1988 AIME Problem 2

Attempt Problem 2 of the 1988 AIME below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1988 AIME solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

For any positive integer

k,

let

f_1(k)

denote the square of the sum of the digits of

k.

For

n\geq2,

let

f_n(k)=f_1(f_{n-1}(k)).

Find

f_{1988}(11).

Answer

Compute the first several iterates until a value repeats

After the transient values, the sequence alternates between two numbers

The iterates begin

4,

16,

49,

169,

256,

followed by

169,

256,

\ldots.

Thus from the fourth iterate onward, even indices give

169

and odd indices give

256.

Since

1988

is even,

f_{1988}(11)=169.

Problem 2 in other years