Difference between revisions of "2010 AIME I Problems/Problem 14"

Problem

For each positive integer n, let $f(n) = \sum_{k = 1}^{100} \lfloor log_{10} (kn) \rfloor$. Find the largest value of n for which $f(n) \le 300$.

Note: $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.

Solution 1

Observe that $f$ is strictly increasing in $n$. We realize that we need $100$ terms to add up to around $300$, so we need some sequence of $2$s, $3$s, and then $4$s.

It follows that $n \approx 100$. Manually checking shows that $f(109) = 300$ and $f(110) > 300$. Thus, our answer is $\boxed{109}$.

Solution 2

Because we want the value for which $f(n)=300$, the average value of the 100 terms of the sequence should be around $3$. For the value of $\lfloor log_{10} (kn) \rfloor$ to be $3$, $1000 \le kn < 10000$. We want kn to be around the middle of that range, and for k to be in the middle of 0 and 100, let $k=50$, so $50n=\frac{10000+1000}{2}=\frac{11000}{2}=5500$, and $n = 110$. $f(110) = 301$, so we want to lower $n$. Testing $109$ yields $300$, so our answer is still $\boxed{109}$.

Solution 3

For any $n$ where the sum is close to $300$, all the terms in the sum must be equal to $2$, $3$ or $4$. Let $M$ be the number of terms less than or equal to $3$ and $N$ be the number of terms equal to $2$ (also counted in $M$). With this definition of $M$ and $N$ the total will be $400 - M - N = 300$, from which $M + N = 100$. Now $M+1$ is the smallest integer $k$ for which $\log(kn) >= 4$ or $kn >= 10000$, thus $$M = \left\lceil\frac{10000}{n}\right\rceil - 1.$$ Similarly, $$N = \left\lceil\frac{1000}{n}\right\rceil - 1.$$ Unless $n$ is a divisor of $10000$, we have $$M = \left\lfloor\frac{10000}{n}\right\rfloor$$ and $$N = \left\lfloor\frac{1000}{n}\right\rfloor = \left\lfloor \frac{M}{10} \right\rfloor.$$ Under the same assumption, $$M + \left\lfloor \frac{M}{10} \right\rfloor = 100,$$ and so $M = 91$. Dividing $10000$ by $91$ we see that $n = 109$. Since $n = 110$ is also not a divisor of $10000$ and with $n = 110$ we get $M + N < 100$ (that is, a sum greater than $300$) and the sum is obviously monotone in $n$, the largest value is $\boxed{109}$.

See also

 2010 AIME I (Problems • Answer Key • Resources) Preceded byProblem 13 Followed byProblem 15 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 All AIME Problems and Solutions
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