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2021 AIME II Problems/Problem 15: Difference between revisions

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Write <math>7f(n)=4g(n)</math>, which simplifies to <math>3k^2+k-10=3n</math>. Notice that we want the <math>LHS</math> expression to be divisible by 3; as a result, <math>k \equiv 1 \pmod{3}</math>. We also want n to be strictly greater than <math>k^2</math>, so <math>k-10>0, k>10</math>. The LHS expression is always even (why?), so to ensure that k and n share the same parity, k should be even. Then the least k that satisfies these requirements is <math>k=16</math>, giving <math>n=258</math>.
Write <math>7f(n)=4g(n)</math>, which simplifies to <math>3k^2+k-10=3n</math>. Notice that we want the <math>LHS</math> expression to be divisible by 3; as a result, <math>k \equiv 1 \pmod{3}</math>. We also want n to be strictly greater than <math>k^2</math>, so <math>k-10>0, k>10</math>. The LHS expression is always even (why?), so to ensure that k and n share the same parity, k should be even. Then the least k that satisfies these requirements is <math>k=16</math>, giving <math>n=258</math>.


Indeed, <math>f(258)=48, g(258)=84</math>, so we're done.
Indeed - if we check our answer, it works. Therefore, the answer is <math>\boxed{258}</math>


-Ross Gao
-Ross Gao

Revision as of 16:10, 22 March 2021

Problem

Let $f(n)$ and $g(n)$ be functions satisfying \[f(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 1 + f(n+1) & \text{ otherwise} \end{cases}\]and \[g(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 2 + g(n+2) & \text{ otherwise} \end{cases}\]for positive integers $n$. Find the least positive integer $n$ such that $\tfrac{f(n)}{g(n)} = \tfrac{4}{7}$.

Solution

Consider what happens when we try to calculate $f(n)$ where n is not a square. If $k^2<n<(k+1)^2$ for (positive) integer k, recursively calculating the value of the function gives us $f(n)=(k+1)^2-n+f((k+1)^2)=k^2+3k+2-n$. Note that this formula also returns the correct value when $n=(k+1)^2$, but not when $n=k^2$. Thus $f(n)=k^2+3k+2-n$ for $k^2<n \leq (k+1)^2$.

If $2 \mid (k+1)^2-n$, $g(n)$ returns the same value as $f(n)$. This is because the recursion once again stops at $(k+1)^2$. We seek a case in which $f(n)<g(n)$, so obviously this is not what we want. We want $(k+1)^2,n$ to have a different parity, or $n, k$ have the same parity. When this is the case, $g(n)$ instead returns $(k+2)^2-n+g((k+2)^2)=k^2+5k+6-n$.

Write $7f(n)=4g(n)$, which simplifies to $3k^2+k-10=3n$. Notice that we want the $LHS$ expression to be divisible by 3; as a result, $k \equiv 1 \pmod{3}$. We also want n to be strictly greater than $k^2$, so $k-10>0, k>10$. The LHS expression is always even (why?), so to ensure that k and n share the same parity, k should be even. Then the least k that satisfies these requirements is $k=16$, giving $n=258$.

Indeed - if we check our answer, it works. Therefore, the answer is $\boxed{258}$

-Ross Gao

See also

2021 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 14 		Followed by
Last Question
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All AIME Problems and Solutions

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