Difference between revisions of "2016 AIME I Problems/Problem 14"
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== Solution == | == Solution == | ||
− | First note that <math>1001 = 143 \cdot 7</math> and <math>429 = 143 \cdot 3</math> so every point of the form <math>(7k, 3k)</math> is on the line. Then consider the line <math>l</math> from <math>(7k, 3k)</math> to <math>(7(k + 1), 3(k + 1))</math>. Translate the line <math>l</math> so that <math>(7k, 3k)</math> is now the origin. | + | First note that <math>1001 = 143 \cdot 7</math> and <math>429 = 143 \cdot 3</math> so every point of the form <math>(7k, 3k)</math> is on the line. Then consider the line <math>l</math> from <math>(7k, 3k)</math> to <math>(7(k + 1), 3(k + 1))</math>. Translate the line <math>l</math> so that <math>(7k, 3k)</math> is now the origin. There is one square and one circle that intersect the line around <math>(0,0)</math>. Then the points on <math>l</math> with an integral <math>x</math>-coordinate are, since <math>l</math> has the equation <math>y = \frac{3x}{7}</math>: |
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+ | <cmath> (0,0), (1, \frac{3}{7}), (2, \frac{6}{7}), (3, 1 + \frac{2}{7}), (4, 1 + \frac{5}{7}), (5, 2 + \frac{1}{7}), (6, 2 + \frac{4}{7}), (7,3). </cmath> | ||
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+ | We claim that the lower right vertex of the square centered at <math>(2,1)</math> lies on <math>l</math>. Since the square has side length <math>\frac{1}{5}</math>, the lower right vertex of this square has coordinates $(2 + \frac{1}{10}, |
Revision as of 16:25, 4 March 2016
Problem
Centered at each lattice point in the coordinate plane are a circle radius and a square with sides of length whose sides are parallel to the coordinate axes. The line segment from to intersects of the squares and of the circles. Find .
Solution
First note that and so every point of the form is on the line. Then consider the line from to . Translate the line so that is now the origin. There is one square and one circle that intersect the line around . Then the points on with an integral -coordinate are, since has the equation :
We claim that the lower right vertex of the square centered at lies on . Since the square has side length , the lower right vertex of this square has coordinates $(2 + \frac{1}{10},