Difference between revisions of "2014 AIME I Problems/Problem 13"
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− | <math>269+275+405+411=1360</math>, a multiple of <math>17</math>. | + | <math>269+275+405+411=1360</math>, a multiple of <math>17</math>. In addition, <math>EG=FH=34</math>, which is <math>17\cdot 2</math>. |
− | Therefore the <math>\text{square}</math> of the "hypotenuse" of a triangle with slope <math>m</math> must be a multiple of <math>17</math>. All of these triples are primitive: | + | Therefore, we suspect the <math>\text{square}</math> of the "hypotenuse" of a triangle with slope <math>m</math> must be a multiple of <math>17</math>. All of these triples are primitive: |
<cmath>17=1^2+4^2</cmath> | <cmath>17=1^2+4^2</cmath> |
Revision as of 13:08, 14 February 2016
Contents
Problem 13
On square , points , and lie on sides and respectively, so that and . Segments and intersect at a point , and the areas of the quadrilaterals and are in the ratio Find the area of square .
Solution
Notice that . This means passes through the centre of the square.
Draw with on , on such that and intersects at the centre of the square .
Let the area of the square be . Then the area of and the area of . This is because is perpendicular to (given in the problem), so is also perpendicular to . These two orthogonal lines also pass through the center of the square, so they split it into 4 congruent quadrilaterals.
Let the side length of the square be .
Draw and intersects at . .
The area of , so the area of .
Let . Then
Consider the area of .
Thus, .
Solving , we get .
Therefore, the area of
Strategy
, a multiple of . In addition, , which is . Therefore, we suspect the of the "hypotenuse" of a triangle with slope must be a multiple of . All of these triples are primitive:
The sides of the square could either be the longer leg or the shorter leg. Substituting :
The three viable answers here are . Since the areas add up to parts, probably the answer should be , a multiple of . And perhaps MAA wants a high answer at the end of the AIME :).
See also
2014 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.