Difference between revisions of "1997 AIME Problems/Problem 1"
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== Problem == | == Problem == | ||
| − | How many of the integers between 1 and 1000, inclusive, can be expressed as the difference of the squares of two nonnegative integers? | + | How many of the integers between 1 and 1000, inclusive, can be expressed as the [[difference of squares|difference of the squares]] of two nonnegative integers? |
== Solution == | == Solution == | ||
| − | { | + | If we let the two squares be <math>a^2 - b^2 = x</math>, then by difference of squares we have <math>(a-b)(a+b) = x</math>. Notice that <math>a-b</math> and <math>a+b</math> have the same [[parity|parities]]. This eliminates all numbers in the form of <math>4n+2</math>: when <math>x=2(2n+1)</math> is factored, one of the factors must be even, but not both, so its factors cannot have the same parity. |
| + | |||
| + | The remaining <math>\boxed{750}</math> numbers, we can describe specific squares which fit the conditions: | ||
| + | *For all odd <math>x = 2n+1</math>, <math>(n+1)^2 - (n^2) = x</math>. | ||
| + | *For all <math>x = 4n</math>, <math>(n+1)^2 - (n-1)^2 = x</math>. | ||
| + | |||
== See also == | == See also == | ||
{{AIME box|year=1997|before=First Question|num-a=2}} | {{AIME box|year=1997|before=First Question|num-a=2}} | ||
| + | |||
| + | [[Category:Intermediate Algebra Problems]] | ||
| + | [[Category:Intermediate Number Theory Problems]] | ||
Revision as of 17:23, 21 November 2007
Problem
How many of the integers between 1 and 1000, inclusive, can be expressed as the difference of the squares of two nonnegative integers?
Solution
If we let the two squares be 
, then by difference of squares we have 
. Notice that 
and 
have the same parities. This eliminates all numbers in the form of 
: when 
is factored, one of the factors must be even, but not both, so its factors cannot have the same parity.
The remaining 
numbers, we can describe specific squares which fit the conditions:
- For all odd
, 
. - For all
, 
.
, 
.
, 
.See also
| 1997 AIME (Problems • Answer Key • Resources) | ||
| Preceded by First Question |
Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
