Difference between revisions of "1984 AIME Problems/Problem 7"
(→Solution 3) |
(→Solution 3) |
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Because then | Because then | ||
<cmath>f(84)=f(f^n(84+5))=(f^n(84+5))-3</cmath> | <cmath>f(84)=f(f^n(84+5))=(f^n(84+5))-3</cmath> | ||
− | From which we'll get a numerical value for <math>f(84). | + | From which we'll get a numerical value for <math>f(84)</math>. |
− | Notice that the value of < | + | Notice that the value of <math>n</math> we expect to find is basically the smallest <math>n</math> such that after <math>f(x)=f(f(n+5)</math> is performed <math>n</math> times and then <math>f(x)=n-3</math> is performed back <math>n</math> times, the result is greater than or equal to <math>1000</math>. |
== See also == | == See also == |
Revision as of 16:16, 19 August 2019
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 since they require iteration of the function.
Soon we realize the for integers
either equal
or
based on it parity. (If short on time, a guess of
or
can be taken now.)
If
is even
if
is odd
.
has even parity, so
.
The result may be rigorously shown through induction.
Solution 3
Assume that is to be performed
times. Then we have
In order to find
, we want to know the smallest value of
such that
Because then
From which we'll get a numerical value for
.
Notice that the value of we expect to find is basically the smallest
such that after
is performed
times and then
is performed back
times, the result is greater than or equal to
.
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 |