Difference between revisions of "1992 AIME Problems/Problem 4"

Revision as of 18:23, 4 July 2013

Problem

In Pascal's Triangle, each entry is the sum of the two entries above it. In which row of Pascal's Triangle do three consecutive entries occur that are in the ratio $3: 4: 5$ ?

Solution

In Pascal's Triangle, we know that the binomial coefficients of the $n$ th row are $\binom{n} {0}, \binom{n}{1}, ..., \binom{n} {n}$ . Let our row be the $n$ th row such that the three consecutive entries are $\binom{n} {r}$ , $\binom{n}{r+1}$ and $\binom{n} {r+2}$ .

After expanding and dividing one entry by another (to clean up the factorials), we see that $\frac 34=\frac{r+1}{n-r}$ and $\frac45=\frac{r+2}{n-r-1}$ . Solving, $n = \boxed{062}$ .

1992 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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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 [[Category:Intermediate Combinatorics Problems]]
 [[Category:Intermediate Number Theory Problems]]
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Difference between revisions of "1992 AIME Problems/Problem 4"

Revision as of 18:23, 4 July 2013

Problem

Solution