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2009 AIME II Problems/Problem 4

Revision as of 12:39, 6 April 2009 by Misof (talk | contribs) (New page: == Problem == A group of children held a grape-eating contest. When the contest was over, the winner had eaten <math>n</math> grapes, and the child in <math>k</math>-th place had eaten <m...)
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Problem

A group of children held a grape-eating contest. When the contest was over, the winner had eaten $n$ grapes, and the child in $k$-th place had eaten $n+2-2k$ grapes. The total number of grapes eaten in the contest was $2009$. Find the smallest possible value of $n$.

Solution

Resolving the ambiguity

The problem statement is confusing, as it can be interpreted in two ways: Either as "there is a $k>1$ such that the child in $k$-th place had eaten $n+2-2k$ grapes", or "for all $k$, the child in $k$-th place had eaten $n+2-2k$ grapes".

The second meaning was apparrently the intended one. Hence we will restate the problem statement in this way:

A group of $c$ children held a grape-eating contest. When the contest was over, the following was true: There was a $n$ such that for each $k$ between $1$ and $c$, inclusive, the child in $k$-th place had eaten exactly $n+2-2k$ grapes. The total number of grapes eaten in the contest was $2009$. Find the smallest possible value of $n$.

Solving this problem

The total number of grapes eaten can be computed as the sum of the arithmetic progression with initial term $n$ (the number of grapes eaten by the child in $1$-st place), difference $d=-2$, and number of terms $c$. We can easily compute that this sum is equal to $c(n-c+1)$.

Hence we have the equation $2009=c(n-c+1)$, and we are looking for a solution $(n,c)$, where both $n$ and $c$ are positive integers, $n\geq 2(c-1)$, and $n$ is maximized. (The condition $n\geq 2(c-1)$ states that even the last child had to eat a non-negative number of grapes.)

The prime factorization of $2009$ is $2009=7^2 \cdot 41$. Hence there are $6$ ways how to factor $2009$ into two positive terms $c$ and $n-c+1$:

  • $c=1$, $n-c+1=2009$, then $n=2009$ is a solution.
  • $c=7$, $n-c+1=7\cdot 41=287$, then $n=293$ is a solution.
  • $c=41$, $n-c+1=7\cdot 7=49$, then $n=89$ is a solution.
  • In each of the other three cases, $n$ will obviously be less than $2(c-1)$, hence these are not valid solutions.

The smallest valid solution is therefore $c=41$, $n=\boxed{089}$.

See Also

2009 AIME II (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
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