Difference between revisions of "1989 AIME Problems/Problem 4"
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== Problem == | == Problem == | ||
| − | If <math>a<b<c<d<e^{}_{}</math> are consecutive positive | + | If <math>a<b<c<d<e^{}_{}</math> are [[consecutive]] [[positive]] [[integer]]s such that <math>b+c+d^{}_{}</math> is a [[perfect square]] and <math>a+b+c+d+e^{}_{}</math> is a [[perfect cube]], what is the smallest possible value of <math>c^{}_{}</math>? |
== Solution == | == Solution == | ||
| − | + | Since the middle term of an [[arithmetic progression]] with an odd number of terms is the average of the series, we know <math>b + c + d = 3c</math> and <math>a + b + c + d + e = 5c</math>. Thus, <math>c</math> must be in the form of <math>3 \cdot x^2</math> based upon the first part and in the form of <math>5^2 \cdot y^3</math> based upon the second part, with <math>x</math> and <math>y</math> denoting an [[integer]]s. <math>c</math> is minimized if it’s [[prime factorization]] contains only <math>3,5</math>, and since there is a cubed term in <math>5^2 \cdot y^3</math>, <math>3^3</math> must be a factor of <math>c</math>. <math>3^35^2 = 675</math>, which works as the solution. | |
== See also == | == See also == | ||
| − | + | {{AIME box|year=1989|num-b=3|num-a=5}} | |
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Revision as of 19:57, 26 February 2007
Problem
If 
are consecutive positive integers such that 
is a perfect square and 
is a perfect cube, what is the smallest possible value of 
?
Solution
Since the middle term of an arithmetic progression with an odd number of terms is the average of the series, we know 
and 
. Thus, 
must be in the form of 
based upon the first part and in the form of 
based upon the second part, with 
and 
denoting an integers. is minimized if it’s prime factorization contains only 
, and since there is a cubed term in , 
must be a factor of . 
, which works as the solution.
See also
| 1989 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 3 |
Followed by Problem 5 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
