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During the month of April, the average temperatures on day $n$ in two cities $A$ and $B$ are denoted by $T A(n)$ and $T B(n)$ respectively. The average temperature in city $\mathrm{A}$ on any day is the average of the average temperatures of the previous two days in city A. The average temperature in city B on any day is the average of the average temperatures of the previous two days in city B. If $T A(1)=30, T A(2)=35, T B(1)=35$ and $T B(2)=30$, which of the following statements must be true?

I. $\mathrm{TA}(4)-\mathrm{TB}(4)>\operatorname{TA}(29)-\mathrm{TB}(29)$
II. $\mathrm{TA}(4)+\mathrm{TB}(4)=\mathrm{TA}(29)+\mathrm{TB}(29)$
III. $|T A(n+1)-T A(n)|=|T B(n+1)-T B(n)|$ for $n>3$
A. II only
B. III only
C. I and II only
D. II and III only
E. I, II and III
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1) from the given we can calculate that the 1 st is true

2) second, let's look at answer choices: only $\mathrm{C}$ and $\mathrm{E}$ left, we need to test the $3 \mathrm{rd}$ case

3) $66,25 / 2-67,5 / 2=-0,75$
$63,25-62,5=0,75$

hence 3 rd is correct
by Diamond (78,454 points)

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