Difference between revisions of "1984 AIME Problems/Problem 7"
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<cmath>\begin{align*}f(999)=f(f(1004))=f(1001)=998\\ | <cmath>\begin{align*}f(999)=f(f(1004))=f(1001)=998\\ | ||
− | </cmath>\begin{align*}f(998)=f(f(1003))=f(1000)=997\\ | + | </cmath>\begin{align*}f(998)=f(f(1003))=f(1000)=997 \\ |
<cmath>\begin{align*}f(997)=f(f(1002))=f(999)=998\\ | <cmath>\begin{align*}f(997)=f(f(1002))=f(999)=998\\ | ||
− | </cmath>\begin{align*}f(996)=f(f(1001))=f(998)=997\\ | + | </cmath>\begin{align*}f(996)=f(f(1001))=f(998)=997 \\ |
Soon we realize the <math>f(k)</math> for integers <math>k<1000</math> either equal <math>998</math> or <math>997</math> based on it parity. (If short on time, a guess of 998 or 997 can be taken now.) | Soon we realize the <math>f(k)</math> for integers <math>k<1000</math> either equal <math>998</math> or <math>997</math> based on it parity. (If short on time, a guess of 998 or 997 can be taken now.) |
Revision as of 18:27, 25 July 2011
Contents
Problem
The function f is defined on the set of integers and satisfies
Find .
Solution 1
Define , where the function
is performed
times. We find that
.
. So we now need to reduce
.
Let’s write out a couple more iterations of this function:
So this function reiterates with a period of 2 for
. It might be tempting at first to assume that
is the answer; however, that is not true since the solution occurs slightly before that. Start at
:
Solution 2
We start by finding values of the function right under 1000 since they require little repetition.
\begin{align*}f(999)=f(f(1004))=f(1001)=998\\ (Error compiling LaTeX. Unknown error_msg)
\begin{align*}f(998)=f(f(1003))=f(1000)=997 \\
\begin{align*}f(997)=f(f(1002))=f(999)=998\\ (Error compiling LaTeX. Unknown error_msg)
\begin{align*}f(996)=f(f(1001))=f(998)=997 \\
Soon we realize the for integers
either equal
or
based on it parity. (If short on time, a guess of 998 or 997 can be taken now.)
If
is even
if
is odd
.
has even parity, so
.
See also
1984 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |