MathDB

4

Part of 2012 Korea National Olympiad

Problems(2)

Maximum with condition a^2 + b^2 + c^2 = 2abc + 1

Source: Korea National 2012 Problem 4

8/19/2012
a,b,ca,b,c are positive numbers such that a2+b2+c2=2abc+1 a^2 + b^2 + c^2 = 2abc + 1 . Find the maximum value of (a−2bc)(b−2ca)(c−2ab) (a-2bc)(b-2ca)(c-2ab)
inequalitiesinequalities proposed
Large subset of pairs satisfying a divisibility condition

Source: Korea National 2012 Problem 8

8/19/2012
Let p≡3(mod4) p \equiv 3 \pmod{4} be a prime. Define T={(i,j)∣i,j∈{0,1,⋯ ,p−1}}∖{(0,0)}T = \{ (i,j) \mid i, j \in \{ 0, 1, \cdots , p-1 \} \} \smallsetminus \{ (0,0) \} . For arbitrary subset S(≠∅)⊂T S ( \ne \emptyset ) \subset T , prove that there exist subset A⊂S A \subset S satisfying following conditions: (a) (xi,yi)∈A(1≤i≤3) (x_i , y_i ) \in A ( 1 \le i \le 3) then p∤x1+x2−y3 p \not | x_1 + x_2 - y_3 or p∤y1+y2+x3 p \not | y_1 + y_2 + x_3 . (b) 8n(A)>n(S) 8 n(A) > n(S)
modular arithmeticpigeonhole principlecombinatorics proposedcombinatorics