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 2 of 4: Sum the cyclic inequalities

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2\sum_{n=1}^{1995}\sqrt{x_n}\ge\sum_{n=1}^{1995}(x_n+1)

Detailed analysis

Add the $1994$ inequalities $2\sqrt{x_n}\ge x_{n+1}+1$ and the final inequality $2\sqrt{x_{1995}}\ge x_1+1$ . The terms $x_1,\ldots,x_{1995}$ on the right occur exactly once, together with $1995$ copies of $1$ , giving the displayed sum.