Monthly Archives: February 2013

Antifreeze and Water

A quart bottle contains a mixture that is antifreeze and water. A half-gallon bottle contains a mixture that is  antifreeze and  water. The contents of the two bottles are poured into a gallon jug. In the form , what is the ratio … Continue reading

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A Plus B

and are real numbers. If the sum of and is two and the sum of the squares of and is , what is the value of the sum of the cubes of and ? Source: mathcontest.olemiss.edu 2/25/2013 SOLUTION Calculate Substitute … Continue reading

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Rectangle and Equilateral Triangle

A rectangle and an equilateral triangle with equal area can both be inscribed in a circle with radius 2 cm. What are the dimensions of the rectangle? Source: NCTM Mathematics Teacher SOLUTION is a triangle with and . Area of … Continue reading

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Isosceles Trapezoid

The isosceles trapezoid is inscribed in a semicircle such that the trapezoid and the semicircle share the same base, , and . Compute . Source: NCTM Mathematics Teacher SOLUTION Triangle is a right triangle because intercepts a semicircle. By the … Continue reading

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Sum of Four Consecutive Integers

How many positive multiples of that are less than are the sum of four consecutive integers? Source: NCTM Mathematics Teacher SOLUTION The pattern is These multiples of when plotted on a number line are separated by an interval of . … Continue reading

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Least Right Side

The sides of a right triangle are all positive integers. Two of the sides are odd numbers that differ by 50. What is the least possible value for the third side? Source: mathcontest.olemiss.edu 2/18/2013 SOLUTION By the Pythagorean theorem The … Continue reading

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Four Points in The Plane

We are given points and in the plane such that , whereas . Compute the measure of in degrees. Source: NCTM Mathematics Teacher SOLUTION is the center of a circle of radius passing through points . The central because is equilateral. … Continue reading

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Circular Table

The circular table in the diagram is pushed against two perpendicular walls. The point on the circumference of the table is a distance from one wall and a distance from the other. What is the radius of the table? Source: … Continue reading

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Ten Slips of Paper

Ten slips of paper, numbered through , are placed in a hat. Three numbers are drawn out, one after another. What is the probability that the three numbers are drawn in increasing order? Source: NCTM Mathematics Teacher SOLUTION First Smaller … Continue reading

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Cards on Forehead Game

Bob, Fred, and Vi play a game in which a card numbered , or is stuck to each of their foreheads. Each player can see the other two players’ numbers but not his or her own. If an observer tells … Continue reading

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