MathDB

4

Part of 2014 JBMO TST - Turkey

Problems(2)

Minimize (a+5)^2+(b-2)^2+(c-9)^2 subject to a constraint

Source: Turkey JBMO TST 2014 P4

6/21/2014
Determine the smallest value of (a+5)2+(b−2)2+(c−9)2(a+5)^2+(b-2)^2+(c-9)^2 for all real numbers a,b,ca, b, c satisfying a2+b2+c2−ab−bc−ca=3a^2+b^2+c^2-ab-bc-ca=3.
inequalities proposedinequalities
A game on a complete graph

Source: Turkey JBMO TST 2014 P8

6/21/2014
Alice and Bob play a game on a complete graph GG with 20142014 vertices. They take moves in turn with Alice beginning. At each move Alice directs one undirected edge of GG. At each move Bob chooses a positive integer number m,m, 1≤m≤10001 \le m \le 1000 and after that directs mm undirected edges of GG. The game ends when all edges are directed. If there is some directed cycle in GG Alice wins. Determine whether Alice has a winning strategy.
algorithmgraph theorycombinatorics proposedcombinatorics