Difference between revisions of "1985 AIME Problems"

m (Problem 3: added)
m (Problem 4: added)
Line 13: Line 13:
  
 
==Problem 4==
 
==Problem 4==
 +
A small  square is constructed inside a square of area 1 by dividing each side of the unit square into <math>n</math> equal parts, and then connecting the  vertices to the division points closest to the opposite vertices. Find the value of <math>n</math> if the the area of the small square is exactly <math>\frac1{1985}</math>.
  
 +
[[1985 AIME Problems/Problem 4 | Solution]]
  
[[1985 AIME Problems/Problem 4 | Solution]]
 
 
==Problem 5==
 
==Problem 5==
  

Revision as of 09:23, 3 December 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

Find $c$ if $a$, $b$, and $c$ are positive integers which satisfy $c=(a + bi)^3 - 107i$, where $i^2 = -1$.

Solution

Problem 4

A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly $\frac1{1985}$.

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

Invalid username
Login to AoPS