Monthly Archives: July 2011

Two Prime Roots

Two prime numbers are roots of the quadratic equation . What value of makes this a true statement? Source: mathcontest.olemiss.edu 11/13/2006 SOLUTION Let and be two prime numbers that are roots of the quadratic equation . We factor the equation … Continue reading

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Water Tank

A tank contains 250 gallons of water. A specific amount of water is drained from the tank and replaced with the same amount of alcohol. The tank is thoroughly mixed. From the new mixture, the same amount of liquid is … Continue reading

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Triangle ADE

Points A, B, C, and D are all on segment AD, in the order listed. Segment AB is congruent to segment CD. Segment BC equals 12 units in length. Also, an additional point E exists that is not on segment … Continue reading

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That’s That!

If THAT = (AH)(HA) and each letter represents a unique digit, determine the four-digit value of THAT. Source: mathcontest.olemiss.edu 2/12/2007 SOLUTION We begin by forming the products (AH)(HA) using the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 … Continue reading

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Triangular Thirty-six

The integer 36 is a square number because it equals 6 x 6. Thirty-six is also a triangular number because it is the sum of the first eight positive whole numbers. Thirty-six is the smallest integer greater than one that … Continue reading

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Four In A Row

Four numbers are displayed in a row. The average of the first two numbers is 8, the average of the middle two numbers is 11.8, and the average of the last two numbers is 7.7. Determine the average of the … Continue reading

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Friendly Numbers

Two integers are said to be amicable or friendly if each number is the sum of all the possible divisors of the other. One such example is 220 and 284. Since the possible divisiors of 220 are 1, 2, 4, … Continue reading

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Michigan Papyrus

The Michigan papyrus 620 is a Greek papyrus written in the second century. It contains three arithmetic problems. One of these problems follows: Four numbers: their sum is 9900; let the second exceed the first by one-seventh of the first; … Continue reading

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Inverts

Find the sum of the reciprocals of two real numbers, given that these two numbers have a sum of 50 and a product of 25. Source: mathcontest.olemiss.edu 5/7/2007 SOLUTION The above figure shows the graph of the line and the … Continue reading

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The Power of Seven

What digits would be found in the tens and ones place of 7 to the 7001st power? Source: mathcontest.olemiss.edu SOLUTION The pattern of the tens and ones place digits is . To determine the tens and ones place digits of … Continue reading

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