MathDB
f(x,y), f(x,0)=ax, f(c,d) = f(h,k) is constant on the line through (c,d),(h,k)

Source: Vietnamese MO (VMO) 1970

August 23, 2018
functionalgebraLinear Function

Problem Statement

The function f(x,y)f(x, y) is defined for all real numbers x,yx, y. It satisfies f(x,0)=axf(x,0) = ax (where aa is a non-zero constant) and if (c,d)(c, d) and (h,k)(h, k) are distinct points such that f(c,d)=f(h,k)f(c, d) = f(h, k), then f(x,y)f(x, y) is constant on the line through (c,d)(c, d) and (h,k)(h, k). Show that for any real bb, the set of points such that f(x,y)=bf(x, y) = b is a straight line and that all such lines are parallel. Show that f(x,y)=ax+byf(x, y) = ax + by, for some constant bb.
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