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1. For each integer a_0 > 1, define the sequence
a_0, a_1, a_2, ... for n >= 0 as
a_{n+1} = sqrt(a_n) if sqrt(a_n) is an integer,
a_n + 3 otherwise.
Determine all values of a_0 such that there exists
a number A such that a_n = A for infinitely many values
of n.
2. Let R be the set of real numbers. Determine all
functions f: R --> R such that, for any real numbers
x and y,
f( f(x) f(y) ) + f( x+y ) = f( xy ).
3. A hunter and an invisible rabbit play a game in the
Euclidean plane. The rabbit's starting point, A_0,
and the hunter's starting point, B_0, are the same.
After (n-1) rounds of the game, the rabbit is at point
A_{n-1} and the hunter is at point B_{n-1}. In the
n-th round of the game, three things occur in order:
i. The rabbit moves invisibly to a point A_n such
that the distance between A_{n-1} and A_n is
exactly 1.
ii. A tracking device reports a point P_n to the
hunter. The only guarantee provided by the
tracking device to the hunter is that the distance
between P_n and A_n is at most 1.
iii. The hunter moves visibly to a point B_n such that
the distance between B_{n-1} and B_n is exactly 1.
Is it always possible, no matter how the rabbit moves,
and no matter what points are reported by the tracking
device, for the hunter to choose her moves so that after
10^9 rounds, she can ensure that the distance between
her and the rabbit is at most 100?
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兩兔傍地走 安能辨我是雄雌
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