Problem 1

Determine all sequences of real numbers

(a_1,a_2,\ldots,a_{1995})

satisfying

2\sqrt{a_n-(n-1)}\ge a_{n+1}-(n-1)

for

n=1,2,\ldots,1994

, and

2\sqrt{a_{1995}-1994}\ge a_1+1

Step 4 of 4: Recover and verify the unique sequence

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x_n=1\Rightarrow a_n=n

Detailed analysis

From $x_n=1$ and $x_n=a_n-(n-1)$ , obtain $a_n=n$ . Conversely, these values give $2\sqrt{a_n-(n-1)}=2=a_{n+1}-(n-1)$ for $n=1,\ldots,1994$ , and $2\sqrt{a_{1995}-1994}=2=a_1+1$ . Thus $a_n=n$ is the unique solution.