1986 AMC 12 Problem 30

Question

The number of real solutions

(x,y,z,w)

of the simultaneous equations

\begin{aligned} 2y&=x+\frac{17}{x},\\ 2z&=y+\frac{17}{y},\\ 2w&=z+\frac{17}{z},\\ 2x&=w+\frac{17}{w}. \end{aligned}

is

1

2

4

8

16

The equations force all four variables to have the same sign

For positive

t,

study

f(t)=\frac{t+\frac{17}{t}}{2}

relative to

\sqrt{17}

and to

t

Each expression

t+\frac{17}{t}

has the same sign as

t,

so all four variables have the same sign. Suppose first that they are positive. By AM-GM, every variable is at least

\sqrt{17}.

For

t\gt\sqrt{17},

\frac{t+\frac{17}{t}}{2}\lt t.

If any variable exceeded

\sqrt{17},

the equations would give the impossible strict cycle

x\gt y\gt z\gt w\gt x.

Hence the only positive solution is

x=y=z=w=\sqrt{17}.

Negating all four variables preserves the system, giving exactly one negative solution, with all variables equal to

-\sqrt{17}.

Thus there are two real solutions.

Therefore the correct answer is B.

Problem 30 in other years