Difference between revisions of "1985 AIME Problems"

 
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Let <math>x_1=97</math>, and for <math>n>1</math> let<math>x_n=\frac{n}{x_{n-1}}</math>. Calculate the product <math>x_1x_2x_3x_4x_5x_6x_7x_8</math>.
 
Let <math>x_1=97</math>, and for <math>n>1</math> let<math>x_n=\frac{n}{x_{n-1}}</math>. Calculate the product <math>x_1x_2x_3x_4x_5x_6x_7x_8</math>.
  
[[USA AIME 1985 Problems/Problem 1 | Solution]]
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[[1985 AIME Problems/Problem 1 | Solution]]
 
==Problem 2==
 
==Problem 2==
When a right triangle is rotated about one leg, the volume of the cone produced is <math>800\pi \text{cm}^3</math>. When the triangle is rotation about the other leg, the volume of the cone produced is <math>1920\pi \text{cm}^3</math>. What is the length (in cm) of the hypotenuse of the triangle?
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When a right triangle is rotated about one leg, the volume of the cone produced is <math>800\pi \;\textrm{cm}^3</math>. When the triangle is rotated about the other leg, the volume of the cone produced is <math>1920\pi \;\textrm{cm}^3</math>. What is the length (in cm) of the hypotenuse of the triangle?
  
[[USA AIME 1985 Problems/Problem 2 | Solution]]
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[[1985 AIME Problems/Problem 2 | Solution]]
 
==Problem 3==
 
==Problem 3==
  
  
[[USA AIME 1985 Problems/Problem 3 | Solution]]
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[[1985 AIME Problems/Problem 3 | Solution]]
 
==Problem 4==
 
==Problem 4==
  
  
[[USA AIME 1985 Problems/Problem 4 | Solution]]
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[[1985 AIME Problems/Problem 4 | Solution]]
 
==Problem 5==
 
==Problem 5==
  
  
[[USA AIME 1985 Problems/Problem 5 | Solution]]
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[[1985 AIME Problems/Problem 5 | Solution]]
 
==Problem 6==
 
==Problem 6==
  
  
[[USA AIME 1985 Problems/Problem 6 | Solution]]
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[[1985 AIME Problems/Problem 6 | Solution]]
 
==Problem 7==
 
==Problem 7==
  
  
[[USA AIME 1985 Problems/Problem 7 | Solution]]
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[[1985 AIME Problems/Problem 7 | Solution]]
 
==Problem 8==
 
==Problem 8==
  
  
[[USA AIME 1985 Problems/Problem 8 | Solution]]
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[[1985 AIME Problems/Problem 8 | Solution]]
 
==Problem 9==
 
==Problem 9==
  
  
[[USA AIME 1985 Problems/Problem 9 | Solution]]
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[[1985 AIME Problems/Problem 9 | Solution]]
 
==Problem 10==
 
==Problem 10==
  
  
[[USA AIME 1985 Problems/Problem 10 | Solution]]
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[[1985 AIME Problems/Problem 10 | Solution]]
 
==Problem 11==
 
==Problem 11==
  
  
[[USA AIME 1985 Problems/Problem 11 | Solution]]
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[[1985 AIME Problems/Problem 11 | Solution]]
 
==Problem 12==
 
==Problem 12==
  
  
[[USA AIME 1985 Problems/Problem 12 | Solution]]
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[[1985 AIME Problems/Problem 12 | Solution]]
 
==Problem 13==
 
==Problem 13==
  
  
[[USA AIME 1985 Problems/Problem 13 | Solution]]
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[[1985 AIME Problems/Problem 13 | Solution]]
 
==Problem 14==
 
==Problem 14==
  
  
[[USA AIME 1985 Problems/Problem 14 | Solution]]
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[[1985 AIME Problems/Problem 14 | Solution]]
 
==Problem 15==
 
==Problem 15==
  
  
[[USA AIME 1985 Problems/Problem 15 | Solution]]
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[[1985 AIME Problems/Problem 15 | Solution]]
----
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==See also==
* [[USA AIME 1985]]
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* [[1985 AIME]]
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* [[American Invitational Mathematics Examination]]
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* [[AIME Problems and Solutions]]
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* [[Mathematics competition resources]]

Revision as of 22:25, 7 November 2006

Problem 1

Let $x_1=97$, and for $n>1$ let$x_n=\frac{n}{x_{n-1}}$. Calculate the product $x_1x_2x_3x_4x_5x_6x_7x_8$.

Solution

Problem 2

When a right triangle is rotated about one leg, the volume of the cone produced is $800\pi \;\textrm{cm}^3$. When the triangle is rotated about the other leg, the volume of the cone produced is $1920\pi \;\textrm{cm}^3$. What is the length (in cm) of the hypotenuse of the triangle?

Solution

Problem 3

Solution

Problem 4

Solution

Problem 5

Solution

Problem 6

Solution

Problem 7

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

Solution

Problem 15

Solution

See also

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