Difference between revisions of "1991 AIME Problems/Problem 2"
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The length of the diagonal is <math>\sqrt{3^2 + 4^2} = 5</math> (a 3-4-5 [[right triangle]]). For each <math>k</math>, <math>\overline{P_kQ_k}</math> is the [[hypotenuse]] of a <math>3-4-5</math> right triangle with sides of <math>3 \cdot \frac{k}{168}, 4 \cdot \frac{k}{168}</math>. Thus, its length is <math>5 \cdot \frac{k}{168}</math>. Let <math>a_k=\frac{5k}{168}</math>. We want to find 2<math>\sum\limits_{k=1}^{168} a_k</math>-5 since we are over counting <math>5</math>. | The length of the diagonal is <math>\sqrt{3^2 + 4^2} = 5</math> (a 3-4-5 [[right triangle]]). For each <math>k</math>, <math>\overline{P_kQ_k}</math> is the [[hypotenuse]] of a <math>3-4-5</math> right triangle with sides of <math>3 \cdot \frac{k}{168}, 4 \cdot \frac{k}{168}</math>. Thus, its length is <math>5 \cdot \frac{k}{168}</math>. Let <math>a_k=\frac{5k}{168}</math>. We want to find 2<math>\sum\limits_{k=1}^{168} a_k</math>-5 since we are over counting <math>5</math>. | ||
− | <math>2\sum\limits_{k=1}^{168} \frac{5k}{168}-5 | + | <math>2\sum\limits_{k=1}^{168} \frac{5k}{168}-5 |
− | = | + | =2\sum\limits_{k=0}^{168} \frac{5k}{168}-5 |
− | = | + | =2\frac{(0+5)\cdot169}{2}-5 |
− | = | + | =168\cdot5 |
− | = | + | =\boxed{840}</math> |
== See also == | == See also == |
Revision as of 00:48, 3 January 2016
Problem
Rectangle has sides of length 4 and of length 3. Divide into 168 congruent segments with points , and divide into 168 congruent segments with points . For , draw the segments . Repeat this construction on the sides and , and then draw the diagonal . Find the sum of the lengths of the 335 parallel segments drawn.
Solution
The length of the diagonal is (a 3-4-5 right triangle). For each , is the hypotenuse of a right triangle with sides of . Thus, its length is . Let . We want to find 2-5 since we are over counting .
See also
1991 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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