Side is extended to point so that triangle is similar to triangle . What is ? Source: NCTM Mathematics Teacher, November 2006 SOLUTION Triangle is similar to triangle Solving Cross multiply Solving Multiply the left … Continue reading

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Spy Code

A spy had to send the 4-digit code to headquarters. For security reasons, he sent instead these separate codes: In each of the codes, at least one of the digits , and occurs in the correct position. What is the … Continue reading

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More Remainders

How many positive integers have a remainder when is divided by ? Source: NCTM Mathematics Teacher, November 2006 SOLUTION If divided by has a remainder , there exists a quotient such that and . which implies that the number of … Continue reading

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Black and White Marbles

Each of two boxes contains marbles, and each marble is either black or white. The total number of black marbles is different from the total number of white marbles. One marble is drawn at random from each box. The probability … Continue reading

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Value of a+b+c

There are positive integers , and that satisfy this system of two equations: What is the value of ? Source: NCTM Mathematics Teacher, November 2006 SOLUTION Multiply¬†the first equation by , the second equation by and add¬†them because is a … Continue reading

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Area of Right Triangle

A right triangle with integer side lengths , and satisfies and . What is the area of the right triangle? Source: NCTM Mathematics Teacher, November 2006 SOLUTION must be positive There are two values of between and for which is … Continue reading

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Function Value

Suppose that is a function such that for every real number . What is the value of ? Source: NCTM Mathematics Teacher, November 2006 SOLUTION When When Multiply Eq. by and Eq. by and add them Answer:

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Divisible by 25

How many 5-digit numbers with all digits nonzero and no digit repeated are divisible by ? Source: NCTM Mathematics Teacher, November 2006 SOLUTION A number is divisible by if it ends in , or . The nonzero digit condition restricts … Continue reading

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Base of Number

If the number in base is represented as in base , what is the base-ten representation of the number in base ? Source: NCTM Mathematics Teacher, November 2006 SOLUTION Solving the quadratic equation for yields . Answer:

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Concentric Circles

The circles in the figure are concentric. The chord is tangent to the inner circle and has length . What is the area of the non-shaded region? Source: NCTM Mathematics Teacher, November 2006 SOLUTION area of large circle = area … Continue reading

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