Problem A. 433. (September 2007)
A. 433. Prove that if a, b, c are real numbers such that a2+b2+c2=1 then .
(Proposed by János Bodnár, Budapest)
Deadline expired on October 15, 2007.
Solution 1. Let , and . For these variables the constraint can be re-written as
The statement to prove is
Case 1: u,v,w0. From the constraint , hence ; it can be obtained similarly that v<3 and w<3. Therefore
Case 2: at least one of u,v,w is negative. Equation (1a) cannot hold if two of them are negative; so exactly one of u,v,w is negative. Then
Solution 2. By the constraint we have 2aba2+b2a2+b2+c2=1 and similarly 2ac1, 2bc1.
From the solution of Tuan Nhat Le
16 students sent a solution. 5 points: Blázsik Zoltán, Huszár Kristóf, Korándi Dániel, Lovász László Miklós, Nagy 235 János, Nagy 314 Dániel, Szűcs Gergely, Tomon István, Tossenberger Anna, Tuan Nhat Le, Wolosz János. 1 point: 1 student. 0 point: 4 students.