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3

Part of 1980 Vietnam National Olympiad

Problems(2)

Midpoints of segments forming triangles with equal areas.

Source: Vietnam MO 1980 P3

3/17/2011
Let PP be a point inside a triangle A1A2A3A_1A_2A_3. For i=1,2,3i = 1, 2, 3, line PAiPA_i intersects the side opposite to AiA_i at BiB_i. Let CiC_i and DiD_i be the midpoints of AiBiA_iB_i and PBiPB_i, respectively. Prove that the areas of the triangles C1C2C3C_1C_2C_3 and D1D2D3D_1D_2D_3 are equal.
geometryanalytic geometry
Maximum of the sum x_ix_{i+1} in terms of p, the sum of x_i.

Source: Vietnam MO 1980 P6

3/17/2011
Let be given an integer n\ge 2 and a positive real number pp. Find the maximum of ∑i=1n−1xixi+1,\displaystyle\sum_{i=1}^{n-1} x_ix_{i+1}, where xix_i are non-negative real numbers with sum pp.
inequalities proposedinequalities