IMO 1995 SL A3

Origin: UKR | Category: Algebra

Problem

Let n be an integer, n \geq3. Let a1, a2, . . . , an be real numbers such that 2 \leqai \leq3 for i = 1, 2, . . ., n. If s = a1 + a2 + \cdot \cdot \cdot + an, prove that a2 1 + a2 2 −a2 a1 + a2 −a3

• a2 2 + a2 3 −a2 a2 + a3 −a4
• \cdot \cdot \cdot + a2 n + a2 1 −a2 an + a1 −a2 \leq2s −2n.

Solution

Write Ai = a2 i +a2 i+1−a2 i+2 ai+ai+1−ai+2 = ai+ai+1+ai+2− 2aiai+1 ai+ai+1−ai+2 . Since 2aiai+1 \geq 4(ai + ai+1 −2) (which is equivalent to (ai −2)(ai+1 −2) \geq0), it follows that Ai \leqai + ai+1 + ai+2 −4  1 + ai+2−2 ai+ai+1−ai+2

\leqai + ai+1 + ai+2 −  1 + ai+2−2

, because 1 \leqai + ai+1 −ai+2 \leq4. Therefore Ai \leqai + ai+1 −2, so n i=1 Ai \leq2s −2n as required.