Monthly Archives: January 2011

Remainder Difference

If  is the remainder when are all divided by  (where  is greater than one), find the value of . Source: mathcontest.olemiss.edu 1/31/2011 SOLUTION Because is the remainder when are divided by , there exist positive integers such that         (1)         (2) … Continue reading

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Ten Digit Square

Find the sum of the square roots of the smallest and largest square numbers (perfect squares) that use each digit once and only once. Source: mathcontest.olemiss.edu 4/23/2007 SOLUTION         nine-digit perfect square       smallest ten-digit perfect square                largest ten-digit … Continue reading

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Reciprocal Difference of One

Two different positive numbers  each differ from their reciprocals by .  What is the sum of  ? Source: mathcontest.olemiss.edu 7/14/2008 SOLUTION There are two cases to consider depending on whether the number is greater than its reciprocal or vice-versa. CASE … Continue reading

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Month of the Planets

You randomly select a letter in the English alphabet.  What is the probability that the chosen letter does not occur in the name of one of the twelve months of the year and in the name of one of the … Continue reading

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How Many Triples?

Let represent a triple of three consecutive integers .  How many different triples exist such that the product of the three integers equals the sum of the integers? You must also provide each such triple. Source: mathcontest.olemiss.edu 11/9/2009 SOLUTION Let’s … Continue reading

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Dodecahedron

A dodecahedron has 12 faces, all of which are regular pentagons.  Three edges meet at each vertex.  An interior diagonal is a segment connecting two vertices such that the segment is not an edge or along a face of the … Continue reading

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Trapezoid

In trapezoid ABCD, base AB has a length of 52 units, base CD has a length of 39 units, side AD has a length of 5 units, and side BC has a length of 12 units.  Find the area of … Continue reading

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Unknown Area

Find the area (to the nearest tenth) of the polygon bounded by the lines , the -axis, and -axis in the coordinate plane. Source: mathcontest.olemiss.edu 2/2/2009 SOLUTION The polygon is bounded by Line represented by Eq. (1), Line represented by Eq. … Continue reading

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Hot Cocoa

It is a cold winter night and you decide to make yourself a hot cup of cocoa.  You begin by making a full cup of cocoa (consisting of water and cocoa mixture).  You drink one-eighth of it.  You then decide … Continue reading

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2003 Expansion

When the following expression is expanded, what will be the coefficient of ? Source: mathcontest.olemiss.edu 7/7/2008 SOLUTION  STEP 1 Product of two factors: . Let be the 9 coefficients, then we have . Let . The coefficients follow the pattern … Continue reading

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