SAT1000 - P880

Algebra Level pending

For function f(x),g(x)f(x), g(x) which have the same domain DD, if there exists h(x)=kx+bh(x)=kx+b (k,bk,b are constant), so that:

m(0,+),x0D,xDx>x0,0<f(x)h(x)<m,0<h(x)g(x)<m,\forall m \in (0,+\infty), \exists x_0 \in D, \forall x \in D \wedge x>x_0, 0<f(x)-h(x)<m, 0<h(x)-g(x)<m,

Then line ll: y=kx+by=kx+b is called the bipartite asymptote for curve y=f(x)y=f(x) and y=g(x)y=g(x).

Here are four groups of functions which are defined at (1,+)(1,+\infty):

  1. f(x)=x2,g(x)=xf(x)=x^2, g(x)=\sqrt{x}.

  2. f(x)=10x+2,g(x)=2x3xf(x)=10^{-x}+2, g(x)=\dfrac{2x-3}{x}.

  3. f(x)=x2+1x,g(x)=xlnx+1lnxf(x)=\dfrac{x^2+1}{x}, g(x)=\dfrac{x \ln x+1}{\ln x}.

  4. f(x)=2x2x+1,g(x)=2(x1ex)f(x)=\dfrac{2x^2}{x+1}, g(x)=2(x-1-e^{-x}).

Which groups of curve y=f(x)y=f(x) and y=g(x)y=g(x) has a bipartite asymptote?

How to submit:

Let p1,p2,p3,p4p_1, p_2, p_3, p_4 be the boolean value of the group 1,2,3,41,2,3,4, if group kk has a bipartite asymptote, pk=1p_k=1, else pk=0p_k=0.

Then submit k=14pk2k1\displaystyle \sum_{k=1}^4 p_k \cdot 2^{k-1}.


Have a look at my problem set: SAT 1000 problems

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