MathDB

3

Part of 1993 Vietnam Team Selection Test

Problems(2)

maximal and the minimal values of x_1 - 2 * x_2 + x_3

Source: Vietnam TST 1993 for the 34nd IMO, problem 3

6/25/2005
Let's consider the real numbers x1,x2,x3,x4x_1, x_2, x_3, x_4 satisfying the condition 12≤x12+x22+x32+x42≤1 \dfrac{1}{2}\le x_1^2+x_2^2+x_3^2+x_4^2\le 1 Find the maximal and the minimal values of expression: A=(x1−2⋅x2+x3)2+(x2−2⋅x3+x4)2+(x2−2⋅x1)2+(x3−2⋅x4)2 A = (x_1 - 2 \cdot x_2 + x_3)^2 + (x_2 - 2 \cdot x_3 + x_4)^2 + (x_2 - 2 \cdot x_1)^2 + (x_3 - 2 \cdot x_4)^2
inequalities unsolvedinequalities
maximal value of n for which we can paint all edges

Source: Vietnam TST 1993 for the 34th IMO, problem 6

6/25/2005
Let nn points A1,A2,…,AnA_1, A_2, \ldots, A_n, (n>2n>2), be considered in the space, where no four points are coplanar. Each pair of points Ai,AjA_i, A_j are connected by an edge. Find the maximal value of nn for which we can paint all edges by two colors – blue and red such that the following three conditions hold: I. Each edge is painted by exactly one color. II. For i=1,2,…,ni = 1, 2, \ldots, n, the number of blue edges with one end AiA_i does not exceed 4. III. For every red edge AiAjA_iA_j, we can find at least one point AkA_k (k≠i,jk \neq i, j) such that the edges AiAkA_iA_k and AjAkA_jA_k are blue.
combinatorics unsolvedcombinatorics