Difference between revisions of "2000 AIME II Problems/Problem 6"

Revision as of 19:28, 11 November 2007

Problem

One base of a trapezoid is $100$ units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio $2: 3$ . Let x be the length of the segment joining the legs of the trapezoid that is parallel to the bases and that divides the trapezoid into two regions of equal area. Find the greatest integer that does not exceed $x^2/100$ .

Solution

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 == Problem ==
-For how many ordered pairs <math>(x,y)</math> of integers is it true that <math>0 < x < y < 10^{6}</math> and that the arithmetic mean of <math>x</math> and <math>y</math> is exactly <math>2</math> more than the geometric mean of <math>x</math> and <math>y</math>?
+One base of a trapezoid is <math>100</math> units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio <math>2: 3</math>. Let x be the length of the segment joining the legs of the trapezoid that is parallel to the bases and that divides the trapezoid into two regions of equal area. Find the greatest integer that does not exceed <math>x^2/100</math>.
 == Solution ==

2000 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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Difference between revisions of "2000 AIME II Problems/Problem 6"

Revision as of 19:28, 11 November 2007

Problem

Solution

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