MathDB

2

Part of 1988 China Team Selection Test

Problems(2)

sum equals product in function

Source: China TST 1988, problem 2

6/27/2005
Find all functions f:Q↦Cf: \mathbb{Q} \mapsto \mathbb{C} satisfying (i) For any x1,x2,…,x1988∈Qx_1, x_2, \ldots, x_{1988} \in \mathbb{Q}, f(x1+x2+…+x1988)=f(x1)f(x2)…f(x1988)f(x_{1} + x_{2} + \ldots + x_{1988}) = f(x_1)f(x_2) \ldots f(x_{1988}). (ii) f(1988)‾f(x)=f(1988)f(x)‾\overline{f(1988)}f(x) = f(1988)\overline{f(x)} for all x∈Qx \in \mathbb{Q}.
functionalgebrafunctional equationalgebra unsolved
quadrilateral PEFG is maximum

Source: China TST 1988, problem 6

6/27/2005
Let ABCDABCD be a trapezium AB//CD,AB // CD, MM and NN are fixed points on AB,AB, PP is a variable point on CDCD. E=DN∩APE = DN \cap AP, F=DN∩MCF = DN \cap MC, G=MC∩PBG = MC \cap PB, DP=λ⋅CDDP = \lambda \cdot CD. Find the value of λ\lambda for which the area of quadrilateral PEFGPEFG is maximum.
geometrytrapezoidgeometry unsolved