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Weekly Problem 32 - 2010

Stage: 3 Short Challenge Level: Challenge Level:1

I is true if and only if $ -1 < x < 1$; II is true if $x> 1$ or if $x< -1$.
By considering the graph of $y = x - x^2$, which intersects the x - axis at (0,0) and (1,0) and has a maximum at ($\frac{1}{2}, \frac{1}{4}$), it may be seen that statement V is true is and only if $0 < x < 1$.
We see from the table below that a maximum of three statements may be true at any one time.
$x < -1$ then II true- 1
$x=-1$ then none true- 0
$-1< x< 0$ then I, III true- 2
$x=0$ then none true- 0
$0< x< 1$ then I, IV, V true- 3
$x=1$ then none true- 0
$x> 1$ then II true- 1

This problem is taken from the UKMT Mathematical Challenges.

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