2021 AIME II Problems/Problem 15
Problem
Let and
be functions satisfying
and
for positive integers
. Find the least positive integer
such that
.
Solution
Consider what happens when we try to calculate where n is not a square. If
for (positive) integer k, recursively calculating the value of the function gives us
. Note that this formula also returns the correct value when
, but not when
. Thus
for
.
If ,
returns the same value as
. This is because the recursion once again stops at
. We seek a case in which
, so obviously this is not what we want. We want
to have a different parity, or
have the same parity. When this is the case,
instead returns
.
Write , which simplifies to
. Notice that we want the
expression to be divisible by 3; as a result,
. We also want n to be strictly greater than
, so
. 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
, giving
.
Indeed, , so we're done.
-Ross Gao
See also
2021 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Last Question | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
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